LOCAL STRONG SOLUTIONS OF THE NONHOMOGENEOUS NAVIER-STOKES SYSTEM WITH CONTROL OF THE INTERVAL OF EXISTENCE

作者：Farwig Reinhard^*; Sohr Hermann; Varnhorn Werner

来源：Topological Methods in Nonlinear Analysis, 2015, 46(2): 999-1012.

DOI：10.12775/tmna.2015.074

• Instationary Navier-Stokes equations
• nonhomogeneous data
• strong solutions
• weak solutions

Consider a bounded domain Omega subset of R-3 with smooth boundary partial derivative Omega, a time interval [0, T), 0 < T <= infinity, and in [0, T) x Omega the nonhomogeneous Navier-Stokes system u(t) - Delta u + u . del u + del p = f, u vertical bar(t)=0 = v0, div u = k, u vertical bar partial derivative Omega = g, with sufficiently smooth data f, v0, 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 [5], and the class of local very weak solutions, see [1], [2], [3], 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 equal 0, g not equal 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 < T* <= T. Our proof is essentially based on a detailed analysis of the corresponding linear system.

• 出版日期2015-12