On generalized vector equilibrium problems

In this paper, we study the generalized vector equilibrium problems in real Hausdorff topological vector space settings. The concepts of weak solutions and strong solutions are introduced. Several new results of existence for the weak solutions and strong solutions of generalized vector equilibrium problems are derived. The new results extend and modify various existence theorems for similar problems.     

Quasi-hydrostatic Primitive Equations for Ocean Global Circulation Models

Global existence of weak and strong solutions to the quasi-hydrostatic primitive equations is studied in this paper. This model, that derives from the full non-hydrostatic model for geophysical fluid dynamics in the zero-limit of the aspect ratio, is more realistic than the classical hydrostatic model, since the traditional approximation that consists in neglecting a part of the Coriolis force is relaxed. After justifying the derivation of the model, the authors provide a rigorous proof of global existence of weak solutions, and well-posedness for strong solutions in dimension three.     

Time dependent linear transport I. Existence,uniqueness, and continuous dependence

This paper concerns the time dependent linear transport equation posed in a multidimensional rectangular parallelepiped with partially reflecting walls. We consider the continuous transport equation and the discrete ordinate equations simultaneously. Our boundary condition, partial specular reflection, includes both vacuum and reflecting boundaries as special cases. We define strong and weak solutions of the problem, strong solutions being solutions in the ordinary sense and weak solutions being distributions, and show that a weak solution is a strong solution if it has space and time derivatives almost everywhere. For weak solutions we establish existence, uniqueness, and continuous dependence upon the initial data and the other functions which define the problem.     

Weak and strong attractors for the 3D Navier-Stokes system

We study in this paper the asymptotic behaviour of the weak solutions of the three-dimensional Navier-Stokes equations. On the one hand, using the weak topology of the usual phase space H (of square integrable divergence free functions) we prove the existence of a weak attractor in both autonomous and nonautonomous cases. On the other, we obtain a conditional result about the existence of the strong attractor, which is valid under an unproved hypothesis. Also, with this hypothesis we obtain continuous weak solutions with respect to the strong topology of H.     

On NAVIER-STOKES and KELVIN-VOIGT Equations in Three Dimensions in Interpolation Spaces

We show for the three-dimensional initial-boundary value problem of the NAVIER -STOKES and KELVIN -VOIGT equation over bounded domains and for the corresponding stationary problem the existence of weak solutions in intermediate spaces between some of the usual spaces for weak solutions of the NAVIER -STOKES equations and spaces for the corresponding strong solutions. The proofs yield estimates of the solutions in terms of the data which are supposed to be in appropriate intermediate spaces. The basic ingredients of the proof are well-known results for weak and strong solutions and some nonlinear interpolation arguments.     

Strong solutions and trajectory attractors to the thin-film equation with absorption

We prove local and global in time existence of non-negative weak solutions to the thin-film equation with absorption and obtain sufficient conditions for extra regularity of these solutions. Moreover, for the class of global strong solutions, we show existence of a trajectory attractor.     

Generalized Vector Equilibrium Problems with Trifunctions

In this paper, we study the existence of strong and weak solutions of the generalized vector equilibrium problems for trifunctions. Two special classes of vector-valued trifunctions are introdcued, called the classes of (SPM) and (GPM), respectively. Some existence results for strong solutions associated to functions of these classes are given.     

Generalized Vector Equilibrium Problems with Trifunctions

In this paper, we study the existence of strong and weak solutions of the generalized vector equilibrium problems for trifunctions. Two special classes of vector-valued trifunctions are introdcued, called the classes of (SPM) and (GPM), respectively. Some existence results for strong solutions associated to functions of these classes are given.     

Global weak solutions for a periodic two-component?μ-Hunter–Saxton system

This paper is concerned with global existence of weak solutions for a periodic two-component?μ-Hunter–Saxton system. We first derive global existence for strong solutions to the system with smooth approximate initial data. Then, we show that the limit of approximate solutions is a global weak solution of the two-component?μ-Hunter–Saxton system.     

MULTIVALUED STOCHASTIC MCKEAN-VLASOV EQUATION

In this work, under global Lipschitz conditions, we prove the existence and uniqueness of strong solutions for multivalued stochastic McK ean-Vlasov equation. Moreover, under continuous and linear growth assumptions, we also obtain the existence of weak solutions.     

Multivalued stochastic McKean-Vlasov equation

In this work, under global Lipschitz conditions, we prove the existence and uniqueness of strong solutions for multivalued stochastic McKean-Vlasov equation. Moreover, under continuous and linear growth assumptions, we also obtain the existence of weak solutions.     

On properties of strong solutions of stochastic equations with respect to semimartingales

The paper is devoted to strong solutions of stochastic integral equations with respect to semimartingales. We study existence (for non-Lipschitz coefficients) and asymptotic behaviour of strong solutions and obtain also a number of results on weak and square mean convergence of these solutions.     

Optimal control problems for the reaction–diffusion–convection equation with variable coefficients

The solvability of optimal control problems is proved on both weak and strong solutions of a boundary value problem for the nonlinear reaction–diffusion–convection equation with variable coefficients. In the second case, the requirements for smoothness of the multiplicative control are reduced. The study of extremal problems is based on the proof of the solvability of the corresponding boundary value problems and on the qualitative analysis of their solutions properties. The large data existence results for weak solutions, the maximum principle as well as the local existence and uniqueness of a strong solution are established. Moreover, an optimal feedback control problem is considered. Using methods of the theory of topological degree for set-valued perturbations (with aspheric image sets) of generalized monotone operators, we obtain sufficient conditions for the solvability of this problem in the class of weak solutions.     

Properties of solutions of stochastic differential equations with standard and fractional Brownian motions

We show that conditions ensuring the existence of strong and weak solutions of stochastic differential equations with standard and fractional Brownian motions guarantee the continuous dependence of these solutions on the initial conditions and right-hand sides. We prove a theorem on the uniform continuity of conditional expectations of strong solutions.     

On F-Implicit Generalized Vector Variational Inequalities

In this paper, we study the F-implicit generalized (weak) case for vector variational inequalities in real topological vector spaces. Both weak and strong solutions are considered. These two sets of solutions coincide whenever the mapping T is single-valued, but not set-valued. We use the Ferro minimax theorem to discuss the existence of strong solutions for F-implicit generalized vector variational inequalities.     

Lq‐estimates for the stationary Oseen equations on the exterior of a rotating obstacle

We study the Dirichlet problem for the stationary Oseen equations around a rotating body in an exterior domain. Our main results are the existence and uniqueness of weak and very weak solutions satisfying appropriate L^q‐estimates. The uniqueness of very weak solutions is shown by the method of cut‐off functions with an anisotropic decay. Then our existence result for very weak solutions is deduced by a duality argument from the existence and estimates of strong solutions. From this and interior regularity of very weak solutions, we finally establish the complete D^1,r‐result for weak solutions of the Oseen equations around a rotating body in an exterior domain, where 4/3<r <4. Here, D^1,r is the homogeneous Sobolev space.     

Weak–strong uniqueness for a class of generalized dissipative weak solutions for non-homogeneous,non-Newtonian and incompressible fluids

We consider a system of PDEs describing a non-homogeneous, non-Newtonian and incompressible fluid in a space-periodic domain. We define the generalized dissipative weak solutions as well as give a proof of its existence, when the boundary data is provided. The main result of this work is the weak–strong uniqueness of the defined solutions.     

Existence of weak and strong solutions of an integrodifferential equation in Banach spaces

A class of evolution integrodifferential equation has been studied over the analytic semigroup of operators in a Banach space. Further the existence of weak and strong solutions in Banach space have been proved and extended to a maximum interval of existence.     

Solutions to the Landau–Lifshitz system with nonhomogenous Neumann boundary conditions arising from surface anisotropy and super-exchange interactions in a ferromagnetic media

We study the solutions to the Landau–Lifshitz system in a bilayered ferromagnetic body when super-exchange and surface anisotropy interactions are present at the interface between the layers. We prove the existence of weak solutions in infinite time and strong solutions in finite time.     

Optimal control of the bidomain system (II): uniqueness and regularity theorems for weak solutions

Motivated by the study of related optimal control problems, weak and strong solution concepts for the bidomain system together with two-variable ionic models are analyzed. A key ingredient for the analysis is the bidomain bilinear form. Global existence of weak and strong solutions as well as stability and uniqueness theorems is proven.