We investigate the existence of weak solutions and their smoothness properties for a generalized Stokes problem. The generalization is twofold: the Laplace operator is replaced by a general second order linear elliptic operator in divergence form and the “pressure” gradient $\nabla p$ is replaced by a linear operator of first order.

Existence of solutions for equations of the nonstationary Stokes system in a bounded domain Ω ⊂ ℝ³ is proved in a class such that velocity belongs to $W_p^{2,1}\left(\Omega \times (0,T)\right)$, and pressure belongs to $W_p^{1,0}\left(\Omega \times (0,T)\right)$ for p > 3. The proof is divided into three steps. First, the existence of solutions with vanishing initial data is proved in a half-space 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...

For a class of random dynamical systems which describe dissipative nonlinear PDEs perturbed by a bounded random kick-force, I propose a “direct proof” of the uniqueness of the stationary measure and exponential convergence of solutions to this measure, by showing that the transfer-operator, acting in the space of probability measures given the Kantorovich metric, defines a contraction of this space.

In this paper the fluid-structure interaction problem is studied on a simplified model of the human vocal fold. The problem is mathematically described and the arbitrary Lagrangian-Eulerian method is applied in order to treat the time dependent computational domain. The viscous incompressible fluid flow and linear elasticity models are considered. The fluid flow and the motion of elastic body is approximated with the aid of finite element method. An attention is paid to the applied stabilization...

Global-in-time existence of solutions for equations of viscous compressible barotropic fluid in a bounded domain Ω ⊂ ${ℝ}^3$ 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 $H^{2+\alpha ,1+\alpha /2}\left(\Omega \times {ℝ}_{+}\right)$ and the density belongs to $H^{1+\alpha ,1/2+\alpha /2}\left(\Omega \times {ℝ}_{+}\right)$, α ∈ (1/2,1).

Global existence of regular solutions to the Navier-Stokes equations for velocity and pressure coupled with the heat convection equation for temperature in a cylindrical pipe is shown. We assume the slip boundary conditions for velocity and the Neumann condition for temperature. First we prove long time existence of regular solutions in [kT,(k+1)T]. Having T sufficiently large and imposing some decay estimates on ${\left|\right|f\left(t\right)\left|\right|}_{L₂\left(\Omega \right)}$, $\left|\right|f_{,x₃}\left(t\right){\left|\right|}_{L₂\left(\Omega \right)}$ we continue the local solutions step by step up to a global one.

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...

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.

We consider the flow of a non-homogeneous viscous incompressible fluid which is known at an initial time. Our purpose is to prove that, when $\Omega$ is smooth enough, there exists a local strong regular solution (which is global for small regular data).