The Sobolev type equations with a nonsurjective operator at the time derivative (Q941267)

Let \(U\) and \(F\) be Banach spaces conjugated to \(Y\) and \(X\), respectively and let \(L,M\in L(U,F)\) be conjugated to \(A,B:X\to Y\), respectively. For \(k\in\mathbb{N}_0\) let \(Y_0=\{0\}\), \(X_k=A^{-1}(Y_k)\), \(Y_{k+1}=B(X_k)\), \(X_\infty=\cup^\infty_{k=0}X_k\), \(Y_\infty=\cup^\infty_{k=0}Y_k\), where \(A^{-1}(Y_k)\) denotes the preimage of the set \(Y_k\), and let \(\text{Im}\,L\) be closed (not necessary equal \(F)\), \(\text{Ker}\,L=\{0\}\). Then the author proves that the Cauchy problem \[ L\dot u(t)=Mu(t)+f(t),\quad u(0)=u_0 \] is uniquely solvable in the class \(C^1(0,T)\cap C([0,T))\) iff \(u_0\in Y^\perp\). This result is applied to the Rossby equation \[ -\Delta u_t+u_{x_1}=0 \] in a bounded domain \(\Omega\) under appropriate initial-boundary conditions.