We describe the Wavelet-Vaguelette Decomposition (WVD) of a linear inverse problem. It is a substitute for the singular value decomposition (SVD) of an inverse problem, and it exists for a class of special inverse problems of homogeneous type { such as numerical di erentiation, inversion of Abel-type transforms, certain convolution transforms, and the Radon Transform. We propose to solve ill-po...

We consider an equation similar to the Navier-Stokes equation. We show that there is initial data that exists in every Triebel-Lizorkin or Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the solution is in no Triebel-Lizorkin or Besov space (and hence in no Lebesgue or Sobolev space). The purpose is to show the limitations of the so called semigroup m...

This chapter is devoted to the generalization of the continuous shearlet transform to higher dimensions as well as to the construction of associated smoothness spaces and to the analysis of their structural properties, respectively. To construct canonical scales of smoothness spaces , so-called shearlet coorbit spaces , and associated atomic decompositions and Banach frames we prove that the ge...

In this article, we prove uniqueness and energy balance for isentropic Euler system driven by a cylindrical Wiener process. Pathwise result is obtained weak solutions having Hölder regularity C α , > 1 ∕ 2 in space satisfying one-sided Lipschitz bound on velocity. We Onsager’s conjecture with stochastic forcing, that is, equation enjoying 3 . Both the results have been more general setting cons...

We study Besov spaces 5°(LP(Í2)), 0<p,q,a<oo,on domains Í2 in Rd . We show that there is an extension operator W which is a bounded mapping from B°(LP(Ü.)) onto B%(Lp(Rd)). This is then used to derive various properties of the Besov spaces such as interpolation theorems for a pair of B%(Lp(il)), atomic decompositions for the elements of 5°(Lp(f2)), and a description of the Besov spaces by means...

In a recent paper [11], Vishik proved the global well-posedness of the two-dimensional Euler equation in the critical Besov space B 2,1. In the present paper we prove that Navier-Stokes system is globally well-posed in B 2,1, with uniform estimates on the viscosity. We prove also a global result of inviscid limit. The convergence rate in L is of order ν. keywords. navier-Stokes equations; Incom...

We show a new Bernstein’s inequality which generalizes the results of CannonePlanchon, Danchin and Lemarié-Rieusset. As an application of this inequality, we prove the global well-posedness of the 2D quasi-geostrophic equation with the critical and supercritical dissipation for the small initial data in the critical Besov space, and local wellposedness for the large initial data. Mathematics Su...