Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation.
Nonhomogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: Regularity of the solution.
Nonhomogeneous Cahn–Hilliard fluids
Nonlinear compressible vortex sheets in two space dimensions
We consider supersonic compressible vortex sheets for the isentropic Euler equations of gas dynamics in two space dimensions. The problem is a free boundary nonlinear hyperbolic problem with two main difficulties: the free boundary is characteristic, and the so-called Lopatinskii condition holds only in a weak sense, which yields losses of derivatives. Nevertheless, we prove the local existence of such piecewise smooth solutions to the Euler equations. Since the a priori estimates for the linearized...
Nonlinear diffusion from a delocalized source : affine self-similarity, time reversal, & nonradial focusing geometries
Nonlinear elliptic problems with incomplete Dirichlet conditions and the stream function solution of subsonic rotational flows past profiles or cascades of profiles
The paper is devoted to the solvability of a nonlinear elliptic problem in a plane multiply connected domain. On the inner components of its boundary Dirichlet conditions are known up to additive constants which have to be determined together with the sought solution so that the so-called trailing stagnation conditions are satisfied. The results have applications in the stream function solution of subsonic flows past groups of profiles or cascades of profiles.
Nonlinear instability in an ideal fluid
Nonlinear instability of double-humped equilibria
Nonlocal problems for the equations of motion of Kelvin-Voigt fluids
Non-uniqueness of almost unidirectional inviscid compressible flow
Our aim is to find roots of the non-unique behavior of gases which can be observed in certain axisymmetric nozzle geometries under special flow regimes. For this purpose, we use several versions of the compressible Euler equations. We show that the main reason for the non-uniqueness is hidden in the energy decomposition into its internal and kinetic parts, and their complementary behavior. It turns out that, at least for inviscid compressible flows, a bifurcation can occur only at flow regimes with...
Numerical analysis of coupling for a kinetic equation
In this paper we introduce a coupled systems of kinetic equations for the linearized Carleman model. We then study the existence theory and the asymptotic behaviour of the resulting coupled problem. In order to solve the coupled problem we propose to use the time marching algorithm. We then develop a convergence theory for the resulting algorithm. Numerical results confirming the theory are then presented.
Numerical analysis of Eulerian multi-fluid models in the context of kinetic formulations for dilute evaporating sprays
The purpose of this article is the analysis and the development of Eulerian multi-fluid models to describe the evolution of the mass density of evaporating liquid sprays. First, the classical multi-fluid model developed in [Laurent and Massot, Combust. Theor. Model.5 (2001) 537–572] is analyzed in the framework of an unsteady configuration without dynamical nor heating effects, where the evaporation process is isolated, since it is a key issue. The classical multi-fluid method consists then in...
Numerical evidence of nonuniqueness in the evolution of vortex sheets
We consider a special configuration of vorticity that consists of a pair of externally tangent circular vortex sheets, each having a circularly symmetric core of bounded vorticity concentric to the sheet, and each core precisely balancing the vorticity mass of the sheet. This configuration is a stationary weak solution of the 2D incompressible Euler equations. We propose to perform numerical experiments to verify that certain approximations of this flow configuration converge to a non-stationary...
Numerical study of a descending sphere in a low Reynolds number strongly stratified fluid.
On 2D Rayleigh-Taylor instabilities
On a certain lubrication slip model.
On a differential inequality for a viscous compressible heat conducting capillary fluid bounded by a free surface
We derive a global differential inequality for solutions of a free boundary problem for a viscous compressible heat concluding capillary fluid. The inequality is essential in proving the global existence of solutions.
On a free boundary barotropic model
On a functional equation arising in the kinetic theory of gases
The problem of finding the summational collision invariants for the Boltzmann equation leads to a functional equation related to the Cauchy equation. The solution of this equation is known under different assumptions on its unknown . Most proofs assume that the equation is pointwise satisfied, while the result needed in kinetic theory concerns the solutions of the equation when the latter is satisfied almost everywhere. The only results of this kind appear to be due to the authors of the present...
On a generalized Stokes problem
We deal with a generalization of the Stokes system. Instead of the Laplace operator, we consider a general elliptic operator and a pressure gradient with small perturbations. We investigate the existence and uniqueness of a solution as well its regularity properties. Two types of regularity are provided. Aside from the classical Hilbert regularity, we also prove the Hölder regularity for coefficients in VMO space.