Local strong solutions of the nonhomogeneous Navier-Stokes system with control of the interval of existence

Abstract

Consider a bounded domain $\Omega\subseteq \mathbb R^3$ with smooth boundary $\partial\Omega$, a time interval $[0,T)$, $0\le T\le \infty$, and in $[0,T) \times\Omega$ the completely nonhomogeneous Navier-Stokes system $u_t - \Delta u+u\cdot \nabla u + \nabla p = f$, $u|_t=0=v_0$, ${\rm div}\,u=k$, $u|_\partial\Omega = g$, with sufficiently smooth data $f,v_0,k,g$. In this general case there are mainly known two classes of weak solutions, the class of global weak solutions, similar as in the well known case $k=0$, $g=0$ which need not be unique, see [R. Farwig, H. Kozono and H. Sohr, Global weak solutions of the Navier-Stokes equations with nonhomogeneous boundary data and divergence, Rend. Sem. Math. Univ. Padova 125 (2011), 51-70], and the class of local very weak solutions, see [H. Amann, Nonhomogeneous Navier-Stokes Equations with Integrable Low-regularity Data, Int. Math. Ser., Kluwer Academic/Plenum Publishing, New York, 2002, 1-26], [H. Amann, Navier-Stokes equations with nonhomogenous Dirichlet data, J. Nonlinear Math. Phys. 10, Suppl. 1 (2003), 1-11], [R. Farwig, G.P. Galdi and H. Sohr, A new class of weak solutions of the Navier-Stokes equations with nonhomogeneous data, J. Math. Fluid Mech. 8 (2006), 423-444], which are uniquely determined but have no differentiability properties and need not satisfy an energy inequality. Our aim is to introduce the new class of local strong solutions in the usual sense for $k\not= 0$, $g\not=0$ satisfying similar regularity and uniqueness properties as in the well known case $k=0$, $g=0$. Further, we obtain precise information through the given data on the interval of existence $[0,T^*)$, $0\le T^*\le T$. Our proof is essentially based on a detailed analysis of the corresponding linear system.