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We study the Cauchy problem for the full compressible Navier-Stokes-Maxwell system with a nonconstant background density in R 3. By means of suitable choosing of symmetrizers and weighted energy estimates with some new developments, we establish the global existence and uniqueness of the classical solution provided that the initial data are near this equilibrium. Furthermore, by using the spectrum analysis on the linearized homogeneous system of the full compressible Navier-Stokes-Maxwell equations and refining the convergence property, we obtain the time-algebraic convergence rates of the perturbed solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the global spherically symmetric solutions for the initial boundary value problem of a coupled compressible Navier–Stokes/Allen–Cahn system which describes the motion of two-phase viscous compressible fluids. We prove the existence and uniqueness of global classical solution, weak solution and strong solution under the assumption of spherically symmetry condition for initial data ρ0 without vacuum state. [ABSTRACT FROM AUTHOR]

We examine an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters, where each cluster is assumed to be composed of identical units. In contrast to previous investigations into such discrete-size fragmentation models, we allow the fragmentation coefficients to vary with time. By formulating the initial-value problem for the system as a non-autonomous abstract Cauchy problem, posed in an appropriately weighted $ \ell^1 $ space, and then applying results from the theory of evolution families, we prove the existence and uniqueness of physically relevant, classical solutions for suitably constrained coefficients. [ABSTRACT FROM AUTHOR]

This research paper introduces novel findings regarding a specific class of Kirchhoff problems. Building upon the theoretical groundwork established in a previous work by Ricceri [15], the paper centers its investigation on the Sierpiński gasket (V, |•|) situated in (Rn-1,- |•|), where n≥ 2. The intrinsic boundary of the Sierpiński gasket, denoted as Vo, comprises n corners. Under certain conditions, we prove that a Kirchhoff problem has two classical solutions in the space H0¹(V). [ABSTRACT FROM AUTHOR]

The current paper is concerned with the stabilization in the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, $ \begin{equation} \begin{cases} u_t = \Delta u-\chi_1 \nabla\cdot (\frac{u}{w} \nabla w)+u(a_1-b_1u-c_1v) , \quad &x\in \Omega\cr v_t = \Delta v-\chi_2 \nabla\cdot (\frac{v}{w} \nabla w)+v(a_2-b_2v-c_2u), \quad &x\in \Omega\cr 0 = \Delta w-\mu w +\nu u+ \lambda v, \quad &x\in \Omega \cr \frac{\partial u}{\partial n} = \frac{\partial v}{\partial n} = \frac{\partial w}{\partial n} = 0, \quad &x\in\partial\Omega, \end{cases} \end{equation}~~~~(1) $ where $ \Omega \subset \mathbb{R}^N $ is a bounded smooth domain, and $ \chi_i $, $ a_i $, $ b_i $, $ c_i $ ($ i = 1, 2 $) and $ \mu, \, \nu, \, \lambda $ are positive constants. In [25], among others, we proved that for any given nonnegative initial data $ u_0, v_0\in C^0(\bar\Omega) $ with $ u_0+v_0\not \equiv 0 $, (1) has a unique globally defined classical solution $ (u(t, x;u_0, v_0), v(t, x;u_0, v_0), w(t, x;u_0, v_0)) $ with $ u(0, x;u_0, v_0) = u_0(x) $ and $ v(0, x;u_0, v_0) = v_0(x) $ in any space dimensional setting with any positive constants $ \chi_i, a_i, b_i, c_i $ ($ i = 1, 2 $) and $ \mu, \nu, \lambda $. In this paper, we assume that the competition in (1) is weak in the sense that $ \frac{c_1}{b_2}<\frac{a_1}{a_2}, \quad \frac{c_2}{b_1}<\frac{a_2}{a_1}. $ Then (1) has a unique positive constant solution $ (u^*, v^*, w^*) $, where$ u^* = \frac{a_1b_2-c_1a_2}{b_1b_2-c_1c_2}, \quad v^* = \frac{b_1a_2-a_1c_2}{b_1b_2-c_1c_2}, \quad w^* = \frac{\nu}{\mu}u^*+\frac{\lambda}{\mu} v^*. $We obtain some explicit conditions on $ \chi_1, \chi_2 $ which ensure that the positive constant solution $ (u^*, v^*, w^*) $ is globally stable, that is, for any given nonnegative initial data $ u_0, v_0\in C^0(\bar\Omega) $ with $ u_0\not \equiv 0 $ and $ v_0\not \equiv 0 $, $ \lim\limits_{t\to\infty}\Big(\|u(t, \cdot;u_0, v_0)-u^*\|_\infty +\|v(t, \cdot;u_0, v_0)-v^*\|_\infty+\|w(t, \cdot;u_0, v_0)-w^*\|_\infty\Big) = 0. $ [ABSTRACT FROM AUTHOR]

The Cauchy problem of compressible quantum Navier–Stokes–Poisson equations in three-dimensional space is considered in this paper. Under some smallness conditions on the initial data, we derive the existence of the global classical solution near the non-constant steady state by using the energy method. Combining the linear decay rate and the energy method, we also prove the algebraic decay rate of the solution toward the non-constant steady state with a small doping profile. [ABSTRACT FROM AUTHOR]

We introduce a notion of a weak solution of the master equation without idiosyncratic noise in mean field game theory and establish its existence, uniqueness up to a constant, and consistency with classical solutions when it is smooth. We work in a monotone setting and rely on Lions' Hilbert space approach. For the first-order master equation without idiosyncratic noise, we also give an equivalent definition in the space of measures and establish the well-posedness. [ABSTRACT FROM AUTHOR]

In this paper, we study the global existence and pointwise behavior of classical solution to one dimensional isentropic Navier-Stokes equations with mixed type boundary condition in half space. Based on classical energy method for half space problem, the global existence of classical solution is established firstly. Through analyzing the quantitative relationships of Green's function between Cauchy problem and initial boundary value problem, we observe that the leading part of Green's function for the initial boundary value problem is composed of three items: delta function, diffusive heat kernel, and reflected term from the boundary. Then applying Duhamel's principle yields the explicit expression of solution. With the help of accurate estimates for nonlinear wave coupling and the elliptic structure of velocity, the pointwise behavior of the solution is obtained under some appropriate assumptions on the initial data. Our results prove that the solution converges to the equilibrium state at the optimal decay rate [Math Processing Error] (1 + t) − 1 2 in [Math Processing Error] L ∞ norm. [ABSTRACT FROM AUTHOR]

We consider a time-fractional Cahn–Hilliard equation where the conventional first-order time derivative is replaced by a Caputo fractional derivative of order |$\alpha \in (0,1)$|⁠. Based on an a priori bound of the exact solution, global existence of solutions is proved and detailed regularity results are included. A finite element method is then analyzed in a spatially discrete case and in a completely discrete case based on a convolution quadrature in time generated by the backward Euler method. Error bounds of optimal order are obtained for solutions with smooth and nonsmooth initial data, thereby extending earlier studies on the classical Cahn–Hilliard equation. Further, by proving a new result concerning the positivity of a discrete time-fractional integral operator, it is shown that the proposed numerical scheme inherits a discrete energy dissipation law at the discrete level. Numerical examples are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper concerns the compressible Navier–Stokes–Korteweg equations. Based on previous work [Li et al., Arch. Ration. Mech. Anal. 232, 557–590 (2019)], we prove that the classical solution with finite energy does not exist in the inhomogeneous Sobolev space for any short time under some natural assumptions on initial data near the vacuum. [ABSTRACT FROM AUTHOR]

For p ~ h → -parabolic equations with continuous coefficients, we study the problem of finding classical solutions satisfying modified initial conditions with generalized data in the form of Gelfand- and Shilovtype distributions. This condition linearly combines the values of the solution at the initial time and at a certain intermediate time point. We establish the conditions for the correct solvability of this problem and deduce the formula for its solution. By using the obtained results, we solve the corresponding problem with impulsive action. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global existence, uniqueness and asymptotic stability of time periodic classical solution for a class of extended Fisher-Kolmogorov equations with delays and general nonlinear term. We establish a general framework to investigate the asymptotic behavior of time periodic solutions for nonlinear extended Fisher-Kolmogorov equations with delays and general nonlinear function, which will provide an effective way to deal with such kinds of problems. The discussion is based on the theory of compact and analytic operator semigroups and maximal regularization method. [ABSTRACT FROM AUTHOR]

We study Boltzmann-Fermi-Dirac equation when quantum effects are taken into account in dilute gas dynamics. By employing new estimates on trilinear terms in collision integral, we prove the global existence of the classical solution to Boltzmann-Fermi-Dirac equation near equilibrium. Furthermore, the limit from Boltzmann-Fermi-Dirac equation to incompressible Navier-Stokes-Fourier equations is justified rigorously, which was formally derived in the thesis of Zakrevskiy [48]. [ABSTRACT FROM AUTHOR]

In this paper, the existence of global classical solutions is justified for the three-dimensional compressible magnetohydrodynamic (MHD) equations with vacuum. The main goal of this paper is to obtain a unique global classical solution on R 3 × [ 0 , T ] with any T ∈ (0 , ∞) , provided that the initial magnetic field in the L 3 -norm and the initial density are suitably small. Note that the first result is obtained under the condition of ρ 0 ∈ L γ ∩ W 2 , q with q ∈ (3 , 6) and γ ∈ (1 , 6) . It should be noted that the initial total energy can be arbitrarily large, the initial density allowed to vanish, and the system does not satisfy the conservation law of mass (i.e., ρ 0 ∉ L 1 ). Thus, the results obtained particularly extend the one due to Li–Xu–Zhang (Li et al. in SIAM J. Math. Anal. 45:1356–1387, 2013), where the global well-posedness of classical solutions with small energy was proved. [ABSTRACT FROM AUTHOR]

We prove a local in time well-posedness result for quasi-linear Hamiltonian Schrödinger equations on T d for any d ≥ 1. For any initial condition in the Sobolev space H s , with s large, we prove the existence and uniqueness of classical solutions of the Cauchy problem associated to the equation. The lifespan of such a solution depends only on the size of the initial datum. Moreover we prove the continuity of the solution map. [ABSTRACT FROM AUTHOR]

In this paper, we consider planar magnetohydrodynamics (MHD) system when the viscous coefficients and heat conductivity depend on specific volume v and temperature 𝜃. For technical reasons, the viscous coefficients and heat conductivity are assumed to be proportional to h(v)𝜃α where h(v) is a non-degenerate smooth function satisfying some additional conditions. We prove the existence and uniqueness of the global-in-time classical solution to the Cauchy problem with general large initial data when |α| is sufficiently small and the coefficient of magnetic diffusion ν is suitably large. Moreover, it is shown that the global solution is asymptotically stable as time tends to infinity. [ABSTRACT FROM AUTHOR]

We use a modified method of characteristics to derive an explicit formula for the unique stable classical solution of a linear mixed problem for a general one-dimensional wave equation in the first quarter plane with time-varying characteristic second derivatives in the boundary condition. We find a criterion for the Hadamard well-posedness of this problem in the form of conditions on the right-hand side of the equation and the initial and boundary data ensuring the unique and stable global solvability of the problem in the set of classical solutions. The well-posedness criterion includes necessary and sufficient smoothness conditions on the initial data of the problem and conditions for the consistency of the boundary condition with the initial conditions and the equation itself. The data smoothness conditions ensure that the solution is twice continuously differentiable, while the consistency conditions are needed for the solution to be smooth across the critical characteristic of the equation. [ABSTRACT FROM AUTHOR]

We consider the Euler–Korteweg system with space periodic boundary conditions x ∈ T d . We prove a local in time existence result of classical solutions for irrotational velocity fields requiring natural minimal regularity assumptions on the initial data. [ABSTRACT FROM AUTHOR]

The Cauchy problem of the Cahn-Hilliard equation with inertial term in multi space dimension is considered. Based on detailed analysis of Green's function, using fixed-point theorem, we get the global existence in time of classical solution with large initial data. Furthermore, we get Lp decay rate of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we show the global existence of classical solutions to the incompressible elastodynamics equations with a damping mechanism on the stress tensor in dimension three for sufficiently small initial data on periodic boxes, that is, with periodic boundary conditions. The approach is based on a time-weighted energy estimate, under the assumptions that the initial deformation tensor is a small perturbation around an equilibrium state and the initial data have some symmetry. [ABSTRACT FROM AUTHOR]

This paper proves the existence of small‐amplitude global‐in‐time unique mild solutions to both the Landau equation including the Coulomb potential and the Boltzmann equation without angular cutoff. Since the well‐known works [45] and [3, 43] on the construction of classical solutions in smooth Sobolev spaces which in particular are regular in the spatial variables, it still remains an open problem to obtain global solutions in an Lx,v∞ framework, similar to that in [49], for the Boltzmann equation with the cutoff assumption in general bounded domains. One main difficulty arises from the interaction between the transport operator and the velocity‐diffusion‐type collision operator in the non‐cutoff Boltzmann and Landau equations; another major difficulty is the potential formation of singularities for solutions to the boundary value problem. In the present work we introduce a new function space with low regularity in the spatial variable to treat the problem in cases when the spatial domain is either a torus or a finite channel with boundary. For the latter case, either the inflow boundary condition or the specular reflection boundary condition is considered. An important property of the function space is that the LT∞Lv2 norm, in velocity and time, of the distribution function is in the Wiener algebra A(Ω) in the spatial variables. Besides the construction of global solutions in these function spaces, we additionally study the large‐time behavior of solutions for both hard and soft potentials, and we further justify the property of propagation of regularity of solutions in the spatial variables. © 2019 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

For parabolic Shilov equations with continuous coefficients, the problem of finding classical solutions that satisfy a modified initial condition with generalized data such as the Gelfand and Shilov distributions is considered. This condition arises in the approximate solution of parabolic problems inverse in time. It linearly combines the meaning of the solution at the initial and some intermediate points in time. The conditions for the correct solvability of this problem are clarified and the formula for its solution is found. Using the results obtained, the corresponding problems with impulse action were solved. [ABSTRACT FROM AUTHOR]

Sufficient conditions for the existence and uniqueness of a classical solution to a nonlocal problem for a system of Sobolev-type differential equations with integral condition are established. By introducing a new unknown function, we reduce the considered problem to an equivalent problem consisting of a nonlocal problem for the system of hyperbolic equations of second order with a functional parameter and an integral relation. We propose the algorithm for finding an approximate solution to the investigated problem and prove its convergence. [ABSTRACT FROM AUTHOR]

This paper concerns the global existence and large time behavior of classical, strong, and weak solutions to the two‐dimensional compressible micropolar equations with large initial data and vacuum. We assume that the shear and angular viscosity coefficients are positive constants and the bulk coefficient is λ=ρβ, where ρ is the density and β > 3/2. It is crucial to derive an upper bound of the density uniformly in the time such that all the classical, strong, and weak solutions converge to the equilibrium state as the time tends to infinity. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of global classical solutions to continuous coagulation–fragmentation equations with unbounded coefficients under the sole assumption that the coagulation rate is dominated by a power of the fragmentation rate, thus improving upon a number of recent results by not requiring any polynomial growth bound for either rate. This is achieved by proving a new result on the analyticity of the fragmentation semigroup and then using its regularizing properties to prove the local and then, under a stronger assumption, the global classical solvability of the coagulation–fragmentation equation considered as a semilinear perturbation of the linear fragmentation equation. Furthermore, we show that weak solutions of the coagulation–fragmentation equation, obtained by the weak compactness method, coincide with the classical local in time solutions provided the latter exist. [ABSTRACT FROM AUTHOR]

In this paper, we study the thermally radiative magnetohydrodynamic equations in 3D, which describe the dynamical behaviors of magnetized fluids that have nonignorable energy and momentum exchange with radiation under the nonlocal thermal equilibrium case. By using exquisite energy estimate, global existence and uniqueness of classical solutions to Cauchy problem in ℝ 3 or T 3 are established when initial data is a small perturbation of some given equilibrium. We can further prove that the rates of convergence of solution toward the equilibrium state are algebraic in ℝ 3 and exponential in T 3 under some additional conditions on initial data. The proof is based on the Fourier multiplier technique. [ABSTRACT FROM AUTHOR]

The initial boundary-value problem for systems of fourth-order partial differential equations with two independent variables is considered. By using a new unknown eigenfunction, the problem under consideration is reduced to an equivalent nonlocal problem for a system of second-order hyperbolic-type integro-differential equations with integral conditions. An algorithm for finding an approximate solution of the resulting equivalent problem is proposed, and its convergence is proved. Conditions for the existence of a unique classical solution of the initial boundary-value problem for systems of fourth-order differential equations are established in terms of the coefficients of the system and the boundary matrices. [ABSTRACT FROM AUTHOR]

In this article we study the large-time behavior of perturbative classical solutions to the Fokker-Planck-Boltzmann equation for non-cutoff hard potentials. When the initial data is a small pertubation of an equilibrium state, global existence and temporal decay estimates of classical solutions are established. [ABSTRACT FROM AUTHOR]

The goal of this paper is to show existence of short-time classical solutions to the so called Master Equation of first order Mean Field Games, which can be thought of as the limit of the corresponding master equation of a stochastic mean field game as the individual noises approach zero. Despite being the equation of an idealistic model, its study is justified as a way of understanding mean field games in which the individual players' randomness is negligible; in this sense it can be compared to the study of ideal fluids. We restrict ourselves to mean field games with smooth coefficients but do not impose any monotonicity conditions on the running and initial costs, and we do not require convexity of the Hamiltonian, thus extending the result of Gangbo and Swiech to a considerably broader class of Hamiltonians. [ABSTRACT FROM AUTHOR]

In this paper, we examine the solvability of a mixed problem with an integral condition for a third-order equation whose principal part contains the wave operator. The existence and uniqueness of a classical solution to this problem are proved by the Riemann method. [ABSTRACT FROM AUTHOR]

The work is devoted to study the existence and uniqueness of the classical solution of the inverse boundary value problem of determining the lowest coefficient in one fourth order equation. The original problem is reduced to an equivalent problem. The existence and uniqueness of the integral equation are proved by means of the contraction mappings principle, and we obtained that this solution is unique for a boundary value problem. Further, using these facts, we prove the existence and uniqueness of the classical solution for this problem. [ABSTRACT FROM AUTHOR]

The semi-geostrophic equations have attracted the attention of the physical and mathematical communities since the work of Hoskins in the 1970s owing to their ability to model the formation of fronts in rotation-dominated flows, and also to their connection with optimal transport theory. In this paper, we study an active scalar equation, whose activity is determined by way of a Neumann-to-Dirichlet map associated to a fully nonlinear second-order Neumann boundary value problem on the infinite strip R 2 × (0 , 1) , that models a semi-geostrophic flow in regime of constant potential vorticity. This system is an expression of an Eulerian semi-geostrophic flow in a coordinate system originally due to Hoskins, to which we shall refer as Hoskins' coordinates. We obtain results on the local-in-time existence and uniqueness of classical solutions of this active scalar equation in Hölder spaces. [ABSTRACT FROM AUTHOR]

In this paper, we prove a localised version of the bounded L 2 -curvature theorem of (Klainerman et al. Invent Math 202(1):91–216, 2015). More precisely, we consider initial data for the Einstein vacuum equations posed on a compact spacelike hypersurface Σ with boundary, and show that the time of existence of a classical solution depends only on an L 2 -bound on the Ricci curvature, an L 4 -bound on the second fundamental form of ∂ Σ ⊂ Σ , an H 1 -bound on the second fundamental form, and a lower bound on the volume radius at scale 1 of Σ . Our localisation is achieved by first proving a localised bounded L 2 -curvature theorem for small data posed on B(0, 1), and then using the scaling of the Einstein equations and a low regularity covering argument on Σ to reduce from large data on Σ to small data on B(0, 1). The proof uses the author's previous works and the bounded L 2 -curvature theorem as black boxes. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a blow-up criterion for classical solutions to the two-dimensional Prandtl equations under a monotonicity assumption in weighted Sobolev spaces. Our proof is based on nonlinear energy estimates inspired by Masmoudi and Wong [Commun. Pure Appl. Math. 68, 1683–1741 (2015)] and maximum principles for parabolic equations. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in proving the existence of a weak martingale solution of the stochastic nonlocal Cahn–Hilliard–Navier–Stokes system driven by a pure jump noise in both 2D and 3D bounded domains. Our goal is achieved by using the classical Faedo–Galerkin approximation, a compactness method and a version of the Skorokhod embedding theorem for nonmetric spaces. In the 2D case, we prove the pathwise uniqueness of the solution and use the Yamada–Watanabe classical result to derive the existence of a strong solution. • We introduce a stochastic nonlocal Cahn–Hilliard–Navier–Stokes model. • Present a Galerkin approximation. • Derive some a priori estimates. • Prove the existence of a martingale weak solution. • Pathwise uniqueness of the solution in 2D. [ABSTRACT FROM AUTHOR]

For a fourth-order generalized Mangeron equation with nonsmooth coefficients defined on a rectangular domain, we consider the Neumann problem with nonclassical conditions that do not require matching conditions. We justify the equivalence of these conditions to classical boundary conditions for the case in which the solution to the problem is sought in an isotropic Sobolev space. The problem is solved by reduction to a system of integral equations whose well-posed solvability is established based on the method of integral representations. The well-posed solvability of the Neumann problem for the generalized Mangeron equation is proved by the method of operator equations. [ABSTRACT FROM AUTHOR]

We consider the planar compressible magnetohydrodynamics (MHD) system for a viscous and heat-conducting ideal polytropic gas, when the viscosity, magnetic diffusion and heat conductivity depend on the specific volume v and the temperature 𝜃. For technical reasons, the viscosity coefficients, magnetic diffusion and heat conductivity are assumed to be proportional to h (v) 𝜃 α where h (v) is a non-degenerate and smooth function satisfying some natural conditions. We prove the existence and uniqueness of the global-in-time classical solution to the initial-boundary value problem when general large initial data are prescribed and the exponent | α | is sufficiently small. A similar result is also established for planar Hall-MHD equations. [ABSTRACT FROM AUTHOR]

The paper presents sufficient conditions for existence of an optimal control of solutions to a non-autonomous degenerate operator-differential evolution equation. We construct families of operators that solve this equation, as well as classical and strong solutions of the multipoint initial-final problem for the equation. We show that there exists a solution of an optimal control problem for a given operator-differential equation with a multipoint initial-final condition. The paper, in addition to the introduction and the bibliography, contains five sections. The first three parts contain information about the solvability of the multipoint initial-final problem for a non-autonomous equation. The fourth section presents the main result of the article; that is, a theorem on existence of optimal control of solutions to a multipoint initial-final problem. In the fifth part, the optimal control problem for the non-autonomous modified Chen – Gurtin model with the multipoint initial-final condition is investigated on the basis of the obtained abstract results. [ABSTRACT FROM AUTHOR]

Anderson–Witting model is a relaxational model equation of the relativistic Boltzmann equation, which sees a wide application in physics. In this paper, we study the existence of classical solutions and its asymptotic behavior when the solution starts sufficiently close to a global relativistic Maxwellian. [ABSTRACT FROM AUTHOR]

We study the global existence and regularity of classical solutions to the 2D incompressible magneto‐micropolar equations with partial dissipation. We establish the global regularity for one partial dissipation case. The proofs of our main results rely on anisotropic Sobolev type inequalities and suitable combination and cancellation of terms. [ABSTRACT FROM AUTHOR]

Abstract We study the Cauchy problem for the strong dissipative fifth-order KdV equations (0.1) ∂ t u + β 1 ∂ x 5 u + β 2 ∂ x 4 u = c 1 u ∂ x u + c 2 u 2 ∂ x u + b 1 ∂ x u ∂ x 2 u + b 2 u ∂ x 3 u , x ∈ R , t ∈ R + , u (0 , x) = u 0 (x) ∈ H s (R). We show that the Cauchy problem (0.1) is locally well-posed in H s (R) for any s ≥ 0. Moreover, as u 0 ∈ H 2 (R) , b 1 = 2 b 2 , c 2 = 0 and β 2 → 0 , we prove that the global solution to (0.1) converges to the global weak solution of the fifth-order KdV equations (0.2) ∂ t u + β 1 ∂ x 5 u = c 1 u ∂ x u + 2 b 2 ∂ x u ∂ x 2 u + b 2 u ∂ x 3 u. This global solution result is consistent with the result by Kenig and Pilod (2015) and Guo et al. (2013), who used short-time structure method. However, we only use the classical viscous disappearing method to get the global solution to (0.2). [ABSTRACT FROM AUTHOR]

We consider the fractional Korteweg--de Vries equation ut+uux-|D| αux = 0 in the range of 1 < α < 1, α ≠ 0. Using basic Fourier techniques in combination with the modified energy method, we extend the existence time of classical solutions with initial data of size ε from 1/ε to a time scale of 1/ε2. This analysis, which is carried out in Sobolev space HN(R), N ≥ 3, answers positively a question posed by Linares, Pilod, and Saut in [SIAM J. Math. Anal., 46 (2014), pp. 1505--1537]. [ABSTRACT FROM AUTHOR]

This article establishes the global existence and regularity for a system of the two-dimensional (2D) magnetohydrodynamic (MHD) equations with only directional hyper-resistivity. More precisely, the equation of b1 (the horizontal component of the magnetic field) involves only vertical hyperdiffusion (given by Λ22β b1⁠) while the equation of b2 (the vertical component) has only horizontal hyperdiffusion (given by Λ12β b2), where Λ1 and Λ2 are directional Fourier multiplier operators with the symbols being ξ1 and ξ2, respectively. We prove that, for β > 1, this system always possesses a unique global-in-time classical solution when the initial data is sufficiently smooth. The model concerned here is rooted in the MHD equations with only magnetic diffusion, which play a significant role in the study of magnetic reconnection and magnetic turbulence. In certain physical regimes and under suitable scaling, the magnetic diffusion becomes partial (given by part of the Laplacian operator). There have been considerable recent developments on the fundamental issue of whether classical solutions of these equations remain smooth for all time. The papers of Cao–Wu–Yuan [ 8 ] and of Jiu–Zhao [ 26 ] obtained the global regularity when the magnetic diffusion is given by the full fractional Laplacian (−Δ)β with β > 1⁠. The main result presented in this article requires only directional fractional diffusion and yet we prove the regularization in all directions. The proof makes use of a key observation on the structure of the nonlinearity in the MHD equations and technical tools on Fourier multiplier operators such as the Hörmander–Mikhlin multiplier theorem. The result presented here appears to be the sharpest for the 2D MHD equations with partial magnetic diffusion. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the global stability of homogeneous equilibria in Keller–Segel–Navier–Stokes equations in scaling-invariant spaces. We prove that for any given 0 < M < 1 + μ 1 with μ 1 being the first eigenvalue of Neumann Laplacian, the initial–boundary value problem of the Keller–Segel–Navier–Stokes system has a unique globally bounded classical solution provided that the initial datum is chosen sufficiently close to (M , M , 0) in the norm of L d / 2 (Ω) × W ˙ 1 , d (Ω) × L d (Ω) and satisfies a natural average mass condition. Our proof is based on the perturbation theory of semigroups and certain delicate exponential decay estimates for the linearized semigroup. Our result suggests a new observation that nontrivial classical solution for Keller–Segel–Navier–Stokes equation can be obtained globally starting from suitable initial data with arbitrarily large total mass provided that volume of the bounded domain is large, correspondingly. [ABSTRACT FROM AUTHOR]

This paper is devoted to the global in time existence of classical solutions to the d-Dimensional (dD) micropolar equations with fractional dissipation. Micropolar equations model a class of fluids with nonsymmetric stress tensor such as fluids consisting of particles suspended in a viscous medium. It remains unknown whether or not smooth solutions of the classical 3D micropolar equations can develop finite-time singularities. The purpose here is to explore the global regularity of solutions for dD micropolar equations under the smallest amount of dissipation. We establish the global regularity for two important fractional dissipation cases. Direct energy estimates are not sufficient to obtain the desired global a priori bounds in each case. To overcome the difficulties, we employ the Besov space techniques. [ABSTRACT FROM AUTHOR]

This paper is devoted to a partial differential equation approach to the problem of construction of Feller semigroups associated with one-dimensional diffusion processes with boundary conditions in theory of stochastic processes. In this paper we investigate the boundary-value problem for a one-dimensional linear parabolic equation of the second order (backward Kolmogorov equation) in curvilinear bounded domain with one of the variants of nonlocal Feller-Wentzell boundary condition. We restrict our attention to the case when the boundary condition has only one term and it is of the integral type. The classical solution of the last problem is obtained by the boundary integral equation method with the use of the fundamental solution of backward Kolmogorov equation and the associated parabolic potentials. This solution is used to construct the Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle leaves the boundary of the domain by jumps. [ABSTRACT FROM AUTHOR]

This paper is concerned with the global well-posedness of strong and classical solutions for the 3D nonhomogeneous incompressible micropolar equations with vacuum. We prove that the problem (1.1)–(1.5) has a unique global strong/classical solution (ρ , u , w) , provided μ 1 is sufficiently large, or ∥ ρ 0 ∥ L ∞ or ∥ ρ 0 1 / 2 u 0 ∥ L 2 2 + ∥ ρ 0 1 / 2 w 0 ∥ L 2 2 is small enough. [ABSTRACT FROM AUTHOR]

In the paper we consider a stationary diffusion problem described by the Poisson equation. The problem is considered in a model domain, chosen as a half disk. Classical Dirichlet boundary conditions are set on the arc of the circle. New nonlocal boundary conditions are set on the bottom base. The first condition means the equality of flows through opposite radii, and the second condition is the proportionality of distribution densities on these radii with a variable coefficient of proportionality. Uniqueness and existence of the classical solution to the problem are proved. An inverse problem for the solution to the Poisson equation and its right-hand part depending only on an angular variable are considered. As an additional condition we use the boundary overdetermination. Inverse problems to the Dirichlet and Neumann problems, and to problems with nonlocal conditions of the equality of flows through the opposite radii are considered. The well-posedness of the formulated inverse problems is proved. [ABSTRACT FROM AUTHOR]