Existence and regularity of a class of weak solutions to the Navier-Stokes equations
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.
scientific article; zbMATH DE number 1046943