Lyapunov Functionals and Local Dissipativity for the Vorticity Equation in L^p and Besov Spaces

Blow-up criterion of strong solutions to the Navier-Stokes equations in Besov spaces with negative indices

A Hölder type inequality in Besov spaces is established and applied to show that every strong solution $u(t,x)$ on (0,T) of the Navier-Stokes equations can be continued beyond $t>T$ provided that the vorticity $ω(t,x)\in L^{\frac 2{2-α}}(0,T;\dot{B}^{-α}_{\infty,\infty}(\mr^3))\cap L^{\frac2{1-α}}(0,T;\dot{B}^{-1-α}_{\infty,\infty}(\mr^3))$ for $0<α<1$.

Continuity of the solution map of the Euler equations in Hölder spaces and weak norm inflation in Besov spaces

We construct an example showing that the solution map of the Euler equations is not continuous in the Hölder space from $C^{1,α}$ to $L^\infty_tC^{1,α}_x$ for any $0<α<1$. On the other hand we show that it is continuous when restricted to the little Hölder subspace $c^{1,α}$. We apply the latter to prove an ill-posedness result for solutions of the vorticity equations in Besov spaces near the critical space $B^1_{2,1}$. As a consequence we show that a sequence of best constants of the Sobolev embedding theorem near the critical function space is not continuous.

Incompressible Boussinesq equations and borderline Besov spaces

We prove local-in-time existence and uniqueness of an inviscid Boussinesq-type system. We assume the density equation contains nonzero diffusion and that our initial vorticity and density belong to a space of borderline Besov type.

On the Global Existence for the Axisymmetric Euler-Boussinesq System in Critical Besov Spaces

This paper is devoted to the global existence and uniqueness results for the three-dimensional Boussinesq system with axisymmetric initial data $v^{0}{\in}B_{2,1}^{5/2}(\RR^3)$ and$ $ρ^{0}{\in}B_{2,1}^{1/2}(\RR^3)\cap L^{p}(\RR^3)$ with $p>6.$ This system couples the incompressible Euler equations with a transport-diffusion equation governing the density. In this case the Beale-Kato-Majda criterion is not known to be valid and to circumvent this difficulty we use in a crucial way some geometric properties of the vorticity.

Self-similar solutions for active scalar equations in Fourier-Besov-Morrey spaces

We are concerned with a family of dissipative active scalar equation with velocity fields coupled via multiplier operators that can be of high-order. We consider sub-critical values for the fractional diffusion and prove global well-posedness of solutions with small initial data belonging to a framework based on Fourier transform, namely Fourier-Besov-Morrey spaces. Since the smallness condition is with respect to the weak norm of this space, some initial data with large $L^{2}$-norm can be considered. Self-similar solutions are obtained depending on the homogeneity of the initial data and cou…

一个模拟具有常涡量的浅水波的新型高度非线性方程

A new highly nonlinear equation modelling shallow-water waves with constant vorticity

本文通过形式渐近展开和摄动理论，从常涡量不可压缩流体方程导出一个新的高度非线性浅水模型，采用比Camassa-Holm更大的最优尺度以避免非局部项，并在Besov空间中建立局部适定性和爆破判据。