Search

We consider an optimal control problem with tracking-type cost functional constrained by the Cattaneo equation, which is a well-known model for delayed heat transfer. In particular, we are interested the asymptotic behaviour of the optimal control problems for a vanishing delay time $\tau \rightarrow 0$. First, we show the convergence of solutions of the Cattaneo equation to the ones of the heat equation. Assuming the same right-hand side and compatible initial conditions for the equations, we prove a linear convergence rate. Moreover, we show linear convergence of the optimal states and optimal controls for the Cattaneo equation towards the ones for the heat equation. We present numerical results for both, the forward and the optimal control problem confirming these linear convergence rates.

Following a restriction argument in the Euclidean space, we derive a geometric invariant formula for a possible viscosity operator for an incompressible fluid flow on an ellipsoid embedded in $\mathbb R^3$. We also give an asymptotic expansion of the formula in terms of the eccentricity associated with the ellipsoid., 12 pages

The Peaceman-Rachford alternating direction implicit (ADI) scheme for linear time-dependent Maxwell equations is analyzed on a heterogeneous cuboid. Due to discontinuities of the material parameters, the solution of the Maxwell equations is less than $H^2$-regular in space. For the ADI scheme, we prove a rigorous time-discrete error bound with a convergence rate that is half an order lower than the classical one. Our statement imposes only assumptions on the initial data and the material parameters, but not on the solution. To establish this result, we analyze the regularity of the Maxwell equations in detail in an appropriate functional analytical framework. The theoretical findings are complemented by a numerical experiment indicating that the proven convergence rate is indeed observable and optimal.

For a smooth bounded domain $\Omega$ and $p \geq q \geq 2$, we establish quantified versions of the classical Friedrichs inequality $\|\nabla u\|_p^p - \lambda_1 \|u\|_q^p \geq 0$, $u \in W_0^{1,p}(\Omega)$, where $\lambda_1$ is a generalized least frequency. We apply one of the obtained quantifications to show that the resonant equation $-\Delta_p u = \lambda_1 \|u\|_q^{p-q} |u|^{q-2} u + f$ coupled with zero Dirichlet boundary conditions possesses a weak solution provided $f$ is orthogonal to the minimizer of $\lambda_1$., Comment: 25 pages

We analyze the functional equation $$F(x+F(x))=-F(x)$$ and reveal its relationship with a system of partial differential equations arising as the hydrodynamic limit of a system of pinned billiard balls on the line. The system of balls must freeze at some time, i.e., no velocity may change after the freezing time. The terminal velocity and the freezing time profiles play the role of boundary conditions for the PDEs (qua terminal conditions, despite being initial conditions from the wave equation perspective pursued here). Solutions to the functional equation provide the link between the freezing time and terminal velocity profiles on the one hand, and the solution to the PDE in the entirety of the space-time domain on the other., Comment: Metadata typos corrected

The eigenvalue problem for the Laplacian on bounded, planar, convex domains with mixed boundary conditions is considered, where a Dirichlet boundary condition is imposed on a part of the boundary and a Neumann boundary condition on its complement. Given two different such choices of boundary conditions for the same domain, we prove inequalities between their lowest eigenvalues. As a special case, we prove parts of a conjecture on the order of mixed eigenvalues of triangles.

We prove the existence and multiplicity of weak solutions for a mixed local-nonlocal problem at resonance. In particular, we consider a not necessarily positive operator which appears in models describing the propagation of flames. A careful adaptation of well known variational methods is required to deal with the possible existence of negative eigenvalues., Comment: arXiv admin note: text overlap with arXiv:2207.14008

In this paper we study local higher regularity properties of a linear elliptic system that is coupled with a system of Maxwell-type. The regularity result is proved by means of a modified finite difference argument. These modified finite differences are based on inner variations combined with a Piola-type transformation in order to preserve the $\curl$-structure in the Maxwell system. The result is applied to the relaxed micromorphic model., Comment: 12 pages

We analyze the topological structure of the Nehari set for a class of functionals depending on a real parameter $\lambda$, and having two degrees of homogeneity. A special attention is paid to the extremal parameter $\lambda^*$, which is the threshold value for the Nehari set to be given by a {\it natural constraint}. The main difficulty arises when $\lambda>\lambda^*$, as the energy functional may be unbounded from below over the Nehari set. In such situation we prove the existence of local minimizers of the functional constrained to this set. We unify and extend previous existence and multiplicity results for critical points of indefinite, $(p,q)$-Laplacian, and Kirchhoff type problems.

We consider a non-local Shr\"odinger problem driven by the fractional Orlicz g-Laplace operator as follows \begin{equation}\label{PP} (-\triangle_{g})^{\alpha}u+g(u)=K(x)f(x,u),\ \ \text{in}\ \mathbb{R}^{d},\tag{P} \end{equation} where $d\geq 3,\ (-\triangle_{g})^{\alpha}$ is the fractional Orlicz g-Laplace operator, $f:\mathbb{R}^d\times\mathbb{R}\rightarrow \mathbb{R}$ is a measurable function and $K$ is a positive continuous function. Employing the Nehari manifold method and without assuming the well-known Ambrosetti-Rabinowitz and differentiability conditions on the non-linear term $f$, we prove that the problem \eqref{PP} has a ground state of fixed sign and a nodal (or sign-changing) solutions.

We consider the Cauchy problem for a second-order nonlinear evolution equation in a Hilbert space. This equation represents the abstract generalization of the Ball integro-differential equation. The general nonlinear case with respect to terms of the equation which include a square of a norm of a gradient is considered. A three-layer semi-discrete scheme is proposed in order to find an approximate solution. In this scheme, the approximation of nonlinear terms that are dependent on the gradient is carried out by using an integral mean. We show that the solution of the nonlinear discrete problem and its corresponding difference analogue of a first-order derivative is uniformly bounded. For the solution of the corresponding linear discrete problem, it is obtained high-order a priori estimates by using two-variable Chebyshev polynomials. Based on these estimates we prove the stability of the nonlinear discrete problem. For smooth solutions, we provide error estimates for the approximate solution. An iteration method is applied in order to find an approximate solution for each temporal step. The convergence of the iteration process is proved., 24 pages

We adapt and extend Yosida's parametrix method, originally introduced for the construction of the fundamental solution to a parabolic operator on a Riemannian manifold, to derive Varadhan-type asymptotic estimates for the transition density of a degenerate diffusion under the weak H\"ormander condition. This diffusion process, widely studied by Yor in a series of papers, finds direct application in the study of a class of path-dependent financial derivatives known as Asian options. We obtain the Varadhan formula \begin{equation} \frac{-2 \log p(t,x;T,y) } { \Psi(t,x;T,y) } \to 1, \qquad \text{as } \quad T-t \to 0^+, \end{equation} where $p$ denotes the transition density and $\Psi$ denotes the optimal cost function of a deterministic control problem associated to the diffusion. We provide a partial proof of this formula, and present numerical evidence to support the validity of an intermediate inequality that is required to complete the proof. We also derive an asymptotic expansion of the cost function $\Psi$, expressed in terms of elementary functions, which is useful in order to design efficient approximation formulas for the transition density.

We prove the decay in the energy space for the solution to the defocusing biharmonic Hartree-Fock equations with mass-supercritical and energy-subcritical Choquard-type nonlinearity in space dimension $d\geq3$. We treat both the free and the perturbed by a potential case. As a direct consequence, we obtain large-data scattering in $H^2(\mathbb R^d)^N, \, N\geq 1$.

In this paper, we consider the finite time blow-up results for a parabolic equation coupled with superlinear source term and local linear boundary dissipation. Using a concavity argument, we derive the sufficient conditions for the solutions to blow up in finite time. In particular, we obtain the existence of finite time blow-up solutions with arbitrary high initial energy. We also derive the upper bound and lower bound of the blow up time., in press on Journal of Mathematical Analysis and Applications, see https://www.sciencedirect.com/science/article/pii/S0022247X22003419

The purpose of this paper is to present the derivation and mathematical analysis of a new asymptotic model that describes the evolution of medium amplitude internal waves propagating between a flat rigid-lid and a highly variable topography. The smallness assumptions on the topographic variation parameter used in [Communications on Pure & Applied Analysis, 2015, 14 (6): 2203-2230] and in [Asymptotic Analysis, vol. 106, no. 2, pp. 61-98, 2018] are now relaxed and the results of the aforementioned papers are improved and generalized to the complex case of large topography variation. Limiting the flow to one-layer, we also emphasize our model's well-posedness in comparison to the original asymptotic model.

We establish a continuous embedding $W^{s(\cdot),2}(\Omega)\hookrightarrow L^{\alpha(\cdot)}(\Omega)$, where the variable exponent $\alpha(x)$ can be close to the critical exponent $2_{s}^*(x)=\frac{2N}{N-2\bar{s}(x)}$, with $\bar{s}(x)=s(x,x)$ for all $x\in\bar{\Omega}$. Subsequently, this continuous embedding is used to prove the multiplicity of solutions for critical nonlocal degenerate Kirchhoff problems with a variable singular exponent. Moreover, we also obtain the uniform $L^{\infty}$-estimate of these infinite solutions by a bootstrap argument.

Delay differential equations form the underpinning of many complex dynamical systems. The forward problem of solving random differential equations with delay has received increasing attention in recent years. Motivated by the challenge to predict the COVID-19 caseload trajectories for individual states in the U.S., we target here the inverse problem. Given a sample of observed random trajectories obeying an unknown random differential equation model with delay, we use a functional data analysis framework to learn the model parameters that govern the underlying dynamics from the data. We show the existence and uniqueness of the analytical solutions of the population delay random differential equation model when one has discrete time delays in the functional concurrent regression model and also for a second scenario where one has a delay continuum or distributed delay. The latter involves a functional linear regression model with history index. The derivative of the process of interest is modeled using the process itself as predictor and also other functional predictors with predictor-specific delayed impacts. This dynamics learning approach is shown to be well suited to model the growth rate of COVID-19 for the states that are part of the U.S., by pooling information from the individual states, using the case process and concurrently observed economic and mobility data as predictors.

In this work, we consider a certain multilayered (thick layer) wave--(thin layer) wave--heat (fluid) interactive PDE system. Such coupled PDE systems have been used in the literature to describe the blood transport process in mammalian vascular systems. In particular, the deformations of the boundary interface (thin layer) are described via the two dimensional elastic equation. The present work constitutes an investigation of the extent of the stabilizing effects of the underlying fluid dissipation -- across the boundary interface -- upon both the thick and thin structural components. (All three PDE components evolve on their respective geometries.) In this regard, our main result is the derivation of uniform decay rates for classical solutions of this multilayered PDE model. To obtain these estimates, necessary a priori inequalities for certain static multilayered PDE models are generated here to ultimately allow an application of a wellknown resolvent criterion for rational decay.

We show unconditional uniqueness of solutions to the Cauchy problem associated with the Benjamin-Ono equation under the periodic boundary condition with initial data given in $H^s$ for $s>1/6$. This improves the previous unconditional uniqueness result in $H^{1/2}$ by Molinet and Pilod (2012). Our proof is based on a gauge transform and integration by parts in the time variable., 20 pages. Minor changes made. To appear in J. Math. Anal. Appl

This paper studies in a quantitatively precise manner the field enhancement due to presence of an emitter of the dipole type near the bow-tie structure of perfectly conducting inclusions in the two-dimensional space. We put special emphasis on field enhancement near vertices of the bow-tie structure, and derive upper and lower bounds of the gradient blow-up there. All three different kinds of symmetries are considered by varying locations and directions of the emitter, and a different estimate is derived for each case., Comment: 3 figures

We investigate solutions to nonlinear elliptic Dirichlet problems of the type \[ \left\{\begin{array}{cl} - {\rm div} A(x,u,\nabla u)= \mu &\qquad \mathrm{ in}\qquad \Omega, u=0 &\qquad \mathrm{ on}\qquad \partial\Omega, \end{array}\right. \] where $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^n$ and $A(x,z,\xi)$ is a Carath\'eodory's function. The growth of~the~monotone vector field $A$ with respect to the $(z,\xi)$ variables is expressed through some $N$-functions $B$ and $P$. We do not require any particular type of growth condition of such functions, so we deal with problems in nonreflexive spaces. When the problem involves measure data and weakly monotone operator, we prove existence. For $L^1$-data problems with strongly monotone operator we infer also uniqueness and regularity of~solutions and their gradients in the scale of Orlicz-Marcinkiewicz spaces.

We consider the Cauchy problem associated to the recently derived higher order hamiltonian model for unidirectional water waves and prove global existence for given data in the Sobolev space $H^s$, $s\geq 1$. We also prove an ill-posedness result by showing that the flow-map is not continuous if the given data has Sobolev regularity $s< 1$. The results obtained in this work are sharp., 18 pages. To appear in JMAA - 2019

This paper deals with a lower bound for the blow-up time for solutions of the fully parabolic chemotaxis system { u t = ∇ ⋅ [ ( u + α ) m 1 − 1 ∇ u − χ u ( u + α ) m 2 − 2 ∇ v ] in Ω × ( 0 , T ) , v t = Δ v − v + u in Ω × ( 0 , T ) under Neumann boundary conditions and initial conditions, where Ω is a general bounded domain in R n with smooth boundary, α > 0 , χ > 0 , m 1 , m 2 ∈ R and T > 0 . Recently, Anderson–Deng [1] gave a lower bound for the blow-up time in the case that m 1 = 1 and Ω is a convex bounded domain. The purpose of this paper is to generalize the result in [1] to the case that m 1 ≠ 1 and Ω is a non-convex bounded domain. The key to the proof is to make a sharp estimate by using the Gagliardo–Nirenberg inequality and an inequality for boundary integrals. As a consequence, the main result of this paper reflects the effect of nonlinear diffusion and need not assume the convexity of Ω.

We prove the global well-posedness of weak solutions for nonlinear wave equations with supercritical source and damping terms on a three-dimensional torus T 3 of the prototype u t t − Δ u + | u t | m − 1 u t = | u | p − 1 u , ( x , t ) ∈ T 3 × R + ; u ( 0 ) = u 0 ∈ H 1 ( T 3 ) ∩ L m + 1 ( T 3 ) , u t ( 0 ) = u 1 ∈ L 2 ( T 3 ) , where 1 ≤ p ≤ min ⁡ { 2 3 m + 5 3 , m } . Notably, p is allowed to be larger than 6.

In this paper, we investigate the critical points of solutions to the prescribed constant mean curvature equation with Neumann and Robin boundary conditions respectively in a bounded smooth convex domain $\Omega$ of $\mathbb{R}^{n}(n\geq2)$. Firstly, we show the non-degeneracy and uniqueness of the critical points of solutions in a planar domain by using the local Chen & Huang's comparison technique and the geometric properties of approximate surfaces at the non-degenerate critical points. Secondly, we deduce the uniqueness and non-degeneracy of the critical points of solutions in a rotationally symmetric domain of $\mathbb{R}^{n}(n\geq3)$ by the projection of higher dimensional space onto two dimensional plane., Comment: 15pages, 4figures

We prove that the initial value problem associated to a nonlocal perturbation of the Benjamin-Ono equation is locally and globally well-posed in Sobolev spaces $H^s(\mathbb{R})$ for any $s>-3/2$ and we establish that our result is sharp in the sense that the flow map of this equation fails to be $C^2$ in $H^s(\mathbb{R})$ for $s0$. We also prove some unique continuation properties of the solution flow in these spaces., Comment: 33 pages

We study an inhomogeneous minimization problems associated to the $p(x)$-Laplacian. We make a thorough analysis of the essential properties of their minimizers and we establish a relationship with a suitable free boundary problem. On the one hand, we study the problem of minimizing the functional $J(v)=\int_\Omega\Big(\frac{|\nabla v|^{p(x)}}{p(x)}+\lambda(x)\chi_{\{v>0\}}+fv\Big)\,dx$. We show that nonnegative local minimizers $u$ are solutions to the free boundary problem: $u\ge 0$ and \begin{equation} \label{fbp-px}\tag{$P(f,p,{\lambda}^*)$} \begin{cases} \Delta_{p(x)}u:=\mbox{div}(|\nabla u(x)|^{p(x)-2}\nabla u)= f & \mbox{in }\{u>0\}\\ u=0,\ |\nabla u| = \lambda^*(x) & \mbox{on }\partial\{u>0\} \end{cases} \end{equation} with $\lambda^*(x)=\Big(\frac{p(x)}{p(x)-1}\,\lambda(x)\Big)^{1/p(x)}$ and that the free boundary is a $C^{1,\alpha}$ surface. On the other hand, we study the problem of minimizing the functional $J_{\varepsilon}(v)= \int_\Omega \Big(\frac{|\nabla v|^{p_\varepsilon(x)}}{p_\varepsilon(x)}+B_{\varepsilon}(v)+f_\varepsilon v\Big)\, dx$, where $B_\varepsilon(s)=\int _0^s\beta_\varepsilon(\tau) \, d\tau$, $\varepsilon>0$, ${\beta}_{\varepsilon}(s)={1 \over \varepsilon} \beta({s \over \varepsilon})$, with $\beta$ a Lipschitz function satisfying $\beta>0$ in $(0,1)$, $\beta\equiv 0$ outside $(0,1)$. We prove that if $u_\varepsilon$ are nonnegative local minimizers, then any limit function $u$ ($\varepsilon\to 0$) is a solution to the free boundary problem $P(f,p,{\lambda}^*)$ with $\lambda^*(x)=\Big(\frac{p(x)}{p(x)-1}\,M\Big)^{1/p(x)}$, $M=\int \beta(s)\, ds$, $p=\lim p_\varepsilon$, $f=\lim f_\varepsilon$, and that the free boundary is a $C^{1,\alpha}$ surface. In order to obtain our results we need to overcome deep technical difficulties and develop new strategies, not present in the previous literature for this type of problems.

In the past years, the phenomenon of fractional regularity has been addressed for a large class of linear and/or quasilinear differential operators, mostly, in terms of certain Besov spaces. As it turned out, for equations governed by the p-Laplacian, in general, the regularity of solutions appears in terms of functional spaces with nonlinear order of smoothness. Moreover, despite its own interest, fractional regularity methods may be used as a tool for the investigation of some Partial Differential Equations which are not usually addressed in this manner. Thus, the purpose of the present paper is to exploit such methods in order to provide some results regarding existence and regularity of solutions to a class nonlocal elliptic equations which are linked to the p-Laplacian. This is done by means of explicit a priori estimates regarding Lebesgue and Nikolskii spaces, which are part of the present contribution. As a consequence, this approach allows a relaxation on some of the standard conditions employed in this class of problems.

An affine manifold is said to be geodesically complete if all affine geodesics extend for all time. It is said to be affine Killing complete if the integral curves for any affine Killing vector field extend for all time. We use the solution space of the quasi-Einstein equation to examine these concepts in the setting of homogeneous affine surfaces.

In this paper, we prove Lipschitz stable determination of the complex-valued electric potential and the direction of the magnetic field appearing in the dynamic Schrodinger equation with static coefficients, by finitely many partial boundary measurements of the solution. This is by means of the Bukhgeim–Klibanov method, based on an appropriate Carleman estimate. Since the time symmetrization of the static magnetic Schrodinger equation around t = 0 is not possible, we preliminarily establish a Carleman inequality specifically designed for this problem.

We give a criterion for the global boundedness of integral operators which are known to be locally bounded. As an application, we discuss the global $L^p$-boundedness for a class of Fourier integral operators. While the local $L^p$-boundedness of Fourier integral operators is known from the work of Seeger, Sogge and Stein, not so many results are available for the global boundedness on $L^p({\mathbb R}^n)$. We give several natural sufficient conditions for them., 13 pages. This paper is a substantially reworked version of our paper arXiv:1510.03807, where we now deduce the result of that paper on the global boundedness for Fourier integral operators from a much more general principle for integral operators

This paper establishes an aspect of Bohr's correspondence principle, i.e. that quantum mechanics converges in the high frequency limit to classical mechanics, for commuting semiclassical unitary operators. We prove, under minimal assumptions, that the semiclassical limit of the convex hulls of the quantum spectrum of a collection of commuting semiclassical unitary operators converges to the convex hull of the classical spectrum of the principal symbols of the operators., Comment: 32 pages. Presentation substantially revised and reorganized for clarity. Main Theorem is now in in the introduction, and non central materialshave been moved to appendix 1 and appendix 2. Small mistake in the application of Cayley transform has been fixed

The paper deals with second order parabolic equations on bounded domains with Dirichlet conditions in arbitrary Euclidean spaces. Their interest comes from being models for describing reaction–diffusion processes in several frameworks. A linear diffusion term in divergence form is included which generates a strongly elliptic differential operator. A further linear part, of integral type, is present which accounts of nonlocal diffusion behaviours. The main result provides a unifying method for studying the existence and localization of solutions satisfying nonlocal associated boundary conditions. The Cauchy multipoint and the mean value conditions are included in this investigation. The problem is transformed into its abstract setting and the proofs are based on the homotopic invariance of the Leray–Schauder topological degree. A bounding function (i.e. Lyapunov-like function) theory is developed, which is new in this infinite dimensional context. It allows that the associated vector fields have no fixed points on the boundary of their domains and then it makes possible the use of a degree argument.

We study the system \begin{align*} \label{prob:star} \tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) - u - f(u) w + \kappa, \\ v_t = \Delta v - v + f(u) w, \\ w_t = \Delta w - w + v, \end{cases} \end{align*} which models the virus dynamics in an early stage of an HIV infection, in a smooth, bounded domain $\Omega \subset \mathbb R^n, n \in \mathbb N,$ for a parameter $\kappa \ge 0$ and a given function $f \in C^1([0, \infty))$ satisfying $f \ge 0$, $f(0) = 0$ and $f(s) \le K_f s^\alpha$ for all $s \ge 1$, some $K_f \gt 0$ and $\alpha \in \mathbb R$. We prove that whenever \begin{align*} \alpha \lt \frac2n, \end{align*} solutions to \eqref{prob:star} exist globally and are bounded. The proof mainly relies on smoothing estimates for the Neumann heat semigroup and (in the case $\alpha \gt 1$) on a functional inequality. Furthermore, we provide some indication why the exponent $\frac2n$ could be essentially optimal., Comment: 12 pages

International audience; Druet [6] proved that if $(f_\gamma)_\gamma$ is a sequence of Moser-Trudinger type nonlinearities with critical growth, and if $(u_\gamma)_\gamma$ solves $$ \begin{cases}&\Delta u_\gamma =f_\gamma(x,u_\gamma)\,,~~ u_\gamma>0\text{ in }\Omega\,,\\&u_\gamma =0\text{ on }\partial\Omega\,,\end{cases}$$ and converges weakly in $H^1_0$ to some $u_\infty$, then the Dirichlet energy is quantified, namely there exists an integer $N\ge 0$ such that the energy of $u_\gamma$ converges to $4\pi N$ plus the Dirichlet energy of $u_\infty$. As a crucial step to get the general existence results of [7], it was more recently proved in [8] that, for a specific class of nonlinearities, the loss of compactness (i.e. $N>0$) implies that $u_\infty\equiv 0$. In contrast, we prove here that there exist sequences $(f_\gamma)_\gamma$ of Moser-Trudinger type nonlinearities which admit a noncompact sequence $(u_\gamma)_\gamma$ of solutions having a nontrivial weak limit.

We study an Eulerian droplet model which can be seen as the pressureless gas system with a source term, a subsystem of this model and the inviscid Burgers equation with source term. The condition for loss of regularity of a solution to Burgers equation with source term is established. The same condition applies to the Eulerian droplet model and its subsystem. The Riemann problem for the Eulerian droplet model is constructively solved by going through the solution of the Riemann problems for the inviscid Burgers equation with a source term and the subsystem, respectively. Under suitable generalized Rankine–Hugoniot relations and entropy condition, the existence of delta-shock solution is established. The existence of a solution to the generalized Rankine–Hugoniot conditions is proven. Some numerical illustrations are presented.

We prove that even irregular convergence of semigroups of operators implies similar convergence of mild solutions of the related semi-linear equations with Lipschitz continuous nonlinearity. This result is then applied to three models originating from mathematical biology: shadow systems, diffusions on thin layers, and dynamics of neurotransmitters, 21 pages, no figures. Minor revision: Some typos fixed

We consider an evolution equation generalising the viscous Burgers equation supplemented by an initial condition which is a homogeneous random field. We develop a non-linear framework enabling us to show the existence and regularity of solutions as well as their long time behaviour.

We study the distributional solutions to the (generalized) Beltrami equation under Sobolev assumptions on the Beltrami coefficients. In this setting, we prove that these distributional solutions are true quasiregular maps and they are smoother than expected, that is, they have second order derivatives in L l o c 1 + e , for some e > 0 .

Multiscale analysis of a degenerate pseudoparabolic variational inequality, modelling the two-phase flow with dynamical capillary pressure in a perforated domain, is the main topic of this work. Regularisation and penalty operator methods are applied to show the existence of a solution of the nonlinear degenerate pseudoparabolic variational inequality defined in a domain with microscopic perforations, as well as to derive a priori estimates for solutions of the microscopic problem. The main challenge is the derivation of a priori estimates for solutions of the variational inequality, uniformly with respect to the regularisation parameter and to the small parameter defining the scale of the microstructure. The method of two-scale convergence is used to derive the corresponding macroscopic obstacle problem.

The main goal of this work is to prove the existence of three different solutions (one positive, one negative and one with nonconstant sign) for the equation $(-\Delta_p)^s u= |u|^{p^{*}_s -2} u +\lambda f(x,u)$ in a bounded domain with Dirichlet condition, where $(-\Delta_p)^s$ is the well known $p$-fractional Laplacian and $p^*_s=\frac{np}{n-sp}$ is the critical Sobolev exponent for the non local case. The proof is based in the extension of the Concentration Compactness Principle for the $p$-fractional Laplacian and Ekeland's variational Principle., Comment: 10 pages. arXiv admin note: text overlap with arXiv:0808.3143, arXiv:0912.3465

We study a system of nonlinear partial differential equations describing the unsteady motions of incompressible chemically reacting non-Newtonian fluids. The system under consideration consists of the generalized Navier-Stokes equations with a power-law type stress-strain relation, where the power-law index depends on the concentration of a chemical, coupled to a convection-diffusion equation for the concentration. This system of equations arises in the rheology of the synovial fluid found in the cavities of synovial joints. We prove the existence of global weak solutions of the non-stationary model by using the Galerkin method combined with generalized monotone operator theory and parabolic De Giorgi-Nash-Moser theory. As the governing equations involve a nonlinearity with a variable power-law index, our proofs exploit the framework of generalized Sobolev spaces with variable integrability exponent., Comment: 25 pages

Consider the adjoint restriction inequality associated with the hypersurface $\{ (\tau, \xi) \in \mathbb{R}^{d+1} : \tau = |\xi|^2 \} \cup \{(\tau, \xi) \in \mathbb{R}^{d+1} : \tau - \tau_0 = |\xi - \xi_0|^2\}$ for any $(\tau_0, \xi_0) \neq 0$. We prove that extremizers do not exist for this inequality and fully characterize extremizing sequences in terms of extremizers for the adjoint restriction inequality for the paraboloid.

In this work we study the degenerate diffusion equation $\partial_{t}=x^{\alpha}a\left(x\right)\partial_{x}^{2}+b\left(x\right)\partial_{x}$ for $\left(x,t\right)\in\left(0,\infty\right)^{2}$, equipped with a Cauchy initial data and the Dirichlet boundary condition at $0$. We assume that the order of degeneracy at 0 of the diffusion operator is $\alpha\in\left(0,2\right)$, and both $a\left(x\right)$ and $b\left(x\right)$ are only locally bounded. We adopt a combination of probabilistic approach and analytic method: by analyzing the behaviors of the underlying diffusion process, we give an explicit construction to the fundamental solution $p\left(x,y,t\right)$ and prove several properties for $p\left(x,y,t\right)$; by conducting a localization procedure, we obtain an approximation for $p\left(x,y,t\right)$ for $x,y$ in a neighborhood of 0 and $t$ sufficiently small, where the error estimates only rely on the local bounds of $a\left(x\right)$ and $b\left(x\right)$ (and their derivatives). There is a rich literature on such a degenerate diffusion in the case of $\alpha=1$. Our work extends part of the existing results to cases with more general order of degeneracy, both in the analysis context (e.g., heat kernel estimates on fundamental solutions) and in the probability view (e.g., wellposedness of stochastic differential equations)., Comment: 37 pages

We construct an efficient approach to deal with the global regularity estimates for a class of elliptic double-obstacle problems in Lorentz and Orlicz spaces. The motivation of this paper comes from the study on an abstract result in the viewpoint of the fractional maximal distributions and this work also extends some regularity results proved in \cite{PN_dist} by using the weighted fractional maximal distributions (WFMDs). We further investigate a pointwise estimates of the gradient of weak solutions via fractional maximal operators and Riesz potential of data. Moreover, in the setting of the paper, we are led to the study of problems with nonlinearity is supposed to be partially weak BMO condition (is measurable in one fixed variable and only satisfies locally small-BMO seminorms in the remaining variables).

We show the well-posedness for a large class of degenerate parabolic equations with an additional singularity and mixed Dirichlet-Neumann boundary conditions on bounded Lipschitz domains. The proof is based on an $L^1$-contraction result. In addition, we analyze systems where degenerate equations are coupled to semilinear reaction diffusion equations. This setting includes mathematical models for biofilm growth which are the motivation for our analysis., 33 pages

The aim of this work is to prove the weak-strong uniqueness principle for the compressible Navier-Stokes-Poisson system on an exterior domain, with an isentropic pressure of the type $p(\varrho)=a\varrho^{\gamma}$ and allowing the density to be close or equal to zero. In particular, the result will be first obtained for an adiabatic exponent $\gamma \in [9/5,2]$ and afterwards, this range will be slightly enlarged via pressure estimates "up to the boundary", deduced relaying on boundedness of a proper singular integral operator.

We prove the existence of an extremal function in the Hardy-Littlewood-Sobolev inequality for the energy associated to an stable operator. To this aim we obtain a concentration-compactness principle for stable processes in R N .

We introduce here a new diffuse interface thermodynamically consistent non-isothermal model for tumor growth in presence of a nutrient in a domain Ω ⊂ R 3 . In particular our system describes the growth of a tumor surrounded by healthy tissues, taking into account changes of temperature, proliferation of cells, nutrient consumption and apoptosis. Our aim consists in proving an existence result for our problem associated to the entropy formulation.

In this paper we study approximations of functions of Sobolev spaces $W^2_{p,\loc}(\Omega)$, $\Omega\subset\mathbb R^n$, by Lipschitz continuous functions. We prove that if $f\in W^2_{p,\loc}(\Omega)$, $1\leq p, Comment: 12 pages