Stability of solutions to generalized Forchheimer equations of any degree
The generalized Forchheimer equation \(g(|u|)u = -\nabla p\) is considered, where \(u\) is the velocity of the fluid in a porous medium, \(\nabla p\) is the pressure gradient, and \(g(s)\) is a polynomial of any degree. This equation can be inverted to \(u = -K(|\nabla p|) \nabla p\). The authors study the initial value problem for the parabolic differential equation in the form \[ \frac{\partial p}{\partial t} = \nabla \cdot (K(|\nabla p|) \nabla p). \] They show that the solution of this equation continuously depends on the boundary data and the coefficients of the Forchheimer polynomial in the spaces \(L^{\alpha}\), \(\alpha \geq 1\), and \(W^{1,2-a}\), \(a = \frac{\mathrm{deg}(g)}{(1 + \mathrm{deg}(g))}\). The results are obtained for the DC case, when the technical degree condition \(\mathrm{deg}(g) \leq \frac{n}{n - 2}\) holds, and for the opposite NDC case.
scientific article; zbMATH DE number 6532741
scientific article