# zbMATH — the first resource for mathematics

Existence and regularity of a class of weak solutions to the Navier-Stokes equations. (English) Zbl 0888.35079
The author considers the Navier-Stokes equations in an arbitrary domain $$\Omega\subset\mathbb{R}^3$$, i.e. $u_t-\nu\Delta u+(u\cdot\nabla)u+ \nabla p= f,\quad\text{div } u=0\quad\text{in }\Omega\times (0,\infty),\quad u(\cdot,0)= a\quad\text{in }\Omega$ together with a homogeneous Dirichlet condition on $$\partial\Omega$$ and a condition at infinity if $$\Omega$$ is unbounded. He constructs a global weak solution $$(u,p)$$ which satisfies $u\in L^p(0, T; W^{2,p}(\Omega)),\quad u_t, \nabla p\in L^p(0, T;L^p(\Omega))$ for $$1< p\leq{5/4}$$ and all $$T<\infty$$ extending an earlier work by Ladyzhenskaya, Beirão de Veiga, and others. This weak solution also belongs to $$L^s(0,T; W^{2,r}(\Omega))$$ provided $$s$$ and $$r$$ satisfy $$1/s +3/(2r)= 2$$ for $$1< r\leq 5/4$$. In the last part of his paper, the author extends well-known regularity results for solutions of the Navier-Stokes equations to the case of an arbitrary domain in $$\mathbb{R}^3$$. For example, he proves that $$u$$ is regular provided that $$\nabla u\in L^s(0, T;L^r(\Omega))$$ with $$1/s+ 3/(2r)= 1$$ and the Stokes operator associated with $$\Omega$$ satisfies a suitable $$L^p- L^q$$-estimate.

##### MSC:
 35Q30 Navier-Stokes equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35D10 Regularity of generalized solutions of PDE (MSC2000)
Full Text: