63 results on '"Maurizio Grasselli"'
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2. On a phase field model for RNA-Protein dynamics
- Author
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Maurizio Grasselli, Luca Scarpa, and Andrea Signori
- Subjects
Computational Mathematics ,Mathematics - Analysis of PDEs ,35D30, 35K35, 35K86, 35Q92, 92C17, 92C50 ,Applied Mathematics ,FOS: Mathematics ,Analysis ,Analysis of PDEs (math.AP) - Abstract
A phase field model which describes the formation of protein-RNA complexes subject to phase segregation is analyzed. A single protein, two RNA species, and two complexes are considered. Protein and RNA species are governed by coupled reaction-diffusion equations which also depend on the two complexes. The latter ones are driven by two Cahn-Hilliard equations with Flory-Huggins potential and reaction terms depending on the solution variables. The resulting nonlinear coupled system is equipped with no-flux boundary conditions and suitable initial conditions. The former ones entail some mass conservation constraints which are also due to the nature of the reaction terms. The existence of global weak solutions in a bounded (two- or) three-dimensional domain is established. In dimension two, some weighted-in-time regularity properties are shown. In particular, the complexes instantaneously get uniformly separated from the pure phases. Taking advantage of this result, a unique continuation property is proven. Among the many technical difficulties, the most significant one arises from the fact that the two complexes are initially nonexistent, so their initial conditions are zero i.e., they start from a pure phase. Thus we must solve, in particular, a system of two coupled Cahn-Hilliard equations with singular potential and nonlinear sources without the usual assumption on the initial datum, i.e., the initial phase cannot be pure. This novelty requires a new approach to estimate the chemical potential in a suitable $L^p(L^2)$-space with $p\in(1,2)$. This technique can be extended to other models like, for instance, the well-known Cahn-Hilliard-Oono equation., 56 pages
- Published
- 2022
3. On a system of coupled Cahn-Hilliard equations
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Andrea Di Primio and Maurizio Grasselli
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Computational Mathematics ,Mathematics - Analysis of PDEs ,Applied Mathematics ,General Engineering ,FOS: Mathematics ,General Medicine ,General Economics, Econometrics and Finance ,Analysis ,Analysis of PDEs (math.AP) - Abstract
We consider a system which consists of a Cahn-Hilliard equation coupled with a Cahn-Hilliard-Oono equation in a bounded domain of $\mathbb{R}^d$, $d = 2, 3$. This system accounts for macrophase and microphase separation in a polymer mixture through two order parameters $u$ and $v$. The free energy of this system is a bivariate interaction potential which contains the mixing entropy of the two order parameters and suitable coupling terms. The equations are endowed with initial conditions and homogeneous Neumann boundary conditions both for $u,v$ and for the corresponding chemical potentials. We first prove that the resulting problem is well posed in a weak sense. Then, in the conserved case, we establish that the weak solution regularizes instantaneously. Furthermore, in two spatial dimensions, we show the strict separation property for $u$ and $v$, namely, they both stay uniformly away from the pure phases $\pm 1$ in finite time. Finally, we investigate the long-time behavior of a finite energy solution showing, in particular, that it converges to a single stationary state., Comment: to appear in Nonlinear Analysis: Real World Applications
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- 2022
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4. On the Mass-Conserving Allen-Cahn Approximation for Incompressible Binary Fluids
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Andrea Giorgini, Maurizio Grasselli, and Hao Wu
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Physics::Fluid Dynamics ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Mathematics::Analysis of PDEs ,35D35, 35K61, 35Q30, 35Q31, 35Q35, 76D27, 76D45 ,Analysis ,Analysis of PDEs (math.AP) - Abstract
This paper is devoted to the global well-posedness of two Diffuse Interface systems modeling the motion of an incompressible two-phase fluid mixture in presence of capillarity effects in a bounded smooth domain $\Omega\subset \mathbb{R}^d$, $d=2,3$. We focus on dissipative mixing effects originating from the mass-conserving Allen-Cahn dynamics with the physically relevant Flory-Huggins potential. More precisely, we study the mass-conserving Navier-Stokes-Allen-Cahn system for nonhomogeneous fluids and the mass-conserving Euler-Allen-Cahn system for homogeneous fluids. We prove existence and uniqueness of global weak and strong solutions as well as their property of separation from the pure states. In our analysis, we combine the energy and entropy estimates, a novel end-point estimate of the product of two functions, a new estimate for the Stokes problem with non-constant viscosity, and logarithmic type Gronwall arguments.
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- 2020
5. Nonlocal Cahn–Hilliard–Navier–Stokes systems with shear dependent viscosity
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Maurizio Grasselli, Sergio Frigeri, and Dalibor Pražák
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Convection ,Applied Mathematics ,Weak solution ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Isothermal process ,Physics::Fluid Dynamics ,010101 applied mathematics ,Shear rate ,Compressibility ,Functional derivative ,Boundary value problem ,Uniqueness ,0101 mathematics ,Analysis ,Mathematics - Abstract
We consider a diffuse interface model for the phase separation of an incompressible and isothermal non-Newtonian binary fluid mixture in three dimensions. The averaged velocity u is governed by a Navier–Stokes system with a shear dependent viscosity controlled by a power p > 2 . This system is nonlinearly coupled through the Korteweg force with a convective nonlocal Cahn–Hilliard equation for the order parameter φ, that is, the (relative) concentration difference of the two components. The resulting equations are endowed with the no-slip boundary condition for u and the no-flux boundary condition for the chemical potential μ. The latter variable is the functional derivative of a nonlocal and nonconvex Ginzburg–Landau type functional which accounts for the presence of two phases. We first prove the existence of a weak solution in the case p ≥ 11 / 5 . Then we extend some previous results on time regularity and uniqueness if p > 11 / 5 .
- Published
- 2018
6. The nonlocal Cahn–Hilliard equation with singular potential: Well-posedness, regularity and strict separation property
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Andrea Giorgini, Maurizio Grasselli, and Ciprian G. Gal
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Applied Mathematics ,Weak solution ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,01 natural sciences ,Physics::Fluid Dynamics ,010101 applied mathematics ,Dimension (vector space) ,Attractor ,Convergence (routing) ,0101 mathematics ,Cahn–Hilliard equation ,Constant (mathematics) ,Nonlinear Sciences::Pattern Formation and Solitons ,Analysis ,Separation property ,Well posedness ,Mathematics - Abstract
We consider the nonlocal Cahn–Hilliard equation with singular potential and constant mobility. Well-posedness and regularity of weak solutions are studied. Then we establish the validity of the strict separation property in dimension two. Further regularity results as well as the existence of regular finite dimensional attractors and the convergence of a weak solution to a single equilibrium are also provided. Finally, regularity results and the strict separation property are also proven for the two-dimensional nonlocal Cahn–Hilliard–Navier–Stokes system with singular potential.
- Published
- 2017
7. Global weak solutions to a diffuse interface model for incompressible two-phase flows with moving contact lines and different densities
- Author
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Hao Wu, Ciprian G. Gal, and Maurizio Grasselli
- Subjects
Physics ,Discretization ,Mechanical Engineering ,Weak solution ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,Order (ring theory) ,01 natural sciences ,Domain (mathematical analysis) ,Physics::Fluid Dynamics ,010101 applied mathematics ,Mathematics (miscellaneous) ,Flow velocity ,Compressibility ,Boundary value problem ,0101 mathematics ,Analysis - Abstract
In this paper, we analyze a general diffuse interface model for incompressible two-phase flows with unmatched densities in a smooth bounded domain $$\Omega \subset {\mathbb {R}}^d$$ ( $$d=2,3$$ ). This model describes the evolution of free interfaces in contact with the solid boundary, namely the moving contact lines. The corresponding evolution system consists of a nonhomogeneous Navier–Stokes equation for the (volume) averaged fluid velocity $${\mathbf {v}}$$ that is nonlinearly coupled with a convective Cahn–Hilliard equation for the order parameter $$\varphi $$ . Due to the nontrivial boundary dynamics, the fluid velocity satisfies a generalized Navier boundary condition that accounts for the velocity slippage and uncompensated Young stresses at the solid boundary, while the order parameter fulfils a dynamic boundary condition with surface convection. We prove the existence of a global weak solution for arbitrary initial data in both two and three dimensions. The proof relies on a combination of suitable approximations and regularizations of the original system together with a novel time-implicit discretization scheme based on the energy dissipation law.
- Published
- 2019
8. Energy stable and convergent finite element schemes for the modified phase field crystal equation
- Author
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Morgan Pierre, Maurizio Grasselli, Dipartimento di Matematica, Politecnico di Milano, Dipartimento di Matematica 'F. Brioschi', Politecnico di Milano [Milan] (POLIMI)-Politecnico di Milano [Milan] (POLIMI), Laboratoire de Mathématiques et Applications (LMA-Poitiers), and Université de Poitiers-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Discretization ,65P40 ,Finite elements ,010103 numerical & computational mathematics ,Space (mathematics) ,01 natural sciences ,Piecewise linear function ,second-order schemes ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Order (group theory) ,0101 mathematics ,Galerkin method ,Mathematics ,Numerical Analysis ,74N05 ,Lojasiewicz inequality MSC 2010: 65M60 ,Applied Mathematics ,Mathematical analysis ,Finite element method ,010101 applied mathematics ,Computational Mathematics ,gradient-like systems ,Modeling and Simulation ,Scheme (mathematics) ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] ,Analysis ,Energy (signal processing) ,82C26 - Abstract
We propose a space semi-discrete and a fully discrete finite element scheme for the modified phase field crystal equation (MPFC). The space discretization is based on a splitting method and consists in a Galerkin approximation which allows low order (piecewise linear) finite elements. The time discretization is a second-order scheme which has been introduced by Gomez and Hughes for the Cahn-Hilliard equation. The fully discrete scheme is shown to be unconditionally energy stable and uniquely solvable for small time steps, with a smallness condition independent of the space step. Using energy estimates, we prove that in both cases, the discrete solution converges to the unique energy solution of the MPFC equation as the discretization parameters tend to 0. Using a Lojasiewicz inequality, we also establish that the discrete solution tends to a stationary solution as time goes to infinity. Numerical simulations illustrate the theoretical results.
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- 2016
9. A convergent convex splitting scheme for a nonlocal cahn-hilliard-oono type equation with a transport term
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Laurence Cherfils, Alain Miranville, Hussein Fakih, Maurizio Grasselli, Laboratoire des Sciences de l'Ingénieur pour l'Environnement - UMR 7356 (LaSIE), Université de La Rochelle (ULR)-Centre National de la Recherche Scientifique (CNRS), Lebanese International University (LIU), Lebanese University [Beirut] (LU), Dipartimento di Matematica, Politecnico di Milano, Dipartimento di Matematica 'F. Brioschi', Politecnico di Milano [Milan] (POLIMI)-Politecnico di Milano [Milan] (POLIMI), Laboratoire de Mathématiques et Applications (LMA-Poitiers), and Université de Poitiers-Centre National de la Recherche Scientifique (CNRS)
- Subjects
[PHYS]Physics [physics] ,Numerical Analysis ,Applied Mathematics ,[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] ,Regular polygon ,Stability (probability) ,Term (time) ,Computational Mathematics ,Exact solutions in general relativity ,Modeling and Simulation ,Kernel (statistics) ,Convergence (routing) ,Neumann boundary condition ,Applied mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Vector field ,[MATH]Mathematics [math] ,Analysis ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] ,Mathematics - Abstract
We devise a rst-order in time convex splitting scheme for a nonlocal Cahn HilliardOono type equation with a transport term and subject to homogeneous Neu-mann boundary conditions. The presence of the transport term is not a minor modi-cation, since, for instance, we lose the unconditional unique solvability and stability. However, we prove the stability of our scheme when the time step is suciently small. Furthermore, we prove the consistency of this scheme and the convergence to the exact solution. Finally, we give some numerical simulations which conrm our theoretical results and demonstrate the performance of our scheme not only for phase separation, but also for crystal nucleation, for several choices of the interaction kernel.
- Published
- 2018
10. The Cahn-Hilliard-Hele-Shaw system with singular potential
- Author
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Maurizio Grasselli, Andrea Giorgini, and Hao Wu
- Subjects
Lyapunov stability ,Applied Mathematics ,Weak solution ,010102 general mathematics ,Mathematical analysis ,State (functional analysis) ,01 natural sciences ,Physics::Fluid Dynamics ,010101 applied mathematics ,Singular solution ,Dissipative system ,Uniqueness ,0101 mathematics ,Cahn–Hilliard equation ,Mathematical Physics ,Analysis ,Separation property ,Mathematics - Abstract
The Cahn–Hilliard–Hele–Shaw system is a fundamental diffuse-interface model for an incompressible binary fluid confined in a Hele–Shaw cell. It consists of a convective Cahn–Hilliard equation in which the velocity u is subject to a Korteweg force through Darcy's equation. In this paper, we aim to investigate the system with a physically relevant potential (i.e., of logarithmic type). This choice ensures that the (relative) concentration difference φ takes values within the admissible range. To the best of our knowledge, essentially all the available contributions in the literature are concerned with a regular approximation of the singular potential. Here we first prove the existence of a global weak solution with finite energy that satisfies an energy dissipative property. Then, in dimension two, we further obtain the uniqueness and regularity of global weak solutions. In particular, we show that any two-dimensional weak solution satisfies the so-called strict separation property, namely, if φ is not a pure state at some initial time, then it stays instantaneously away from the pure states. When the spatial dimension is three, we prove the existence of a unique global strong solution, provided that the initial datum is regular enough and sufficiently close to any local minimizer of the free energy. This also yields the local Lyapunov stability of the local minimizer itself. Finally, we prove that under suitable assumptions any global solution converges to a single equilibrium as time goes to infinity.
- Published
- 2018
11. Regularity results for the nonlocal Cahn-Hilliard equation with singular potential and degenerate mobility
- Author
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Ciprian G. Gal, Maurizio Grasselli, and Sergio Frigeri
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Applied Mathematics ,010102 general mathematics ,Degenerate energy levels ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,Strong solutions ,Bounded function ,0101 mathematics ,Cahn–Hilliard equation ,Analysis ,Separation property ,Mathematics ,Mathematical physics - Abstract
We consider the nonlocal Cahn-Hilliard equation with singular potential and degenerate mobility in a bounded domain Ω ⊂ R d , d ≤ 3 . We first prove the existence of maximal strong solutions in weighted (in time) L p spaces. Then we establish further regularity properties of the solution through maximal regularity theory. Finally, we revisit the separation property in an appendix.
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- 2018
- Full Text
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12. Erratum: 'On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems' [Comm. Pure Appl. Anal. 15 (2016), 299-317]
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Francesco Della Porta and Maurizio Grasselli
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010101 applied mathematics ,Mathematics - Analysis of PDEs ,Applied Mathematics ,010102 general mathematics ,FOS: Mathematics ,General Medicine ,0101 mathematics ,01 natural sciences ,Analysis ,Analysis of PDEs (math.AP) - Abstract
In this note, we want to highlight and correct an error in the paper "On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems" [Comm. Pure Appl. Anal. 15 (2016), 299-317] written by the authors.
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- 2018
- Full Text
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13. Robust exponential attractors for the modified phase-field crystal equation
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Hao Wu and Maurizio Grasselli
- Subjects
Physics ,Number density ,Phase field crystal ,Applied Mathematics ,Hölder condition ,Exponential function ,Mathematics - Analysis of PDEs ,Attractor ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Phase function ,35Q82, 37L25, 74N05, 82C26 ,Analysis ,Analysis of PDEs (math.AP) ,Mathematical physics - Abstract
We consider the modified phase-field crystal (MPFC) equation that has recently been proposed by P. Stefanovic et al. This is a variant of the phase-field crystal (PFC) equation, introduced by K.-R. Elder et al., which is characterized by the presence of an inertial term $\beta\phi_{tt}$. Here $\phi$ is the phase function standing for the number density of atoms and $\beta\geq 0$ is a relaxation time. The associated dynamical system for the MPFC equation with respect to the parameter $\beta$ is analyzed. More precisely, we establish the existence of a family of exponential attractors $\mathcal{M}_\beta$ that are H\"older continuous with respect to $\beta$., Comment: arXiv admin note: text overlap with arXiv:1306.5857
- Published
- 2015
14. A nonlinear model for marble sulphation including surface rugosity: theoretical and numerical results
- Author
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Roberto Natalini, Elena Bonetti, Maurizio Grasselli, Francesco Freddi, and Cecilia Cavaterra
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Surface (mathematics) ,bulk-surface PDE system ,Rugosity ,Computer simulation ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,General Medicine ,01 natural sciences ,010101 applied mathematics ,35M33, 65M06, 76R50 ,Mathematics - Analysis of PDEs ,Feature (computer vision) ,Nonlinear model ,Global existence and uniqueness ,FOS: Mathematics ,0101 mathematics ,numerical simulation ,Analysis ,Well posedness ,sulphation phenomena ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
We consider an evolution system describing the phenomenon of marble sulphation of a monument, accounting of the surface rugosity. We first prove a local in time well posedness result. Then, stronger assumptions on the data allow us to establish the existence of a global in time solution. Finally, we perform some numerical simulations that illustrate the main feature of the proposed model.
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- 2017
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15. Strong solutions for two-dimensional nonlocal Cahn–Hilliard–Navier–Stokes systems
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Pavel Krejčí, Sergio Frigeri, and Maurizio Grasselli
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Applied Mathematics ,Open problem ,Weak solution ,010102 general mathematics ,Mathematical analysis ,Dynamical system ,01 natural sciences ,Physics::Fluid Dynamics ,010101 applied mathematics ,Bounded function ,Attractor ,Compressibility ,Uniqueness ,0101 mathematics ,Navier–Stokes equations ,Analysis ,Mathematics - Abstract
A well-known diffuse interface model for incompressible isothermal mixtures of two immiscible fluids consists of the Navier–Stokes system coupled with a convective Cahn–Hilliard equation. In some recent contributions the standard Cahn–Hilliard equation has been replaced by its nonlocal version. The corresponding system is physically more relevant and mathematically more challenging. Indeed, the only known results are essentially the existence of a global weak solution and the existence of a suitable notion of global attractor for the corresponding dynamical system defined without uniqueness. In fact, even in the two-dimensional case, uniqueness of weak solutions is still an open problem. Here we take a step forward in the case of regular potentials. First we prove the existence of a (unique) strong solution in two dimensions. Then we show that any weak solution regularizes in finite time uniformly with respect to bounded sets of initial data. This result allows us to deduce that the global attractor is the union of all the bounded complete trajectories which are strong solutions. We also demonstrate that each trajectory converges to a single equilibrium, provided that the potential is real analytic and the external forces vanish.
- Published
- 2013
16. Nonlocal phase-field systems with general potentials
- Author
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Giulio Schimperna and Maurizio Grasselli
- Subjects
Field (physics) ,Applied Mathematics ,Mathematical analysis ,Ode ,Dynamical Systems (math.DS) ,Function (mathematics) ,Type (model theory) ,Nonlinear system ,Mathematics - Analysis of PDEs ,Singular solution ,Attractor ,FOS: Mathematics ,35B41, 35B41, 35Q79, 37L30, 80A22 ,Discrete Mathematics and Combinatorics ,Mathematics - Dynamical Systems ,Analysis ,Separation property ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We consider a phase-field model of Caginalp type where the free energy depends on the order parameter $\chi$ in a nonlocal way. Therefore, the resulting system consists of the energy balance equation coupled with a nonlinear and nonlocal ODE for $\chi$. Such system has been analyzed by several authors, in particular when the configuration potential is a smooth double-well function. More recently, the first author has established the existence of a finite-dimensional global attractor in the case of a potential defined on $(-1,1)$ and singular at the endpoints. Here we examine both the case of regular potentials as well as the case of physically more relevant singular potentials (e.g., logarithmic). We prove well-posedness results and the eventual global boundedness of solutions uniformly with respect to the initial data. In addition, we show that the separation property holds in the case of singular potentials. Thanks to these results, we are able to demonstrate the existence of a finite-dimensional global attractor in the present cases as well.
- Published
- 2013
17. Cahn–Hilliard–Navier–Stokes systems with moving contact lines
- Author
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Ciprian G. Gal, Alain Miranville, Maurizio Grasselli, Florida International University [Miami] (FIU), Dipartimento di Matematica, Politecnico di Milano, Dipartimento di Matematica 'F. Brioschi', Politecnico di Milano [Milan] (POLIMI)-Politecnico di Milano [Milan] (POLIMI), Laboratoire de Mathématiques et Applications (LMA-Poitiers), and Université de Poitiers-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Convection ,Applied Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,01 natural sciences ,Domain (mathematical analysis) ,Physics::Fluid Dynamics ,010101 applied mathematics ,Classical mechanics ,Flow velocity ,Flow (mathematics) ,Bounded function ,Convergence (routing) ,Compressibility ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Boundary value problem ,0101 mathematics ,Analysis ,Mathematics - Abstract
We consider a well-known diffuse interface model for the study of the evolution of an incompressible binary fluid flow in a two or three-dimensional bounded domain. This model consists of a system of two evolution equations, namely, the incompressible Navier-Stokes equations for the average fluid velocity u coupled with a convective Cahn–Hilliard equation for an order parameter $$\phi $$ . The novelty is that the system is endowed with boundary conditions which account for a moving contact line slip velocity. The existence of a suitable global energy solution is proven and the convergence of any such solution to a single equilibrium is also established.
- Published
- 2016
18. On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems
- Author
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Francesco Della Porta and Maurizio Grasselli
- Subjects
Physics ,Darcy's law ,Applied Mathematics ,Weak solution ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,01 natural sciences ,Physics::Fluid Dynamics ,010101 applied mathematics ,Viscosity ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Compressibility ,Nabla symbol ,Boundary value problem ,Uniqueness ,0101 mathematics ,Cahn–Hilliard equation ,Analysis ,Analysis of PDEs (math.AP) - Abstract
The phase separation of an isothermal incompressible binary fluid in a porous medium can be described by the so-called Brinkman equation coupled with a convective Cahn-Hilliard (CH) equation. The former governs the average fluid velocity $\mathbf{u}$, while the latter rules evolution of $\varphi$, the difference of the (relative) concentrations of the two phases. The two equations are known as the Cahn-Hilliard-Brinkman (CHB) system. In particular, the Brinkman equation is a Stokes-like equation with a forcing term (Korteweg force) which is proportional to $\mu\nabla\varphi$, where $\mu$ is the chemical potential. When the viscosity vanishes, then the system becomes the Cahn-Hilliard-Hele-Shaw (CHHS) system. Both systems have been studied from the theoretical and the numerical viewpoints. However, theoretical results on the CHHS system are still rather incomplete. For instance, uniqueness of weak solutions is unknown even in 2D. Here we replace the usual CH equation with its physically more relevant nonlocal version. This choice allows us to prove more about the corresponding nonlocal CHHS system. More precisely, we first study well-posedness for the CHB system, endowed with no-slip and no-flux boundary conditions. Then, existence of a weak solution to the CHHS system is obtained as a limit of solutions to the CHB system. Stronger assumptions on the initial datum allow us to prove uniqueness for the CHHS system. Further regularity properties are obtained by assuming additional, though reasonable, assumptions on the interaction kernel. By exploiting these properties, we provide an estimate for the difference between the solution to the CHB system and the one to the CHHS system with respect to viscosity.
- Published
- 2016
19. Phase-field crystal equation with memory
- Author
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Maurizio Grasselli, Andrea Giorgini, and Monica Conti
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Applied Mathematics ,Mathematical analysis ,Hölder condition ,Pattern formation ,010103 numerical & computational mathematics ,01 natural sciences ,Exponential function ,010101 applied mathematics ,Phase space ,Attractor ,Dissipative system ,Periodic boundary conditions ,0101 mathematics ,Balanced flow ,Analysis ,Mathematics - Abstract
Phase-field crystal models are used to describe several pattern formation phenomena like crystallization of liquid, diffusion defects and glass formation. The prototypical equation is obtained as the conserved gradient flow associated with a free-energy functional of Swift–Hohenberg type. Here we consider a variant of the phase-field crystal equation proposed by P. Galenko et al. to account for fast dynamics. This version is characterized by the fact that the gradient flow depends on the past history of the particle density through a memory kernel k e , e > 0 being a relaxation time. Therefore the resulting nonlinear evolution equation is an integro-differential equation of sixth order which is equipped with the physically relevant periodic boundary conditions. We show that this problem generates a dissipative dynamical system on a suitable phase space. Moreover, we prove the existence of a robust family of exponential attractors which is Holder continuous with respect to e.
- Published
- 2016
20. Global and Trajectory Attractors for a Nonlocal Cahn–Hilliard–Navier–Stokes System
- Author
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Maurizio Grasselli and Sergio Frigeri
- Subjects
Weak solution ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,01 natural sciences ,Physics::Fluid Dynamics ,010101 applied mathematics ,Mathematics - Analysis of PDEs ,Dimension (vector space) ,Attractor ,FOS: Mathematics ,35Q30, 37L30, 45K05, 76T99 ,Vector field ,Ball (mathematics) ,Uniqueness ,Boundary value problem ,0101 mathematics ,Nonlinear Sciences::Pattern Formation and Solitons ,Trajectory (fluid mechanics) ,Analysis ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
The Cahn-Hilliard-Navier-Stokes system is based on a well-known diffuse interface model and describes the evolution of an incompressible isothermal mixture of binary fluids. A nonlocal variant consists of the Navier-Stokes equations suitably coupled with a nonlocal Cahn-Hilliard equation. The authors, jointly with P. Colli, have already proven the existence of a global weak solution to a nonlocal Cahn-Hilliard-Navier-Stokes system subject to no-slip and no-flux boundary conditions. Uniqueness is still an open issue even in dimension two. However, in this case, the energy identity holds. This property is exploited here to define, following J.M. Ball's approach, a generalized semiflow which has a global attractor. Through a similar argument, we can also show the existence of a (connected) global attractor for the convective nonlocal Cahn-Hilliard equation with a given velocity field, even in dimension three. Finally, we demonstrate that any weak solution fulfilling the energy inequality also satisfies an energy inequality. This allows us to establish the existence of the trajectory attractor also in dimension three with a time dependent external force., 35 pages; presented at INDI2011 in Gargnano
- Published
- 2012
21. Analysis of the Cahn–Hilliard Equation with a Chemical Potential Dependent Mobility
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Riccarda Rossi, Maurizio Grasselli, Alain Miranville, and Giulio Schimperna
- Subjects
Physics::Fluid Dynamics ,Applied Mathematics ,Mathematical analysis ,Attractor ,Mathematics::Analysis of PDEs ,Applied mathematics ,Nonlinear Sciences::Cellular Automata and Lattice Gases ,Cahn–Hilliard equation ,Nonlinear Sciences::Pattern Formation and Solitons ,Analysis ,Mathematics - Abstract
The aim of this paper is to study the well-posedness and the existence of global attractors for a family of Cahn–Hilliard equations with a mobility depending on the chemical potential. Such models ...
- Published
- 2011
22. Singularly perturbed 1D Cahn–Hilliard equation revisited
- Author
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Ahmed Bonfoh, Alain Miranville, and Maurizio Grasselli
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Singular perturbation ,Applied Mathematics ,Mathematical analysis ,Attractor ,Hölder condition ,Periodic boundary conditions ,Cahn–Hilliard equation ,Analysis ,Mathematical physics ,Exponential function ,Mathematics - Abstract
We consider a singular perturbation of the one-dimensional Cahn–Hilliard equation subject to periodic boundary conditions. We construct a family of exponential attractors \({\{{\mathcal M}_\epsilon\}, \epsilon\geq 0}\) being the perturbation parameter, such that the map \({\epsilon \mapsto {\mathcal M}_\epsilon}\) is Holder continuous. Besides, the continuity at \({\epsilon=0}\) is obtained with respect to a metric independent of \({\epsilon.}\) Continuity properties of global attractors and inertial manifolds are also examined.
- Published
- 2010
23. Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D
- Author
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Ciprian G. Gal and Maurizio Grasselli
- Subjects
Applied Mathematics ,Mathematical analysis ,Boundary (topology) ,Fractal dimension ,Physics::Fluid Dynamics ,Rate of convergence ,Flow (mathematics) ,Incompressible flow ,Bounded function ,Attractor ,Boundary value problem ,Mathematical Physics ,Analysis ,Mathematics - Abstract
We consider a model for the flow of a mixture of two homogeneous and incompressible fluids in a two-dimensional bounded domain. The model consists of a Navier–Stokes equation governing the fluid velocity coupled with a convective Cahn–Hilliard equation for the relative density of atoms of one of the fluids. Endowing the system with suitable boundary and initial conditions, we analyze the asymptotic behavior of its solutions. First, we prove that the initial and boundary value problem generates a strongly continuous semigroup on a suitable phase-space which possesses the global attractor A . Then we establish the existence of an exponential attractors E . Thus A has finite fractal dimension. This dimension is then estimated from above in terms of the physical parameters. Moreover, assuming the potential to be real analytic and in absence of volume forces, we demonstrate that each trajectory converges to a single equilibrium. We also obtain a convergence rate estimate in the phase-space metric.
- Published
- 2010
24. The Caginalp phase-field system with coupled dynamic boundary conditions and singular potentials
- Author
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Giulio Schimperna, Alain Miranville, and Maurizio Grasselli
- Subjects
Physics ,Logarithm ,Applied Mathematics ,Mathematical analysis ,Attractor ,Phase (waves) ,Discrete Mathematics and Combinatorics ,Boundary value problem ,Type (model theory) ,Singular boundary method ,Analysis - Abstract
We study in this paper the well-posedness and the asymptotic behavior, in terms of global attractors, of the Caginalp system with coupled dynamic boundary conditions and possibly singular potentials (e.g., of logarithmic type).
- Published
- 2010
25. Robust exponential attractors for singularly perturbed Hodgkin–Huxley equations
- Author
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Cecilia Cavaterra and Maurizio Grasselli
- Subjects
Singular perturbation ,Global attractors ,Quantitative Biology::Neurons and Cognition ,Hyperbolic equations ,Applied Mathematics ,Mathematical analysis ,Hölder condition ,Hodgkin–Huxley model ,Exponential function ,Phase space ,Bounded function ,Attractor ,Exponential attractors ,Hyperbolic partial differential equation ,Analysis ,Mathematics - Abstract
Here we consider a singular perturbation of the Hodgkin–Huxley system which is derived from the Lieberstein's model. We study the associated dynamical system on a suitable bounded phase space, when the perturbation parameter e (i.e., the axon specific inductance) is sufficiently small. We prove the existence of bounded absorbing sets as well as of smooth attracting sets. We deduce the existence of a smooth global attractor A e . Finally we prove the main result, that is, the existence of a family of exponential attractors { E e } which is Holder continuous with respect to e.
- Published
- 2009
26. Robust Exponential Attractors for Singularly Perturbed Phase-Field Equations with Dynamic Boundary Conditions
- Author
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Ciprian G. Gal, Maurizio Grasselli, and Alain Miranville
- Subjects
Mathematics Subject Classification ,Applied Mathematics ,Attractor ,Mathematical analysis ,Heat equation ,Boundary value problem ,Mixed boundary condition ,Type (model theory) ,Analysis ,Robin boundary condition ,Exponential function ,Mathematics - Abstract
We consider a singularly perturbed phase-field model of Caginalp type which is thermally isolated and whose order parameter φ is subject to a dynamic boundary condition. More precisely, we indicate by e a (small) coefficient multiplying ∂tu in the heat equation, u being the temperature, and we construct a family of exponential attractors which is robust as e goes to 0. This is physically meaningful since the limiting problem is the viscous Cahn-Hilliard equation for the sole φ with a dynamic boundary condition. The upper semicontinuity of the global attractor is also analyzed. The paper extends and revisits some results previously obtained by A. Miranville et al. Mathematics Subject Classification (2000). Primary 35B41, 35K55, 37L30; Sec
- Published
- 2008
27. The non-isothermal Allen-Cahn equation with dynamic boundary conditions
- Author
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Maurizio Grasselli and Ciprian G. Gal
- Subjects
Applied Mathematics ,Mathematical analysis ,Mixed boundary condition ,Robin boundary condition ,Poincaré–Steklov operator ,symbols.namesake ,Dirichlet boundary condition ,Free boundary problem ,Neumann boundary condition ,symbols ,Discrete Mathematics and Combinatorics ,Cauchy boundary condition ,Boundary value problem ,Analysis ,Mathematics - Abstract
We consider a model of nonisothermal phase transitions taking place in a bounded spatial region. The order parameter $\psi$ is governed by an Allen-Cahn type equation which is coupled with the equation for the temperature $\theta$. The former is subject to a dynamic boundary condition recently proposed by some physicists to account for interactions with the walls. The latter is endowed with a boundary condition which can be a standard one (Dirichlet, Neumann or Robin) or a dynamic one of Wentzell type. We thus formulate a class of initial and boundary value problems whose local existence and uniqueness is proven by means of a fixed point argument. The local solution becomes global owing to suitable a priori estimates. Then we analyze the asymptotic behavior of the solutions within the theory of infinite-dimensional dynamical systems. In particular, we demonstrate the existence of the global attractor as well as of an exponential attractor.
- Published
- 2008
28. Asymptotic behavior of population dynamics models with nonlocal distributed delays
- Author
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Cecilia Cavaterra and Maurizio Grasselli
- Subjects
education.field_of_study ,Applied Mathematics ,Population ,Monotonic function ,Dynamical system ,Exponential function ,Exponential growth ,Attractor ,Dissipative system ,Neumann boundary condition ,Discrete Mathematics and Combinatorics ,Statistical physics ,education ,Analysis ,Mathematics - Abstract
We consider an integrodifferential reaction-diffusion system on a multidimensional spatial domain, subject to homogeneous Neumann boundary conditions. This system finds applications in population dynamics and it is characterized by nonlocal delay terms depending on both the temporal and the spatial variables. The distributed time delay effects are represented by memory kernels which decay exponentially but they are not necessarily monotonically decreasing. We first show how to construct a (dissipative) dynamical system on a suitable phase-space. Then we discuss the existence of the global attractor as well as of an exponential attractor.
- Published
- 2008
29. Asymptotic behavior of a nonisothermal viscous Cahn–Hilliard equation with inertial term
- Author
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Hana Petzeltová, Giulio Schimperna, and Maurizio Grasselli
- Subjects
State variable ,Global attractors ,Phase-field models ,Bounded absorbing sets ,Applied Mathematics ,Mathematical analysis ,Dynamical system ,Nonlinear system ,Attractor ,Dissipative system ,Neumann boundary condition ,Łojasiewicz–Simon inequality ,Boundary value problem ,Cahn–Hilliard equation ,Analysis ,Convergence to stationary solutions ,Mathematics - Abstract
We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn–Hilliard equation characterized by the presence of an inertial term χtt, χ being the order parameter, which is linearly coupled with an evolution equation for the (relative) temperature ϑ. The latter can be of hyperbolic type if the Cattaneo–Maxwell heat conduction law is assumed. The state variables and the chemical potential are subject to the homogeneous Neumann boundary conditions. We first provide conditions which ensure the well-posedness of the initial and boundary value problem. Then, we prove that the corresponding dynamical system is dissipative and possesses a global attractor. Moreover, assuming that the nonlinear potential is real analytic, we establish that each trajectory converges to a single steady state by using a suitable version of the Łojasiewicz–Simon inequality. We also obtain an estimate of the decay rate to equilibrium.
- Published
- 2007
30. Convergence to equilibrium for a parabolic–hyperbolic phase-field system with dynamical boundary condition
- Author
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Songmu Zheng, Hao Wu, and Maurizio Grasselli
- Subjects
Applied Mathematics ,Mathematical analysis ,Boundary (topology) ,Extended Simon–Łojasiewicz inequality ,Bounded function ,Neumann boundary condition ,Initial value problem ,Parabolic–hyperbolic phase-field system ,Boundary value problem ,Uniqueness ,Hyperbolic partial differential equation ,Dynamical boundary condition ,Analysis ,Mathematics ,Analytic function - Abstract
This paper is concerned with the well-posedness and the asymptotic behavior of solutions to the following parabolic–hyperbolic phase-field system (0.1) { ( θ + χ ) t − Δ θ = 0 , χ t t + χ t − Δ χ + ϕ ( χ ) − θ = 0 , in Ω × ( 0 , + ∞ ) , subject to the Neumann boundary condition for θ (0.2) ∂ ν θ = 0 , on Γ × ( 0 , + ∞ ) , the dynamical boundary condition for χ (0.3) ∂ ν χ + χ + χ t = 0 , on Γ × ( 0 , + ∞ ) , and the initial conditions (0.4) θ ( 0 ) = θ 0 , χ ( 0 ) = χ 0 , χ t ( 0 ) = χ 1 , in Ω , where Ω is a bounded domain in R 3 with smooth boundary Γ, ν is the outward normal direction to the boundary and ϕ is a real analytic function. In this paper we first establish the existence and uniqueness of a global strong solution to (0.1)–(0.4). Then, we prove its convergence to an equilibrium as time goes to infinity and we provide an estimate of the convergence rate.
- Published
- 2007
- Full Text
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31. Long Time Behavior of Solutions to the Caginalp System with Singular Potential
- Author
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Hana Petzeltová, Maurizio Grasselli, and Giulio Schimperna
- Subjects
Work (thermodynamics) ,Nonlinear system ,Phase transition ,Applied Mathematics ,Bounded function ,Mathematical analysis ,Order (group theory) ,Interval (mathematics) ,Analysis ,Stationary state ,Separation property ,Mathematics - Abstract
We consider a nonlinear parabolic system which governs the evolution of the (relative) temperature θ and of an order parameter χ. This system describes phase transition phenomena like, e.g., melting-solidification processes. The equation ruling χ is characterized by a singular potential W which forces χ to take values in the interval [−1, 1]. We provide reasonable conditions on W which ensure that, from a certain time on, χ stays uniformly away from the pure phases 1 and −1. Combining this separation property with the Lojasiewicz-Simon inequality, we show that any smooth and bounded trajectory uniformly converges to a stationary state and we give an estimate of the decay rate. ∗This work was partially supported by the Italian MIUR PRIN Research Projects Modellizzazione Matematica ed Analisi dei Problemi a Frontiera Libera and Aspetti Teorici e Applicativi di Equazioni a Derivate Parziali, and by the Italian MIUR FIRB Research Project Analisi di Equazioni a Derivate Parziali, Lineari e Non Lineari: Aspetti Metodologici, Modellistica, Applicazioni †The work of H.P. was supported by the Grant A1019302 of GA AV CR ‡The work of G.S. was partially supported by the HYKE Research Training Network
- Published
- 2006
32. A reaction-diffusion equation with memory
- Author
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Maurizio Grasselli and Vittorino Pata
- Subjects
Physics ,Diffusion equation ,Applied Mathematics ,Burgers' equation ,symbols.namesake ,Dirichlet boundary condition ,symbols ,Discrete Mathematics and Combinatorics ,Heat equation ,Fokker–Planck equation ,Boundary value problem ,Convection–diffusion equation ,Cahn–Hilliard equation ,Analysis ,Mathematical physics - Abstract
We consider a one-dimensional reaction-diffusion type equation with memory, originally proposed by W.E. Olmstead et al. to model the velocity $u$ of certain viscoelastic fluids. More precisely, the usual diffusion term $u_{x x}$ is replaced by a convolution integral of the form $\int_0^\infty k(s) u_{x x}(t-s)ds$, whereas the reaction term is the derivative of a double-well potential. We first reformulate the equation, endowed with homogeneous Dirichlet boundary conditions, by introducing the integrated past history of $u$. Then we replace $k$ with a time-rescaled kernel $k_\varepsilon$, where $\varepsilon>0$ is the relaxation time. The obtained initial and boundary value problem generates a strongly continuous semigroup $S_\varepsilon(t)$ on a suitable phase-space. The main result of this work is the existence of the global attractor for $S_\varepsilon(t)$, provided that $\varepsilon$ is small enough.
- Published
- 2006
33. On the hyperbolic relaxation of the one-dimensional Cahn–Hilliard equation
- Author
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Stefania Gatti, Alain Miranville, Vittorino Pata, and Maurizio Grasselli
- Subjects
strongly continuous semigroups ,Global attractors ,Inertial frame of reference ,absorbing sets ,Applied Mathematics ,Mathematical analysis ,Absorbing sets ,robust exponential attractors ,Absorbing set (random dynamical systems) ,Strongly continuous semigroups ,Robust exponential attractors ,Cahn-Hilliard equation ,Exponential function ,global attractors ,Phase space ,Attractor ,Relaxation (physics) ,Cahn–Hilliard equation ,Boundary value problem ,Analysis ,Mathematics - Abstract
We consider the one-dimensional Cahn–Hilliard equation with an inertial term ɛ u t t , for ɛ ⩾ 0 . This equation, endowed with proper boundary conditions, generates a strongly continuous semigroup S ɛ ( t ) which acts on a suitable phase-space and possesses a global attractor. Our main result is the construction of a robust family of exponential attractors { M ɛ } , whose common basins of attraction are the whole phase-space.
- Published
- 2005
34. An inverse source problem for the Lamé system with variable coefficients
- Author
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Masahiro Yamamoto, Maurizio Grasselli, and Masaru Ikehata
- Subjects
Body force ,Applied Mathematics ,Bounded function ,Isotropy ,Linear elasticity ,Linear system ,Mathematical analysis ,Boundary (topology) ,Inverse problem ,Lipschitz continuity ,Analysis ,Mathematics - Abstract
According to the linear theory of elasticity, we consider a bounded, compressible, and isotropic body whose mechanical behavior is described by the Lame system with density and Lame coefficients depending on the space variables. Assuming null surface displacement on the whole boundary, we discuss the inverse problem of determining a body force by observation of surface traction on a suitable sub-boundary along a sufficiently large time interval. Our main results are a Lipschitz stability estimate and a reconstruction formula.
- Published
- 2005
35. Asymptotic behavior of a parabolic-hyperbolic system
- Author
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Vittorino Pata and Maurizio Grasselli
- Subjects
Physics ,Asymptotic analysis ,Applied Mathematics ,Bounded function ,Attractor ,Hyperbolic function ,Order (ring theory) ,General Medicine ,Type (model theory) ,Coupling (probability) ,Analysis ,Hyperbolic equilibrium point ,Mathematical physics - Abstract
We consider a parabolic equation nonlinearly coupled with a damped semilinear wave equation. This system describes the evolution of the relative temperature $\vartheta$ and of the order parameter $\chi$ in a material subject to phase transitions in the framework of phase-field theories. The hyperbolic dynamics is characterized by the presence of the inertial term $\mu\partial_{t t}\chi$ with $\mu>0$. When $\mu=0$, we reduce to the well-known phase-field model of Caginalp type. The goal of the present paper is an asymptotic analysis from the viewpoint of infinite-dimensional dynamical systems. We first prove that the model, endowed with appropriate boundary conditions, generates a strongly continuous semigroup on a suitable phase-space $\mathcal V_0$, which possesses a universal attractor $\mathcal A_\mu$. Our main result establishes that $\mathcal A_\mu$ is bounded by a constant independent of $\mu$ in a smaller phase-space $\mathcal V_1$. This bound allows us to show that the lifting $\mathcal A_0$ of the universal attractor of the parabolic system (corresponding to $\mu=0$) is upper semicontinuous at $0$ with respect to the family $\{\mathcal A_\mu,\mu>0\}$. We also construct an exponential attractor; that is, a set of finite fractal dimension attracting all the trajectories exponentially fast with respect to the distance in $\mathcal V_0$. The existence of an exponential attractor is obtained in the case $\mu=0$ as well. Finally, a noteworthy consequence is that the above results also hold for the damped semilinear wave equation, which is obtained as a particular case of our system when the coupling term vanishes. This provides a generalization of a number of theorems proved in the last two decades.
- Published
- 2004
36. Longtime behavior of a viscoelastic Timoshenko beam
- Author
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Vittorino Pata, Giovanni Prouse, and Maurizio Grasselli
- Subjects
Timoshenko beam theory ,Physics ,Nonlinear system ,Applied Mathematics ,Mathematical analysis ,Attractor ,Dissipative system ,Discrete Mathematics and Combinatorics ,Absorbing set (random dynamical systems) ,Dynamical system ,Rotation ,Hyperbolic partial differential equation ,Analysis - Abstract
We consider a Timoshenko model of a viscoelastic beam fixed at the endpoints and subject to nonlinear external forces. The model consists of two coupled second order linear integrodifferential hyperbolic equations that govern the evolution of the lateral displacement $u$ and the total rotation angle $\phi$. We prove that these equations generate a dissipative dynamical system, whose trajectories are eventually confined in a uniform absorbing set, the dissipativity being due to the memory mechanism solely. This fact allows us to state the existence of a uniform compact attractor.
- Published
- 2003
37. Regularity results for a Cahn-Hilliard-Navier-Stokes system with shear dependent viscosity
- Author
-
Dalibor Pražák and Maurizio Grasselli
- Subjects
Physics ,Shear (geology) ,Applied Mathematics ,Navier stokes ,Mechanics ,Analysis - Published
- 2014
38. Hyperbolic Phase-Field Dynamics with Memory
- Author
-
Horacio G. Rotstein and Maurizio Grasselli
- Subjects
Cauchy problem ,Well-posed problem ,Field (physics) ,Heat flux ,Applied Mathematics ,Mathematical analysis ,Neumann boundary condition ,Boundary value problem ,Hyperbolic partial differential equation ,Analysis ,Mathematics ,Energy functional - Abstract
We consider a non-conserved phase-field system which consists of two nonlinearly coupled hyperbolic integrodifferential equations. This model is derived from two basic assumptions: the heat flux law is of Gurtin–Pipkin type and the response of the order parameter to the variation of the free energy functional is delayed. These hypotheses might be a reasonable attempt to describe, for instance, the melt of He 4 crystals. A suitable initial and boundary value problem is then associated with the system and its well-posedness is analyzed in detail.
- Published
- 2001
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39. On the dissipativity of a hyperbolic phase-field system with memory
- Author
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Vittorino Pata and Maurizio Grasselli
- Subjects
Applied Mathematics ,Mathematical analysis ,Phase (waves) ,Analysis ,Mathematics - Published
- 2001
40. Nonsmooth kernels in a phase relaxation problem with memory
- Author
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Fabio Luterotti, Giovanna Bonfanti, Pierluigi Colli, and Maurizio Grasselli
- Subjects
Mathematical optimization ,Condensed matter physics ,Applied Mathematics ,Phase (matter) ,Relaxation (approximation) ,Analysis ,Mathematics - Published
- 1998
41. Longtime behavior of a homogenized model in viscoelastodynamics
- Author
-
Maurizio Grasselli and Vittorino Pata
- Subjects
Physics ,Body force ,Metric space ,Integro-differential equation ,Applied Mathematics ,Attractor ,Displacement field ,Mathematical analysis ,Discrete Mathematics and Combinatorics ,Equations of motion ,Type (model theory) ,Analysis ,Convolution - Abstract
A material with heterogeneous structure at microscopic level is considered. The microscopic mechanical behavior is described by a stress-strain law of Kelvin-Voigt type. It has been shown that a homogenization process leads to a macroscopic stress-strain relation containing a time convolution term which accounts for memory effects. Consequently, the displacement field $\mathbf{u}$ obeys to a Volterra integrodifferential motion equation. The longtime behavior of $\mathbf{u}$ is here investigated proving the existence of a uniform attractor when the body forces vary in a suitable metric space.
- Published
- 1998
42. Weak solution to hyperbolic Stefan problems with memory
- Author
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Pierluigi Colli, Maurizio Grasselli, and Gianni Gilardi
- Subjects
Nonlinear system ,Applied Mathematics ,Weak solution ,Mathematical analysis ,Uniqueness ,Type (model theory) ,Analysis ,Mathematics - Abstract
A model for Stefan problems in materials with memory is considered. This model is mainly characterized by a nonlinear Volterra integrodifferential equation of hyperbolic type. Colli and Grasselli proved the uniqueness of a weak solution under the natural assumptions on data and the existence of a strong solution for smoother data. Taking advantage of these two results and assuming just the hypotheses ensuring uniqueness, the existence of a weak solution is here shown.
- Published
- 1997
43. An inverse problem in population dynamics
- Author
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Maurizio Grasselli
- Subjects
education.field_of_study ,Control and Optimization ,Dynamics (mechanics) ,Mathematical analysis ,Population ,Inverse problem ,Parabolic partial differential equation ,Computer Science Applications ,Convolution ,Past history ,Matrix (mathematics) ,Signal Processing ,Diffusion (business) ,education ,Analysis ,Mathematics - Abstract
The evolution of population densities of two interacting species in presence of diffusion phenomena is governed by a system of semilinear Volterra integrodifferential parabolic equations. In this system there are time convolution integrals, accounting for past history effects, which are essentially characterized by kernels depending on time only. These delay kernels can be viewed as entries of a 2x2 matrix K. The inverse problem of determining K via suitable population measurements is analyzed.
- Published
- 1997
44. Non-isothermal viscous Cahn--Hilliard equation with inertial term and dynamic boundary conditions
- Author
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Hao Wu, Maurizio Grasselli, and Cecilia Cavaterra
- Subjects
Inertial frame of reference ,Applied Mathematics ,Weak solution ,Mathematical analysis ,General Medicine ,35B40, 35B41, 37L99, 80A22 ,Isothermal process ,Term (time) ,Mathematics - Analysis of PDEs ,Attractor ,Convergence (routing) ,FOS: Mathematics ,Boundary value problem ,Cahn–Hilliard equation ,Analysis ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
We consider a non-isothermal modified Cahn--Hilliard equation which was previously analyzed by M. Grasselli et al. Such an equation is characterized by an inertial term and a viscous term and it is coupled with a hyperbolic heat equation. The resulting system was studied in the case of no-flux boundary conditions. Here we analyze the case in which the order parameter is subject to a dynamic boundary condition. This assumption requires a more refined strategy to extend the previous results to the present case. More precisely, we first prove the well-posedness for solutions with bounded energy as well as for weak solutions. Then we establish the existence of a global attractor. Finally, we prove the convergence of any given weak solution to a single equilibrium by using a suitable Lojasiewicz--Simon inequality.
- Published
- 2013
- Full Text
- View/download PDF
45. Longtime behavior of nonlocal Cahn-Hilliard equations
- Author
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Maurizio Grasselli and Ciprian G. Gal
- Subjects
Applied Mathematics ,Mathematical analysis ,Dynamical Systems (math.DS) ,Dynamical system ,Exponential function ,Mathematics - Analysis of PDEs ,Bounded function ,Attractor ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Dynamical Systems ,Cahn–Hilliard equation ,Constant (mathematics) ,35R09, 37L30, 82C24 ,Analysis ,Stationary state ,Separation property ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
Here we consider the nonlocal Cahn-Hilliard equation with constant mobility in a bounded domain. We prove that the associated dynamical system has an exponential attractor, provided that the potential is regular. In order to do that a crucial step is showing the eventual boundedness of the order parameter uniformly with respect to the initial datum. This is obtained through an Alikakos-Moser type argument. We establish a similar result for the viscous nonlocal Cahn-Hilliard equation with singular (e.g., logarithmic) potential. In this case the validity of the so-called separation property is crucial. We also discuss the convergence of a solution to a single stationary state. The separation property in the nonviscous case is known to hold when the mobility degenerates at the pure phases in a proper way and the potential is of logarithmic type. Thus, the existence of an exponential attractor can be proven in this case as well.
- Published
- 2012
46. Convergence to equilibrium of solutions of the backward Euler schemefor asymptotically autonomous second-order gradient-like systems
- Author
-
Morgan Pierre and Maurizio Grasselli
- Subjects
Combinatorics ,Sequence ,Critical point (thermodynamics) ,Applied Mathematics ,Bounded function ,Mathematical analysis ,Order (ring theory) ,Nabla symbol ,Function (mathematics) ,Constant (mathematics) ,Backward Euler method ,Analysis ,Mathematics - Abstract
Following a result of Chill and Jendoubi in the continuous case, we study the asymptotic behavior of sequences $(U^n)_n$ in $R^d$ which satisfy the following backward Euler scheme: $\varepsilon\frac{(U^{n+1}-2U^n+U^{n-1}}{\Delta t^2} +\frac{U^{n+1}-U^n}{\Delta t}+\nabla F(U^{n+1})=G^{n+1}, n\ge 0, $ where $\Delta t>0$ is the time step, $\varepsilon\ge 0$, $(G^{n+1})_n$ is a sequence in $ R^d$ which converges to $0$ in a suitable way, and $F\in C^{1,1}_{l o c}(R^d, R)$ is a function which satisfies a Łojasiewicz inequality. We prove that the above scheme is Lyapunov stable and that any bounded sequence $(U^n)_n$ which complies with it converges to a critical point of $F$ as $n$ tends to $\infty$. We also obtain convergence rates. We assume that $F$ is semiconvex for some constant $c_F\ge 0$ and that $1/\Delta t
- Published
- 2012
47. Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials
- Author
-
Maurizio Grasselli and Sergio Frigeri
- Subjects
Polynomial ,Applied Mathematics ,Weak solution ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Nonlinear Sciences::Cellular Automata and Lattice Gases ,01 natural sciences ,010101 applied mathematics ,Physics::Fluid Dynamics ,Mathematics - Analysis of PDEs ,Attractor ,Compressibility ,FOS: Mathematics ,35Q30, 37L30, 45K05, 76T99 ,Boundary value problem ,Uniqueness ,Ball (mathematics) ,0101 mathematics ,Navier–Stokes equations ,Nonlinear Sciences::Pattern Formation and Solitons ,Analysis ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
Here we consider a Cahn-Hilliard-Navier-Stokes system characterized by a nonlocal Cahn-Hilliard equation with a singular (e.g., logarithmic) potential. This system originates from a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids. We have already analyzed the case of smooth potentials with arbitrary polynomial growth. Here, taking advantage of the previous results, we study this more challenging (and physically relevant) case. We first establish the existence of a global weak solution with no-slip and no-flux boundary conditions. Then we prove the existence of the global attractor for the 2D generalized semiflow (in the sense of J.M. Ball). We recall that uniqueness is still an open issue even in 2D. We also obtain, as byproduct, the existence of a connected global attractor for the (convective) nonlocal Cahn-Hilliard equation. Finally, in the 3D case, we establish the existence of a trajectory attractor (in the sense of V.V. Chepyzhov and M.I. Vishik)., Comment: 32 pages
- Published
- 2012
- Full Text
- View/download PDF
48. Long-time behavior for a hydrodynamic model on nematic liquid crystal flows with asymptotic stabilizing boundary condition and external force
- Author
-
Maurizio Grasselli and Hao Wu
- Subjects
Applied Mathematics ,Weak solution ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Boundary (topology) ,Orientation (vector space) ,Physics::Fluid Dynamics ,Computational Mathematics ,symbols.namesake ,Mathematics - Analysis of PDEs ,Flow (mathematics) ,Liquid crystal ,Dirichlet boundary condition ,FOS: Mathematics ,symbols ,Boundary value problem ,35B40, 35Q35, 76A15, 76D05 ,Analysis ,Stationary state ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper, we consider a simplified Ericksen-Leslie model for the nematic liquid crystal flow. The evolution system consists of the Navier-Stokes equations coupled with a convective Ginzburg-Landau type equation for the averaged molecular orientation. We suppose that the Navier-Stokes equations are characterized by a no-slip boundary condition and a time-dependent external force g(t), while the equation for the molecular director is subject to a time-dependent Dirichlet boundary condition h(t). We show that, in 2D, each global weak solution converges to a single stationary state when h(t) and g(t) converge to a time-independent boundary datum h_\infty and 0, respectively. Estimates on the convergence rate are also obtained. In the 3D case, we prove that global weak solutions are eventually strong so that results similar to the 2D case can be proven. We also show the existence of global strong solutions, provided that either the viscosity is large enough or the initial datum is close to a given equilibrium., 38 pages
- Published
- 2011
49. Parabolic Perturbation of a Nonlinear Hyperbolic Problem Arising in Physiology
- Author
-
Maurizio Grasselli and Pierluigi Colli
- Subjects
Nonlinear system ,Mathematical model ,Applied Mathematics ,Transport coefficient ,Weak solution ,Mathematical analysis ,Perturbation (astronomy) ,Initial value problem ,Uniqueness ,Sliding filament theory ,Analysis ,Mathematics - Abstract
We study a transport-diffusion initial value problem where the diffusion codlicient is "small" and the transport coefficient is a time function depending on the solution in a nonlinear and nonlocal way. We show the existence and the uniqueness of a weak solution of this problem. Moreover we discuss its asymptotic behaviour as the diffusion coefficient goes to zero, obtaining a well-posed first-order nonlinear hyperbolic problem. These problems arise from mathematical models of muscle contraction in the framework of the sliding filament theory.
- Published
- 1993
- Full Text
- View/download PDF
50. Phase-field systems with nonlinear coupling and dynamic boundary conditions
- Author
-
Maurizio Grasselli, Alain Miranville, Cecilia Cavaterra, and Ciprian G. Gal
- Subjects
Quadratic growth ,Applied Mathematics ,Mathematical analysis ,Mixed boundary condition ,Dynamical system ,Robin boundary condition ,symbols.namesake ,Dirichlet boundary condition ,symbols ,Neumann boundary condition ,Boundary value problem ,Uniqueness ,Analysis ,Mathematics - Abstract
We consider phase-field systems of Caginalp type on a three-dimensional bounded domain. The order parameter fulfills a dynamic boundary condition, while the (relative) temperature is subject to a homogeneous boundary condition of Dirichlet, Neumann or Robin type. Moreover, the two equations are nonlinearly coupled through a quadratic growth function. Here we extend several results which have been proven by some of the authors for the linear coupling. More precisely, we demonstrate the existence and uniqueness of global solutions. Then we analyze the associated dynamical system and we establish the existence of global as well as exponential attractors. We also discuss the convergence of given solutions to a single equilibrium.
- Published
- 2010
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