A solvability condition for the first boundary value problem for a linearized Navier--Stokes system
The article is devoted to studying the first boundary value problem in the quadrant \(R^{++}_{4}\) for the linearized Navier-Stokes system \[ \begin{gathered} \frac{\partial u}{\partial t} -\triangle u-[u,\mathbf\omega] +\nabla {\mathbf p}=f(t,x), \\ \text{div } u=f_{4}(t,x), \\ u|_{t=0}=0, \quad u|_{x_{3}=0}=0. \end{gathered} \] Here \(t > 0\), \(x_{3} > 0\), \({\mathbf\omega}=(0,0,\omega)\), \(\omega\geq 0\). Let \(W^{r}_{p,\gamma}(R^{++}_{4})\) be the class of vector-valued functions \(v(t,x)=(v_{1}(t,x),v_{2}(t,x),v_{3}(t,x))\) satisfying \(e^{-\gamma t}v_{j}(t,x)\in W^{r}_{p}(R^{++}_{4})\), \(j=1,2,3\), \(\gamma>0\), \(1\leq p<+\infty\), \(r=(r_0,r_1,r_2,r_3)\). The author establishes necessary and sufficient conditions for solvability of the above problem in the classes \(u(t,x)\in W_{p,\gamma}^{1,2}(R^{++}_{4}), \nabla{\mathbf p}(t,x)\in L_{p,\gamma}(R^{++}_{4})\). He also obtains the corresponding a priori estimates.
sufficient and necessary solvability conditions
scientific article; zbMATH DE number 1285460