We prove existence of weak solutions to nonlinear parabolic systems with p-Laplacians terms in the principal part. Next, in the case of diagonal systems an ${L}_{\infty}$-estimate for weak solutions is shown under additional restrictive growth conditions. Finally, ${L}_{\infty}$-estimates for weakly nondiagonal systems (where nondiagonal elements are absorbed by diagonal ones) are proved. The ${L}_{\infty}$-estimates are obtained by the Di Benedetto methods.

Existence of weak solutions and an ${L}_{\infty}$-estimate are shown for nonlinear nondegenerate parabolic systems with linear growth conditions with respect to the gradient. The ${L}_{\infty}$-estimate is proved for equations with coefficients continuous with respect to x and t in the general main part, and for diagonal systems with coefficients satisfying the Carathéodory condition.

For a class of semi-abstract evolution equations for sections on vector bundles on a three-dimensional compact manifold we prove that for initial values with certain symmetries strong solutions exist for all times. In case these solutions become small after some time, strong solutions exist also for small perturbations of these initial values. Many systems from fluid mechanics are included in this class.

The existence for the Cauchy-Neumann problem for the Stokes system in a bounded domain $\Omega \subset {\mathbb{R}}^{3}$ is proved in a class such that the velocity belongs to ${W}_{r}^{2,1}\left(\Omega \times (0,T)\right)$, where r > 3. The proof is divided into three steps. First, the existence of solutions is proved in a half-space for vanishing initial data by applying the Marcinkiewicz multiplier theorem. Next, we prove the existence of weak solutions in a bounded domain and then we regularize them. Finally, the problem with nonvanishing initial data is considered....

The local-in-time existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion is proved. We show the existence of solutions with lowest possible regularity for this problem such that $u\in {W}_{r}^{2,1}\left({\tilde{\Omega}}^{T}\right)$ with r>3. The existence is proved by the method of successive approximations where the solvability of the Cauchy-Neumann problem for the Stokes system is applied. We have to underline that in the ${L}_{p}$-approach the Lagrangian coordinates must be used. We are looking...

Existence and uniqueness of local solutions for the initial-boundary value problem for the equations of an ideal relativistic fluid are proved. Both barotropic and nonbarotropic motions are considered. Existence for the linearized problem is shown by transforming the equations to a symmetric system and showing the existence of weak solutions; next, the appropriate regularity is obtained by applying Friedrich's mollifiers technique. Finally, existence for the nonlinear problem is proved by the method...

The local existence and the uniqueness of solutions for equations describing the motion of viscous compressible heat-conducting fluids in a domain bounded by a free surface is proved. First, we prove the existence of solutions of some auxiliary problems by the Galerkin method and by regularization techniques. Next, we use the method of successive approximations to prove the local existence for the main problem.

Global-in-time existence of solutions for equations of viscous compressible barotropic fluid in a bounded domain Ω ⊂ ${\mathbb{R}}^{3}$ with the boundary slip condition is proved. The solution is close to an equilibrium solution. The proof is based on the energy method. Moreover, in the ${L}_{2}$-approach the result is sharp (the regularity of the solution cannot be decreased) because the velocity belongs to ${H}^{2+\alpha ,1+\alpha /2}\left(\Omega \times {\mathbb{R}}_{+}\right)$ and the density belongs to ${H}^{1+\alpha ,1/2+\alpha /2}\left(\Omega \times {\mathbb{R}}_{+}\right)$, α ∈ (1/2,1).

CONTENTS1. Introduction.......................................................................52. Notation and auxiliary results............................................93. Statement of the problem (1.1)-(1.3)..............................204. The problem (3.14).........................................................225. Auxiliary results in ${D}_{\vartheta}$...............................................346. Existence of solutions of (3.14) in ${H}_{\mu}^{k}\left({D}_{\vartheta}\right)$............417. Green function................................................................528....

We consider the motion of a viscous compressible barotropic fluid in ${\mathbb{R}}^{3}$ bounded by a free surface which is under constant exterior pressure. For a given initial density, initial domain and initial velocity we prove the existence of local-in-time highly regular solutions. Next assuming that the initial density is sufficiently close to a constant, the initial pressure is sufficiently close to the external pressure, the initial velocity is sufficiently small and the external force vanishes we prove...

Existence and uniqueness of global regular special solutions to Navier-Stokes equations with boundary slip conditions in axially symmetric domains is proved. The proof of global existence relies on the global existence results for axially symmetric solutions which were obtained by Ladyzhenskaya and Yudovich-Ukhovskiĭ in 1968 who employed the problem for vorticity. In this paper the equations for vorticity also play a crucial role. Moreover, the boundary slip conditions imply appropriate boundary...

The existence of solutions to the Dirichlet problem for the compressible linearized Navier-Stokes system is proved in a class such that the velocity vector belongs to ${W}_{r}^{2,1}$ with r > 3. The proof is done in two steps. First the existence for local problems with constant coefficients is proved by applying the Fourier transform. Next by applying the regularizer technique the existence in a bounded domain is shown.

The local existence of solutions for the compressible Navier-Stokes equations with the Dirichlet boundary conditions in the ${L}_{p}$-framework is proved. Next an almost-global-in-time existence of small solutions is shown. The considerations are made in Lagrangian coordinates. The result is sharp in the ${L}_{p}$-approach, because the velocity belongs to ${W}_{r}^{2,1}$ with r > 3.

Using the Il'in integral representation of functions, imbedding theorems for weighted anisotropic Sobolev spaces in 𝔼ⁿ are proved. By the weight we assume a power function of the distance from an (n-2)-dimensional subspace passing through the domain considered.

The aim of this paper is to prove the existence of solutions to the Poisson equation in weighted Sobolev spaces, where the weight is the distance to some distinguished axis, raised to a negative power. Therefore we are looking for solutions which vanish sufficiently fast near the axis. Such a result is useful in the proof of the existence of global regular solutions to the Navier-Stokes equations which are close to axially symmetric solutions.

The existence of solutions to an initial-boundary value problem to the heat equation in a bounded domain in ℝ³ is proved. The domain contains an axis and the existence is proved in weighted anisotropic Sobolev spaces with weight equal to a negative power of the distance to the axis. Therefore we prove the existence of solutions which vanish sufficiently fast when approaching the axis. We restrict our considerations to the Dirichlet problem, but the Neumann and the third boundary value problems can...

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