On the stationary motion of granulated media (Q1123088)

Let \(\Omega\) be an open, bounded domain in \({\mathbb{R}}^ 3\), locally situated on one side of its boundary \(\Gamma\), a differentiable manifold of class \(C^ 2\). In this paper we consider the following system of equations \[ (1)\quad -v\Delta u+(u\cdot \nabla)u-\eta \omega \times u=- \nabla p+f,\quad in\quad \Omega,\quad \nabla \cdot u=0,\quad in\quad \Omega, \] \[ (u\cdot \nabla)\omega +F(p)\omega =g,\quad in\quad \Omega,\quad u=0,\quad on\quad \Gamma, \] which describes the stationary motion of a granulated medium with constant density. Here, the vector fields \(u=(u_ 1,u_ 2,u_ 3)\) and \(\omega =(\omega_ 1,\omega_ 2,\omega_ 3)\) denote the velocity and the angular velocity of rotation of the particles, respectively. The scalar p denotes the pressure. The quantities u(x), \(\omega\) (x) and p(x) are the unknowns, in problem (1). The positive constants \(\eta\), v are the Magnus and viscosity coefficients. The given vector fields \(f=(f_ 1,f_ 2,f_ 3)\) and \(g=(g_ 1,g_ 2,g_ 3)\) denote the exterior mass forces and the density of momentum of the forces, respectively. The function \(F=F(p)\) describes the friction between the particles. In the sequel we will assume that F(\(\xi)\), \(\xi\in {\mathbb{R}}\), is a real continuous function, for which there exist two constants \(m>0\) and \(p_ 0\in {\mathbb{R}}\) such that (2) F(\(\xi)\)\(\geq m\), if \(\xi \geq p_ 0\). This condition includes in particular the physically important case (3) \(F(\xi)=n+k\xi\), where \(n>0\) is a shift cohesion constant and \(k>0\) is the friction constant. Under assumption (2), we succeed in proving the existence of a solution u,\(\omega\),p such that p(x)\(\geq p0\), \(\forall x\in \Omega\). The lower bound F(p(x))\(\geq m\), \(\forall x\in \Omega\), follows then as a consequence. More precisely, we will prove the following result: Theorem: Let \(f\in {\mathbb{L}}^ q(\Omega)\), \(q>3\), \(g\in {\mathbb{L}}^{\infty}(\Omega)\), and let F be a real continuous function verifying (2). Fix a constant a, such that \(a\geq p_ 0\). Then, there exists a solution u,\(\omega\),p of problem (1) such that \(\min_{x\in {\bar \Omega}}p(x)=a\). Moreover, \(u\in {\mathbb{W}}^ 2_ q(\Omega)\), \(p\in W^ 1_ q(\Omega)\), \(\omega \in {\mathbb{L}}^{\infty}(\Omega)\).