Existence and regularity of the pressure for the stochastic Navier-Stokes equations
The authors study the stochastic Navier-Stokes equation with domain \(D \times (0,T)\) where \(D\) is a connected and bounded open subset of \(\mathbb R^d\) with \(d=\)2, 3 or 4. The existence of a velocity \(u\) and a pressure \(p\) solution to this problem is proved when the body force \(F\) is a measurable sublinear mapping from \((0,T] \times (L^2(D))^d\) into \((H^{-1}(D))^d\). The solution is obtained in two steps: first the authors consider a velocity \(u\) satisfying the ``variational N-S equation'' (in which the pressure is eliminated) and then they associate a pressure \(p\) to such a \(u\) by using a generalization of de Rham's theorem to processes. Finally, they prove that with a body force \(F\) with values in the dual \(V'\) of the space \(V=\{v \in (H_0^1(D))^d: \nabla \cdot v=0 \}\), in general it is not possible to solve the stochastic Navier-Stokes equation.
scientific article; zbMATH DE number 2067291