Leray self-similarity equations in fluid dynamics

In the present note, we show that, as a priori bounds, the vorticity dynamics derived from Leray's backward self-similarity hypothesis admits only trivial solution in viscous as well as inviscid flows. By analogy, there is no non-zero solution in the forward self-similar equation. Since the Navier-Stokes or Euler equations are invariant under space translation in the whole space, our analysis establishes that technically flawed arguments have been exploited in a number of past papers, notably in Necas, Ruzicka & Sverak (1996); Tsai (1998); and Pomeau (2016), where the presumed decays or bounds at infinity are ill-defined and non-existent. Furthermore, an effort has been made to exemplify an inappropriate application of the familiar extremum principles in the theory of linear elliptic equation. In the appendix, we give a counterexample to the Sobolev inequality and, hence illustrate the nature of self contradiction. In the totality comparison of Lp norms, its scope of application is not significant.Comment: 14 pages; 5 more references; add one appendi

On the Global Regularity of a Helical-decimated Version of the 3D Navier-Stokes Equations

We study the global regularity, for all time and all initial data in $H^{1/2}$, of a recently introduced decimated version of the incompressible 3D Navier-Stokes (dNS) equations. The model is based on a projection of the dynamical evolution of Navier-Stokes (NS) equations into the subspace where helicity (the $L^2-$scalar product of velocity and vorticity) is sign-definite. The presence of a second (beside energy) sign-definite inviscid conserved quadratic quantity, which is equivalent to the $H^{1/2}-$Sobolev norm, allows us to demonstrate global existence and uniqueness, of space-periodic solutions, together with continuity with respect to the initial conditions, for this decimated 3D model. This is achieved thanks to the establishment of two new estimates, for this 3D model, which show that the $H^{1/2}$ and the time average of the square of the $H^{3/2}$ norms of the velocity field remain finite. Such two additional bounds are known, in the spirit of the work of H. Fujita and T. Kato \cite{kato1,kato2}, to be sufficient for showing well-posedness for the 3D NS equations. Furthermore, they are directly linked to the helicity evolution for the dNS model, and therefore with a clear physical meaning and consequences

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity approaches zero, is one of the most fundamental issues in mathematical fluid mechanics. The problem is classified into two categories: the case when the physical boundary is absent, and the case when the physical boundary is present and the effect of the boundary layer becomes significant. The aim of this article is to review recent progress on the mathematical analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final publication is available at http://www.springerlink.co

Regularity of weak solutions of the compressible isentropic Navier-Stokes equation

Regularity and uniqueness of weak solution of the compressible isentropic Navier-Stokes equations is proven for small time in dimension $N=2,3$ under periodic boundary conditions. In this paper, the initial density is not required to have a positive lower bound and the pressure law is assumed to satisfy a condition that reduces to $\gamma>1$ when $N=2,3$ and $P(\rho)=a\rho^{\gamma}$. In a second part we prove a condition of blow-up in slightly subcritical initial data when $\rho\in L^{\infty}$. We finish by proving that weak solutions in \T^{N} turn out to be smooth as long as the density remains bounded in L^{\infty}(L^{(N+1+\e)\gamma}) with \e>0 arbitrary small