Smooth solutions of the vector Burgers equation in nonsmooth domains
For the equation \(u_t + u \cdot \nabla u = \Delta u\) in dimensions two and three, the initial boundary value problem \(u{|}_{x \in \partial \Omega } = 0\), \(u{|}_{t = 0} = 0\) is considered for an arbitrary open domain \(\Omega \), possibly nonsmooth. Using domain-independent elliptic inequalities of \textit{Xie} [Indiana Univ. Math. J. 40, No. 4, 1185-1192 (1991; Zbl 0736.35012)], existence and uniqueness of smooth solutions on nonsmooth domains is established in a way which is assumed to be eventually transferable to the Navier-Stokes equation.
scientific article; zbMATH DE number 1153442
scientific article