We obtain a ‘stability estimate’ for strong solutions of the Navier–Stokes system, which is an Lα-version, 1 < α < ∞, of the estimate that Serrin [Se] used in obtaining uniqueness of weak solutions to the Navier-Stokes system. By applying this estimate, we obtain new results in stability and uniqueness of solutions, and non-blowup conditions for strong solutions.

Introduction Section 1: Preliminaries 1.1 The Navier-Stokes equations 1.2 Classical, mild and weak solutions 1.3 Navier meets Fourier Section 2: Functional setting of the equations 2.1 The Littlewood-Paley decomposition 2.2 The Besov spaces 2.3 The paraproduct rule 2.4 The wavelet decomposition 2.5 Other useful function spaces Section 3: Existence theorems 3.1 The fixed point theorem 3.2 Scalin...

We prove global existence for a nonlinear Smoluchowski equation (a nonlinear FokkerPlanck equation) coupled with Navier-Stokes equations in 2d. The proof uses a deteriorating regularity estimate in the spirit of [5] (see also [1]) Key wordsNonlinear Fokker-Planck equations, Navier-Stokes equations, Smoluchowski equation, micro-macro interactions. AMS subject classification 35Q30, 82C31, 76A05.

We tackle the issue of the inviscid limit of the incompressible Navier-Stokes equations when the Navier slip-with-friction conditions are prescribed on the impermeable boundaries. We justify an asymptotic expansion which involves a weak amplitude boundary layer, with the same thickness as in Prandtl’s theory and a linear behavior. This analysis holds for general regular domains, in both dimensi...

We consider the zero viscosity limit of long time averages of solutions of damped and driven Navier-Stokes equations in R 2. We prove that the rate of dissipation of enstrophy vanishes. Stationary statistical solutions of the damped and driven Navier-Stokes equations converge to renormalized stationary statistical solutions of the damped and driven Euler equations. These solutions obey the enst...

We prove the global regularity for both of the 3D Navier-Stokes equations and the 3D Euler equations on R for initial data v0 ∈ H (R). 1 Main Result We are concerned on the following Navier-Stokes equations(Euler equations for ν = 0) describing the homogeneous incompressible fluid flows in R. (NS)ν 

On compactness of solutions to the compressible isentropic Navier-Stokes equations when the density is not square integrable Abstract. We show compactness of bounded sets of weak solutions to the isentropic compressible Navier-Stokes equations in three space dimensions under the hypothesis that the adiabatic constant γ > 3/2.

In this note we establish the existence and uniqueness of solutions for optimal control problems for the 2D Navier-Stokes equations in a 2D-channel. Our approach is based on infinite-dimensional optimization ; the cost functional is shown to be strictly convex. Generalization to other control problems as well as a gradient algorithm are presented. Existence et unicité du contrôle optimal des éq...

In the paper compressible, stationary Navier-Stokes equations are considered. A framework for analysis of such equations is established. In particular, the well-posedness for inhomogeneous boundary value problems of elliptic-hyperbolic type is shown. Analysis is performed for small perturbations of the so-called approximate solutions, i.e., the solutions take form (1.12). The approximate soluti...

In this paper we deal with some controllability problems for systems of the Navier– Stokes and Boussinesq kind with distributed controls supported in small sets. Our main aim is to control N-dimensional systems (N + 1 scalar unknowns in the case of the Navier–Stokes equations) with N − 1 scalar control functions. In a first step, we present some global Carleman estimates for suitable adjoint pr...