We consider the Stokes problem with slip type boundary conditions in the half-space R+, with n > 2. The weighted Sobolev spaces yield the functional framework. We study generalized and strong solutions and then the case with very low regularity of data on the boundary. We apply the method of decomposition introduced in our previous work (see [7]), where it is necessary to solve particular probl...

Abstract. We continue our analysis of the Cauchy problem for viscous system of conservation, under natural assumptions. We examine in which way does the existence time depend upon the viscous tensor B(u). In particular, we consider singular limits, where the rank of the symbol B(u; ξ) drops at the limit. This covers a lot of situations, for instance that of the limit of the Navier-Stokes-Fourie...

These equations are the generalized equations of several dispersionless equations. A complete table for p ≤ 10 is provided.

We present a review of the semi-Lagrangian method for advection-diffusion and incompressible Navier-Stokes equations discretized with high-order methods. In particular, we compare the strong form where the departure points are computed directly via backwards integration with the auxiliary form where an auxiliary advection equation is solved instead; the latter is also referred to as Operator In...

This note details the derivation of the generalized advective-diffusive equation system for the 2D space-time incompressible Navier-Stokes governing equations, in conservation form.

In this article, we consider the incompressible Navier-Stokes equations with linearly growing initial data U0 := u0(x) −Mx. Here M is an n × n matrix, trM = 0, M2 is symmetric and u0 ∈ L2(Rn) ∩ Ln(Rn). Under these conditions, we consider v(t) := u(t) − eu0, where u(x) := U(x) −Mx and U(x) is the mild solution of the incompressible Navier-Stokes equations with linearly growing initial data. We s...

Let G be the (open) set of Ḣ 1 2 (R) divergence free vector fields generating global smooth solutions to the three dimensional incompressible Navier-Stokes equations. We prove that any element of G can be perturbed by an arbitrarily large, smooth divergence free vector field which varies slowly in one direction, and the resulting vector field (which remains arbitrarily large) is an element of G...

The contravariant form of the Navier–Stokes equations in a fixed curvilinear coordinate system is well known. However, when the curvilinear coordinate system is time-varying, such as when a body-fitted grid is used to compute the flow over a compliant surface, considerable care is needed to handle the momentum term correctly. The present paper derives the complete contravariant form of the Navi...

The Cauchy problem for the Navier–Stokes system for vorticity on plane is considered. If the Fourier transform of the initial data decays as a power at infinity, then at any positive time the Fourier transform of the solution decays exponentially, i.e. the solution is analytic.

The goal of this lecture is to show that homogenization is a very efficient tool in the modeling of complex phenomena in heterogeneous media. In the first lecture we considered a model problem of diffusion for which the homogenized operator was of the same type (still a diffusion equation). In this context, homogenization is really a matter of defining and computing effective diffusion tensors....