Optimal smoothing and decay estimates for viscously damped conservation laws, with applications to the 2-D Navier-Stokes equation
The authors study a generalization of the vorticity formulation of the 2D-Navier-Stokes equations, that is \(w_t-\Delta w=\text{div}(\underline f(x,w)w)\) with suitable assumptions on the vector field \(\underline f\). They optimize the constants in the \(L_p-L_q\)-estimate \[ |w(.,t)|_q\leq C\cdot t^{-{n\over 2}({1\over p}-{1\over q})}|w(.,0)|_p,\quad 1\leq p\leq q, \] and give even decay results for \(w\) in \(L_1\), as well as spatial estimates. The results are sharp to some extent, yet the authors missed some earlier papers, where qualitatively similar results can be found, e.g. \textit{M. E. Schonbek} [Nonlinear Anal., Theory Methods, Appl. 10, 943-956 (1986; Zbl 0617.35060)] and \textit{Y. Giga}, \textit{T. Miyakawa} and \textit{H. Osada} [Arch. Ration. Mech. Anal. 104, No. 3, 223-250 (1988; Zbl 0666.76052)].
scientific article; zbMATH DE number 880074
scientific article