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Hardy spaces of solenoidal vector fields, with applications to the Navier-Stokes equations. (English) Zbl 0883.35088
In this very detailed paper the author investigates decay properties of weak solutions both for stationary and nonstationary Navier-Stokes equations. The first part of the paper is concerned with the Cauchy problem to the nonstationary Navier-Stokes equations $\frac{\partial u}{\partial t}+u\cdot\nabla u-\Delta u+\nabla p=0,\quad\nabla\cdot u=0,\quad x\in\mathbb{R}^n,\;t>0,\quad u|_{t=0}=a.\tag{1}$ Using some specific properties of Hardy spaces the author proves that for a suitable initial condition $$a$$ the weak solution to the problem (1) satisfies the relations $$\lim|u(t)|_{L^1(\mathbb{R}^n)}=0$$, $$\lim|u(t)|_{H^1(\mathbb{R}^n)}=0$$ as $$t\to\infty$$. The stationary problem $u\cdot\nabla u-\Delta u+\nabla p=\nabla F,\quad\nabla\cdot u=0,\quad\text{in }\mathbb{R}^n,\quad\lim u=0\text{ as }|x|\to\infty\tag{2}$ is considered in the second part of the paper. It is shown that if $$n\geqslant 3$$ and $$F$$ is small in $$L^{n/2}$$ then the problem (2) has a weak solution $$u$$ such that $u\in L^n(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n),\quad\nabla u\in L^{n/2}(\mathbb{R}^n)\cap L^\infty (\mathbb{R}^n).$ More explicit results are obtained for $$n\geq 4$$.

##### MSC:
 35Q30 Navier-Stokes equations 35B40 Asymptotic behavior of solutions to PDEs 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
##### Keywords:
Hardy spaces; weak solution; decay
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