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The dam problem for nonlinear Darcy’s laws and nonlinear leaky boundary conditions. (English) Zbl 0889.35125
This paper is concerned with the dam problem for nonlinear Darcy’s laws and nonlinear leaky boundary conditions on the bottom of the reservoirs. Existence of a weak solution is established and in a simple case the exact unique solution is given.
Let us illustrate the problem. Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\) \((N\geq 2)\). \(\Omega\) represents a porous medium. The boundary \(\partial \Omega\) is divided into three parts: an impervious part \(S_1\), a part in contact with air \(S_2\), and finally a part covered by fluid \(S_3\) which is relatively open in \(\partial \Omega\). Assuming that the flow in \(\Omega\) has reached a steady state the authors are concerned with finding the pressure \(p\) of the fluid and the wet part of the porous medium \(A\) of \(\Omega\). The fluid velocity \(v\) and its pressure \(p\) in \(A\) are related by the nonlinear Darcey’s law: \[ |v|^{m-1} v=-a \nabla (p+x_N), \] where \(m,a\) are positive numbers and \(x= (x_1, \dots, x_N)\) denotes points in \(\mathbb{R}^N\). By putting \(u=p + x_N\) and \(q= (1/m)+1\), one gets \[ \text{div} \bigl(|\nabla u|^{q-2} \nabla u\bigr) \bigl(= -a^{-1/m} \text{div} (v)\bigr) =0\quad \text{in }A, \] where it is assumed that the fluid is incompressible. Let \(\varphi\) be a nonnegative Lipschitz function representing the pressure on \(S_2 \cup S_3\). The authors assume that \(\varphi=0\) on \(S_2\), and \(\varphi= h-x_N\) on \(S_3\) where \(h\) denotes the level of the reservoir covering \(S_3\). If one puts \(\psi= \varphi+ x_N\), then the nonlinear leaky boundary condition reads as \[ |\nabla u|^{q-2} {\partial u \over \partial v} =\beta(x, \psi-u) \quad \text{ on } S_3 \] with some function \(\beta\) on \(S_3 \times \mathbb{R}\) satisfying some conditions. After some arguments, the weak formulation of the problem reads as:
Find \((u,g)\in W^{1,q} (\Omega) \times L^\infty (\Omega)\) such that \[ \text{(i)} \quad u\geq x_N,\;0\leq g\leq 1,\;g(u-x_N) =0\quad \text{a.e. in } \Omega, \qquad \text{(ii)} \quad u=x_N \text{ on } S_2, \]
\[ \text{(iii)} \quad \int_\Omega \bigl(|\nabla u|^{q-2} \nabla u\cdot \nabla\xi -g\xi_{X_N} \bigr)dx \leq\int_{S_3} \beta(x, \psi-u) \cdot \xi d \sigma (x) \] for any \(\xi\in W^{1,q} (\Omega)\) with \(\xi\geq 0\) on \(S_2\).

MSC:
35R35 Free boundary problems for PDEs
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
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