Stability of stationary solutions to the Navier–Stokes equations in the Besov space

We consider the stability of the stationary solution w of the Navier–Stokes equations in the whole space ${\mathbb{R}}^n$ for $n\ge 3$. It is clarified that if w is small in ${\dot{B}}_{p_{*},q^{\prime }}^{-1+\frac{n}{p_{*}}}$ for and , then for every small initial disturbance $a\in {\dot{B}}_{p_0,q}^{-1+\frac{n}{p_0}}$ with and ($1/q+1/q^{\prime }=1$), there exists a unique solution $v(t)$ of the nonstationary Navier–Stokes equations on (0, ∞) with $v(0)=w+a$ such that ${\parallel v(t)-w\parallel }_{L^r}=O(t^{-\frac{n}{2}(\frac{1}{n}-\frac{1}{r})})$ and ${\parallel v(t)-w\parallel }_{{\dot{B}}_{p,q}^s}=O(t^{-\frac{n}{2}(\frac{1}{n}-\frac{1}{p})-\frac{s}{2}})$ as $t\to \infty$, for , , and small $s>0$.     

The two-dimensional stationary Navier–Stokes equations in toroidal Besov spaces

We consider the stationary Navier–Stokes equations in the two-dimensional torus ${\mathbb{T}}^2$. For any $\epsilon >0$, we show the existence, uniqueness, and continuous dependence of solutions in homogeneous toroidal Besov spaces ${\dot{B}}_{p+\epsilon ,q}^{-1+\frac{2}{p}}({\mathbb{T}}^2)$ for given small external forces in ${\dot{B}}_{p+\epsilon ,q}^{-3+\frac{2}{p}}({\mathbb{T}}^2)$ when . These spaces become closer to the scaling invariant ones if the difference ε becomes smaller. This well-posedness is proved by using the embedding property and the para-product estimate in homogeneous Besov spaces. In addition, for the case $(p,q)\in (\{2\}\times (2,\infty ])\cup ((2,\infty ]\times [1,\infty ])$, we can show the ill-posedness, even in the scaling invariant spaces. Actually in such cases of p and q, we can prove that ill-posedness by showing the discontinuity of a certain solution map from ${\dot{B}}_{p,q}^{-3+\frac{2}{p}}({\mathbb{T}}^2)$ to ${\dot{B}}_{p,q}^{-1+\frac{2}{p}}({\mathbb{T}}^2)$.     

The Incompressible Limits of Compressible Navier-Stokes Equations in the Whole Space with General Initial Data

It is showed that, as the Mach number goes to zero, the weak solution of the compressible Navier-Stokes equations in the whole space with general initial data converges to the strong solution of the incompressible Navier-Stokes equations as long as the later exists. The proof of the result relies on the new modulated energy functional and the Strichartz＇s estimate of linear wave equation.     

The Exact Limits and Improved Decay Estimates for All Order Derivatives of Global Weak Solutions of Three Incompressible Fluid Dynamics Equations

First of all, the author accomplishes the exact limits for all order derivatives of the global weak solutions of the $n$-dimensional incompressible magnetohydrodynamics equations, the $n$-dimensional incompressible Navier-Stokes equations and the two-dimensional incompressible dissipative quasi-geostrophic equation. Secondly, by making use of the exact limits, he establishes the improved decay estimates with sharp rates for all order derivatives of the global weak solutions, for all sufficiently large $t$. The author proves these results by making use of existing ideas, existing results and several new, novel ideas.