Existence of weak solutions for nonlinear systems involving several -Laplacian operators.
Existence of weak solutions for steady flows of electrorheological fluid with Navier-slip type boundary conditions
We prove the existence of weak solutions for steady flows of electrorheological fluids with homogeneous Navier-slip type boundary conditions provided . To prove this, we show Poincaré- and Korn-type inequalities, and then construct Lipschitz truncation functions preserving the zero normal component in variable exponent Sobolev spaces.
Existence of weak solutions to a class of degenerate semiconductor equations modeling avalanche generation.
Existence of weak solutions to doubly degenerate diffusion equations
We prove existence of weak solutions to doubly degenerate diffusion equations by Faedo-Galerkin approximation for general domains and general nonlinearities. More precisely, we discuss the equation in an abstract setting, which allows to choose function spaces corresponding to bounded or unbounded domains with Dirichlet or Neumann boundary conditions. The function can be an inhomogeneity or a nonlinearity involving terms of the form or . In the appendix, an introduction to weak differentiability...
Finite Difference Schemes on Nonuniform Meshes for Parabolic Problems With Generalized Solutions
Finite element approximation of kinetic dilute polymer models with microscopic cut-off
We construct a Galerkin finite element method for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ ,d= 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation....
Finite element approximation of kinetic dilute polymer models with microscopic cut-off
We construct a Galerkin finite element method for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ , d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation....
Finite-dimensional Pullback Attractors for Non-autonomous Newton-Boussinesq Equations in Some Two-dimensional Unbounded Domains
We study the existence and long-time behavior of weak solutions to Newton-Boussinesq equations in two-dimensional domains satisfying the Poincaré inequality. We prove the existence of a unique minimal finite-dimensional pullback -attractor for the process associated to the problem with respect to a large class of non-autonomous forcing terms.
Generalized n-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range
Let n ≥ 2 and let Ω ⊂ ℝn be an open set. We prove the boundedness of weak solutions to the problem where ϕ is a Young function such that the space W 01 L Φ(Ω) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h ∈ L Φ(Ω) is a non-trivial continuous function and µ ≥ 0 is a small parameter. We consider two classical cases: the case of Ω being an open bounded set and the case of Ω =...
Generalized solutions for nonlinear elliptic equations.
Generalized solutions to the gKdV equation.
Global Attractors for a Class of Semilinear Degenerate Parabolic Equations on
We prove the existence of global attractors for the following semilinear degenerate parabolic equation on : ∂u/∂t - div(σ(x)∇ u) + λu + f(x,u) = g(x), under a new condition concerning the variable nonnegative diffusivity σ(·) and for an arbitrary polynomial growth order of the nonlinearity f. To overcome some difficulties caused by the lack of compactness of the embeddings, these results are proved by combining the tail estimates method and the asymptotic a priori estimate method.
Global existence and extinction of weak solutions to a class of semiconductor equations with fast diffusion terms.
Global weak solutions of the Navier-Stokes equations with nonhomogeneous boundary data and divergence
Gradient estimates for the p(x)-Laplacian equation in
Under some assumptions on the function p(x), we obtain global gradient estimates for weak solutions of the p(x)-Laplacian type equation in .
Gradient estimates for weak solutions of -harmonic equations.
Gradient estimation of a -harmonic map.
Harnack inequality for nonlinear weighted equations.
Incompressible Maxwell-Boussinesq approximation: Existence, uniqueness and shape sensitivity
Integrability for solutions to quasilinear elliptic systems
In this paper we prove an estimate for the measure of superlevel sets of weak solutions to quasilinear elliptic systems in divergence form. In some special cases, such an estimate allows us to improve on the integrability of the solution.