Piotr Bogusław Mucha

On a new approach to the issue of existence and regularity for the steady compressible Navier-Stokes equations

We prove the existence of weak solutions to the steady compressible Navier–Stokes equations in the isentropic regime. We consider the two-dimensional problem with the slip boundary conditions. The proof is based on a new idea of the construction of approximative solutions which guarantees that the sought density belongs to the L∞ (up to the boundary). Our approach improves the known methods, reducing the number of technical tricks.

On the inviscid limit of the Navier–Stokes equations for flows with large flux

We investigate the inviscid limit of solutions of the evolutionary and stationary Navier–Stokes equations in two-dimensional bounded domains. The system is considered with the slip boundary conditions admitting flows across the boundary. Under a geometrical restriction on the shape of the domain, we prove the L∞-bound on the vorticity of the velocity for any large boundary data. This estimate enables us to prove the existence of solutions to the Navier–Stokes equations. Moreover the bound is independent of the viscosity and guarantees a strong convergence to the Eulerian limit. The result for the nonsteady case is global in time.

Weak solutions to equations of steady compressible heat conducting fluids

We consider the steady compressible Navier–Stokes–Fourier system in a bounded three-dimensional domain. We prove the existence of a solution for arbitrarily large data under the assumption that the pressure p(ϱ, θ) ~ ϱθ + ϱγ for assuming either the slip or no-slip boundary condition for the velocity and the Newton boundary condition for the temperature. The regularity of solutions is determined by the basic energy estimates, constructed for the system.

On a Lp-estimate for the linearized compressible Navier–Stokes equations with the Dirichlet boundary conditions☆

Abstract In this paper a special L p -estimate for the linearized compressible Navier–Stokes in the Lagrangian coordinates for the Dirichlet boundary conditions is obtained. The constant in the estimate does not depend on the length of time interval [0, T ]. The result is essential to obtain an existence for regular solutions for the nonlinear problem with the lowest class of regularity in L p -spaces.

On Navier–Stokes Equations with Slip Boundary Conditions in an Infinite Pipe

The paper examines steady Navier–Stokes equations in a two-dimensional infinite pipe with slip boundary conditions. At both inlet and outlet, the velocity of flow is assumed to be constant. The main results show the existence of weak and regular solutions with no restrictions of smallness of the flux vector, also simply connectedness of the domain is not required.

The Cauchy problem for the compressible Navier-Stokes equations in the L p -framework

The global in time existence of regular solutions to the compressible Navier-Stokes equations in the whole space is obtained. The solutions are close to nontrivial equilibrium solutions. Moreover, the result is sharp in the Lp-framework, the velocity of the fluid belongs to Wr2,1 and the initial density to W1r with r > 3.

Chemically reacting mixtures in terms of degenerated parabolic setting

The paper analyzes basic mathematical questions for a model of chemically reacting mixtures. We derive a model of several (finite) component compressible gas taking rigorously into account the thermodynamical regime. Mathematical description of the model leads to a degenerate parabolic equation with hyperbolic deviation. The thermodynamics implies that the diffusion terms are non-symmetric, not positively defined, and cross-diffusion effects must be strongly marked. The mathematical goal is to establish the existence of weak solutions globally in time for arbitrary number of reacting species. A key point is an entropy-like estimate showing possible renormalization of the system.

A NEW LOOK AT EQUILIBRIA IN STEFAN-TYPE PROBLEMS IN THE PLANE

We study steady states of Stefan-type problems in the plane with the Gibbs–Thomson correction involving a general anisotropic energy density function. By a local analysis we prove the global result showing that the solution is the Wulff shape. The key element is a stability result which enables us to approximate singular models by regular ones.

On the Existence of Traveling Waves in the 3D Boussinesq System

We extend earlier work on traveling waves in premixed flames in a gravitationally stratified medium, subject to the Boussinesq approximation. For three-dimensional channels not aligned with the gravity direction and under the Dirichlet boundary conditions in the fluid velocity, it is shown that a non-planar traveling wave, corresponding to a non-zero reaction, exists, under an explicit condition relating the geometry of the crossection of the channel to the magnitude of the Prandtl and Rayleigh numbers, or when the advection term in the flow equations is neglected.

A caricature of a singular curvature flow in the plane

We study a singular parabolic equation of the total variation type in one dimension. The problem is a simplification of the singular curvature flow. We show the existence and uniqueness of weak solutions. We also prove the existence of weak solutions to the semi-discretization of the problem as well as convergence of the approximating sequences. The semi-discretization shows that facets must form. For a class of initial data we are able to study in detail the facet formation and interactions and their asymptotic behaviour. We note that our qualitative results may be interpreted with the help of a special composition of multivalued operators.