Hemivariational inequalities for stationary Navier--Stokes equations
The authors study a class of ``inequality problems'' for Navier-Stokes type operators. To be precise, they consider for space dimension \(d=2,3\) the following problem: \[ -\nu \operatorname{rot} \operatorname{rot}u+ \operatorname{rot}u\times u+\nabla h=f,\qquad \operatorname{div} u=0 \quad\text{in }\Omega\subset\mathbb{R}^d \] where \(u=\{u_i\}^d_{i=1}\) denotes the velocity of the fluid and \(f=p+\frac{1}{2}| u| ^2\) denotes the dynamic pressure. The authors suppose the boundary conditions: \[ h\in\partial j(x,u_N)\quad \text{and}\quad u_\tau=0 \text{ on }\partial\Omega. \] Here \(\partial j\) denotes the Clarke subdifferential of a locally Lipschitz function \(j(a,\cdot)\). \(u_N\) is the normal and \(u_\tau\) denotes the tangential component of u on the boundary. The hemivariational inequality to this problem is given by: Find a function \(u\) in \(V\) such that: \[ \nu\int_\Omega \operatorname{rot}u\cdot \operatorname{rot}v\, dx+\int_\Omega (\operatorname{rot}u\times u)\cdot v \,dx+\int_\Gamma j^0(x,u_n(x); v_n(x))\,d\sigma(x)\geq \int\_\Omega f\cdot v \,dx \] for every \(v\in V\). The space \(V\) is given by the closure of the space \[ W=\{w\in C^\infty(\Omega;\mathbb{R}^d: \operatorname{div}w=0\text{ in }\Omega,\;w_\tau=0 \text{on }\partial\Omega\} \] in the norm of \(H^1(\Omega;\mathbb{R}^d)\). \(j^0(x,\xi;\eta)\) denotes the directional derivative of \(j(x,\cdot)\) at the point \(\xi\in\mathbb{R}\). The authors establish some existence and uniqueness results for weak solutions of the hemivariational inequality and present an application for their results.
scientific article; zbMATH DE number 2173094