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


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