A necessary condition for solvability of a class of boundary value problems for the linearized Navier-Stokes system (Q2713891)

In the domain \(\Omega=\{(t,x): t>0\), \(x=(x_1,x_2,x_2)\), \(x_3>0\}\), the following boundary value problem is considered: NEWLINE\[NEWLINE u_t -\Delta u +\nabla p=f(t,x), \quad \text{div} u=f_4(t,x) NEWLINE\]NEWLINE NEWLINE\[NEWLINE u|_{t=0}=0,\qquad B(D_x)u|_{x_3=0}=0, NEWLINE\]NEWLINE where \(B(D_x)=\sum_{i=1}^3B_jD_{x_j}u+B_4u\) and \(B_j\) are constant \(3\times 3\)-matrices. It is assumed that the Lopatinskij condition is fulfilled. Under some additional conditions on the matrices \(B_j\) (\(j=1,2,3,4\)), the author establishes that the equality NEWLINE\[NEWLINE \int_{{\mathbb R}_3^+} f_4(t,x) dx=0\;\;(f_4(t,x)\in C_0^{\infty}(\Omega)) NEWLINE\]NEWLINE is a necessary condition for solvability of the above problem in the class \(e^{-\gamma t}u\in W_2^{1,2}(\Omega)\) \((\gamma>0)\) and \(e^{-\gamma t}\nabla p\in L_2(\Omega)\); i.e., the generalized derivatives occurring in the Navier-Stoker equation belongs to the space \(L_2(\Omega)\) with weight \(e^{-\gamma t}\).NEWLINENEWLINEFor the entire collection see [Zbl 0956.00039].