Existence Theorem For Weak Solutions Research Articles

Existence, blow-up, and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions

This paper is devoted to the study of a system of nonlinear equations with nonlinear boundary conditions. First, on the basis of the Faedo–Galerkin method and standard arguments of density corresponding to the regularity of initial conditions, we establish two local existence theorems of weak solutions. Next, we prove that any weak solutions with negative initial energy will blow up in finite time. Finally, the exponential decay property of the global solution via the construction of a suitable Lyapunov functional is presented. Copyright © 2013 John Wiley & Sons, Ltd.

Weak Solutions for a Second Order Dirichlet Boundary Value Problem on Time Scale

We adopt the Leray-Schauder degree theory and critical point theory to consider a second order Dirichlet boundary value problem on time scales and obtain some existence theorems of weak solutions for the previous problem.

Existence and uniqueness theorem on weak solutions to the parabolic–elliptic Keller–Segel system

In Rn (n⩾3), we first define a notion of weak solutions to the Keller–Segel system of parabolic–elliptic type in the scaling invariant class Ls(0,T;Lr(Rn)) for 2/s+n/r=2 with n/2<r<n. Any condition on derivatives of solutions is not required at all. The local existence theorem of weak solutions is established for every initial data in Ln/2(Rn). We prove also their uniqueness. As for the marginal case when r=n/2, we show that if n⩾4, then the class C([0,T);Ln/2(Rn)) enables us to obtain the only weak solution.

On the Feedback Stabilization of Boussinesq Equations

On the Feedback Stabilization of Boussinesq EquationsThis paper provides feedback stabilization results, preserving the invariance of a given convex set, for the Boussinesq system, in 2D and 3D. The proofs use an existence theorem for weak solutions.

The existence of solutions for a class of semilinear elliptic systems

In this paper, some new existence theorems of weak solutions for a class of semilinear elliptic systems are obtained by means of the local linking theorem and the saddle point theorem.

Differential mixed variational inequalities in finite dimensional spaces

In this paper, we introduce and study a class of differential mixed variational inequalities in finite dimensional Euclidean spaces. Under various conditions, we obtain linear growth and bounded linear growth of the solution set for the mixed variational inequalities. Moreover, we present some conclusions which enrich the literature on the mixed variational inequalities and generalize the corresponding results of [4]. In particular we prove existence theorems for weak solutions of a differential mixed variational inequality in the weak sense of Carathéodory by using a result on differential inclusions involving an upper semicontinuous set-valued map with closed convex values. Also by employing the results from differential inclusions we establish a convergence result on Euler time-dependent procedure for solving initial-value differential mixed variational inequalities.

HOPF SOLUTIONS TO A FLUID-ELASTIC INTERACTION MODEL

In this paper, we give a global existence theorem of weak solutions to model equations governing interaction fluid structure in a two-dimensional layer, cf. Refs. 8 and 14. To our knowledge this is the first existence theorem of global in time solutions for such model. The interest of our result is double because, first, we change the original initial value problem by deleting one initial condition, second, we construct a solution through the classical Galerkin method for which several computing codes have been constructed.

On weak solutions for $p$-Laplacian equations with weights

Here we summarize the results obtained in the forthcoming paper \\cite{patraf}, in which we prove existence and non--existence theorems of weak solutions of quasilinear singular elliptic equations with weights. We also establish regularity and qualitative properties of the solutions.

An existence theorem for weak solutions for a class of elliptic partial differential systems in general Orlicz–Sobolev spaces

I prove the existence of a weak solution for the Dirichlet problem of a class of elliptic partial differential systems − ∂ A α i ∂ x α ( x , u ( x ) , D u ( x ) ) + B i ( x , u ( x ) , D u ( x ) ) = 0 in general Orlicz–Sobolev spaces W 0 1 L M ( Ω , R N ) , where i = 1 , … , N , α = 1 , … , n , u : Ω → R N is a vector-valued function, and the summation convention is used throughout with i , j running from 1 to N and α , β running from 1 to n .

Existence of weak solutions for a class of nonlinear diffusion problems

The paper concerns the existence theorem of weak solutions for the initial-boundary value problem of a nonlinear diffusion equation with convection. The equation may be regarded as a generalized non-Newtonian polytropic filtration equation. By doing the necessary B V estimate and other estimates for approximating solutions, we establish the existence theorem.

A mathematical model of the motion of a nonlinear viscous fluid with the condition of slip on the boundary

INTRODUCTION Many mathematical papers are dedicated to various questions of resolvability of the Navier–Stokes equations. The number of papers devoted to mathematical models of non-Newtonian fluids, whose rheological correlations obey the principle of objectivity of the materials behavior, is essentially less (see [1]). Moreover, several authors (see, e. g., [2]) note the importance of the investigation of problems with boundary conditions which differ from the classic sticking conditions. This paper is devoted to the proof of the existence theorem for weak solutions to the boundary value problem which describes the stationary motion of a nonlinear viscous fluid with the condition of slip on the boundary. Note that various mathematical models of nonlinear viscous fluids were investigated earlier by V. G. Litvinov; the results obtained in this paper essentially improve some results of monograph [3]. Assume that the fluid under consideration entirely fills a container with hard walls represented by a bounded domainΩ ⊂ Rn, n ∈ {2, 3}, with the Lipschitz boundary S. It is well-known that the stationary motion of any fluid obeys the motion equation in the Cauchy form

An existence theorem for weak solutions for a class of elliptic partial differential systems in Orlicz spaces

We prove the existence of a weak solution of the Dirichlet problem for a class of elliptic partial differential systems in separable Orlicz–Sobolev spaces.

On Solvability of Some Initial-Boundary Problems for Mathematical Models of the Motion of Nonlinearly Viscous and Viscoelastic Fluids

In the paper, the settings of initial-boundary and initial value problems arising in a number of models of movement of nonlinearly viscous or viscoelastic incompressible fluid are considered, and existence theorems for these problems are presented. In particular, the settings of initial-boundary value problems appearing in the regularized model of the movement of viscoelastic fluid with Jeffris constitutive relation are described. The theorems for the existence of weak and strong solutions for these problems in bounded domains are given. The initial value problem for a nonlinearly viscous fluid on the whole space is considered. The estimates on the right-hand side and initial conditions under which there exist local and global solutions of this problem are presented. The modification of Litvinov's model for laminar and turbulent flows with a memory is described. The existence theorem for weak solutions of initial-boundary value problem appearing in this model is given.

One-sided multiple resonance for quasilinear elliptic partial differential equations

In this article, we apply the Galerkin approximation method to obtain an existence theorem of weak solutions for 2mth order quasilinear elliptic partial differential resonance equations −Q(u)+f(x,u) =G on a bounded open connected subset Ω of in which the nonlinearity f(x,u) has no growth restriction in u in one of the directions (u→ ∞ and u→ −∞ ) and belongs to o(|u| p−1 ) (p> N/m) in the other, and G∈ [ m,p (Ω )]* may satisfy a Landesman–Lazer condition.

An Existence Theorem for Weak Solutions of Stochastic Differential Equations with Discontinuous Right-Hand Sides and with Reflection at the Boundary

An Existence Theorem for Weak Solutions of Stochastic Differential Equations with Discontinuous Right-Hand Sides and with Reflection at the Boundary

Solvability of a quasilinear elliptic resonance equation on the N-torus

In this article, we apply Galerkin-type techniques and Brouwer's fixed point theorem to obtain existence theorems of weak solutions for a quasilinear elliptic resonance equation Q(u)+f(x,u)=G on the N-torus in which the Landesman-Lazer condition for G may be excluded, the null space of Q may not be only constant functions, and the nonlinearity ƒ may grow superlinearly in u as|u|→∞.

Solvability of the Boundary-Value Problem for a Mathematical Model of Steady-State Flows of Nonlinear-Viscous Fluids

We establish an existence theorem for weak solutions of the boundary-value problem for steady-state equations describing both laminar and turbulent flows of nonlinear-viscous fluids.

On the Existence Theorem for Weak Solutions of Increased Smoothness of the Multidimensional Navier7ndash;Stokes Equations

Hopf [1] defined the notion of a weak solution of the 3D Navier–Stokes equations and proved the existence of such solutions. Later, it was proved that, under certain conditions, the weak solutions (the 3D viscous fluid flow velocities) have second derivatives 5/4th-power integrable over the space-time cylinder [2–4]. The present paper deals with some development of these results. We consider viscous fluid flows in an n-dimensional space with velocities 2π-periodic in each of the space variables and show that if the initial velocity is square integrable over the period cube, its derivatives are integrable, and the mass forces satisfy similar integrability conditions over the obvious space-time parallelepiped, then there exist weak solutions whose second derivatives are (4/3 − e)th-power integrable over the space-time parallelepiped for any e > 0.

On the Solvability of a Quasilinear Elliptic Resonance Problem near Its First Eigenvalue

In this article, we apply Galerkin-type techniques and Brouwer's fixed point theorem to obtain existence theorems of weak solutions for quasilinear elliptic partial differential resonance equations in which the Landesman–Lazer condition may be excluded.

Analysis and Approximation of Conservation Laws with Source Terms

We consider a conservation law of the form (CL)I>ut + f(u)x = ax , where $a(\cdot)$ is a bounded piecewise smooth source term and f an even convex function. We first characterize the solution to the Riemann problem through a new Lax-type formula. Then we prove that for $a(\cdot)$ fixed, the semigroup associated with (CL)is an L1 contraction, and we obtain an existence theorem for weak solutions to (CL). We conclude by constructing Godunov-type difference schemes and proving that these schemes are $L^\infty$ stable and have stable steady solutions similar in structure to those of (CL). Some numerical tests are reported.