arXiv subjects
Publications and source records attributed to Ondřej Kreml.
We study the generalized Oldroyd model with viscosity depending on the shear stress behaving like $μ(\mathbf{D}) \sim |\mathbf{D}|^{p-2}$ ($p>\frac 65$) regularized by a nonlinear stress diffusion. Using the Lipschitz truncation method we are able to prove global existence of weak solution to the corresponding system of partial differential equations.
We study the incompressible limit of solutions to the compressible barotropic Navier-Stokes system in the exterior of a bounded domain undergoing a simple translation. The problem is reformulated using a change of coordinates to fixed exterior domain. Using the spectral analysis of the wave propagator, the dispersion of acoustic waves is proved by the means of the RAGE theorem. The solution to the incompressible Navier-Stokes equations is identified as a limit.
We show that 1-D rarefaction wave solutions are unique in the class of bounded entropy solutions to the multidimensional compressible Euler system. Such a result may be viewed as a counterpart of the recent examples of non-uniqueness of the shock wave solutions to the Riemann problem, where infinitely many solutions are constructed by the method of convex integration.
In this paper we extend and complement some recent results by Chiodaroli, De Lellis and Kreml on the well-posedness issue for weak solutions of the compressible isentropic Euler system in $2$ space dimensions with pressure law $p(ρ)=ρ^γ$, $γ\geq 1$. First we show that every Riemann problem whose one-dimensional self-similar solution consists of two shocks admits also infinitely many two-dimensional admissible bounded weak solutions (not containing vacuum) generated by the method of De Lellis and Székelyhidi. Moreover we prove that for some of these Riemann problems and for $1\leq γ< 3$ such solutions have greater energy dissipation rate than the self-similar solution emanating from the same Riemann data. We therefore show that the maximal dissipation criterion proposed by Dafermos does not favour the classical self-similar solutions.
We consider the compressible (barotropic) Navier-Stokes system on time-dependent domains, supplemented with slip boundary conditions. Our approach is based on penalization of the boundary behaviour, viscosity, and the pressure in the weak formulation. Global-in-time weak solutions are obtained.
We study steady flow of a second grade fluid past an obstacle in three space dimensions. We prove existence of solution in weighted Lebesgue spaces with anisotropic weights and thus existence of the wake region behind the obstacle. We use properties of the fundamental Oseen tensor together with results achieved in \cite{Koch} and properties of solutions to steady transport equation to get up to arbitrarily small $\ep$ the same decay as the Oseen fundamental solution.