Subdifferential boundary value problems for the nonstationary Navier-Stokes equations (Q5942177)

Let \(V\) and \(H\) be separable Hilbert spaces with dual spaces \(V'\) and \(H'\) such that \(V\) is densely and compactly embedded in \(H\) and \(H'\) is identified with \(H\). The norms of \(V,V'\) and \(H\) are denoted by \(\|\cdot \|, \|\cdot \|_*\) and \(|\cdot|\) respectively, and \((\cdot, \cdot)\) stands for the inner product of \(H\) as well as the \(V-V'\) duality. The following abstract Cauchy problem is considered in \(V'\). \[ y'(t)+ Ay(t)+ B \bigl[y(t) \bigr]+\partial \varphi\bigl(y(t) \bigr)\ni f(t),\quad y(0)=y_0.\tag{1} \] Here \(y'(t)=dy(t)/dt\) and \(A:V\to V'\) is a linear continuous symmetric operator such that \((Ay,y)\geq \nu\|y\|^2+ \alpha|y|^2\), \(\nu>0\), \(\alpha\in \mathbb{R}\), \(B[y]=B(y,y): V\to V'\) is a weakly continuous mapping given by the bilinear continuous mapping \(B(u,v):V\times V\to V'\) such that \((B(u,v), v)=0\) for all \(u,v\in V\), and \(\partial\varphi: V\to V'\) is the subdifferential of a lower semicontinuous convex functional \(\varphi: V\to (-\infty, +\infty]\). It is shwon in Theorem 2.1 that if \(B\) satisfies \[ \biggl|\bigl(B(w,v), w\bigr)\biggr |\leq K_1\|w\|^{1+ \theta} |w|^{1-\theta} \|v\|\quad\forall w,v\in V,\;\theta\in(0,1),\;K_1>0, \] then for any \(y_0\in H\) and \(f\in L^2(0,T;V')\), (1) has at least one weak solution. Furthermore, it is shown in Theorem 2.2 that if \(A\) and \(B\) satisfy \[ |Av|\leq K_2\|v\|_U,\quad \bigl|B[v]\bigr|\leq K_1\|v\|^2_U \quad\forall v\in U, \] \[ \biggl|\bigl(B(w,v), w\bigr) \biggl|\leq K_3\|w\|^{1+ \theta}|w|^{1-\theta} \|v \|^\gamma |v|^{1-\gamma} \forall w,v\in V,\theta,\gamma \in[0,1/2], \;K_3>0, \] and if \(y_0\in U\), \(\partial\varphi(y_0) \cap\widetilde H\neq \emptyset\), \(f,f'\in L^2(0,T;V')\), \(f|_{t=0} \in\widetilde H\), then there exists a unique strong solution of (1). Here \(\widetilde H\) and \(U\) are separable Hilbert spaces such that \(H\subseteq \widetilde H\), \((\cdot, \cdot)_{\widetilde H}= (\cdot,\cdot)_H\), \(U\) is continuously and densely embedded in \(V\). The method of proofs rely on the Galerkin approximation and the standard energy estimates. The abstract results are applied to Navier-Stokes equations in bounded domains in \(\mathbb{R}^d\), \(d=2,3\), with some nonlinear (local as well as nonlocal) boundary conditions, which are formulated as variational inequalities described in terms of the subdifferential operators.