Generic solvability of the equations of Navier-Stokes.(English)Zbl 0676.35073

The authors study the existence of unique strong solutions for arbitrary times $$T>0$$ of the initial boundary value problem for the Navier-Stokes equations on bounded domains in $${\mathbb{R}}^ 3$$. For given suitable initial data $$u_ 0$$ they investigate the question whether any body force $$f\in L^ p(0,T;L^ p(\Omega)^ 3)$$, $$p\in [2,\infty)$$, $$\Omega$$ being the domain in $${\mathbb{R}}^ 3$$, leads to a unique strong solution for arbitrary $$T>0$$. The set of such f is open in $$L^ p(0,T;L^ p(\Omega)^ 3)$$. The authors prove in this paper that the set of these forces f lies dense in $$L^ p(0,T;L^ p(\Omega)^ 3)$$ with respect to the norm of $$L^ s(0,T;L^ q(\Omega)^ 3)$$ for all $$s,q\in (1,\infty)$$ with $$4<2/s+3/q.$$ Therefore, in this sense the set of forces f in $$L^ p(0,T;L^ p(\Omega)^ 3)$$ forms a large subset, i.e. the problem is generically solvable.
Reviewer: G.Warnecke

MSC:

 35Q30 Navier-Stokes equations 35K45 Initial value problems for second-order parabolic systems 35K55 Nonlinear parabolic equations 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) 76D05 Navier-Stokes equations for incompressible viscous fluids 35B65 Smoothness and regularity of solutions to PDEs