Hysteresis in filtration through porous media (Q5938140)

The paper deals with an evolution problem for filtration through porous media, accounting for hysteresis in the saturation versus pressure constitutive relation. The memory effect in the constitutive relation consists not only of a rate-independent component but also of a rate-dependent one. For any fixed \(s^0\in \mathbb R\) a hysteresis operator \(\mathcal G(.,s^0):C^0([0,T])\to C^0([0,T])\) is defined. Let \(\beta:[\hat {s},1]\to \mathbb R,\;0<\hat{s}<1 \) be an unbounded maximal monotone graph. The ordinary differential inclusion \(\quad \alpha \dfrac{ds}{dt}+\mathcal G(s,s^0)+\beta(s)\ni u\) a.e. in \([0,T]\), \(s(0)=s^0\) is considered. If the operator \(\mathcal G\) is bounded and Lipschitz continuous, then for any \(u\in L^2(0,T)\) there exists its unique solution \(s\in H^1(0,T)\). This defines the Lipschitz continuous solution operator \(\mathcal F:L^2(0,T)\to H^1(0,T)\). This result is applied to solving the weak formulation of the filtration problem in a form NEWLINE\[NEWLINE\iint_Q [s_t(u-v)+k(s)\nabla(u+\rho gz).\nabla (u-v)] dx dt\leq 0, \;v\in K,\;\alpha s_t+\mathcal G(s,s^0)+\beta(s)\ni u \text{ a.e. in } Q.NEWLINE\]NEWLINE The admissible functions in the variational inequality are from the convex set \(K=\{u\in L^2(0,T;H^1(\Omega)):(\gamma_0u)^+=P\) on \(\Gamma \times (0,T)\}\), where \(\gamma_0\) is the trace operator and \(P\in C^0([0,T]\), \(H^1(\Omega))\cap H^1(0,T;L^2(\Omega))\), \(P\geq 0.\) The operator \(\mathcal G +\beta\) is the inverse of a hysteresis operator \(\mathcal F\) which satisfies the property NEWLINE\[NEWLINE\int_0^T[v(.,t)-v(.,0)]\dfrac {\partial \mathcal F(v)}{\partial t} dt \geq 0\quad \text{a.e. in} \Omega,\;v:Q\to \mathbb R \text{ suitable regular}.NEWLINE\]NEWLINE Using the approximation by an implicit time discretization the existence theorem for the solution \((u,s)\) is verified.