Global existence of strong solutions to a groundwater flow problem

Abstract

In this paper we study the initial boundary value problem for the system $Δv= u_{x_1},\ u_t-\mbox{div}\left(\left((a|\mathbf{q}|+m)I+(b-a)\frac{\mathbf{q}\otimes\mathbf{q}}{|\mathbf{q}|}\right)\nabla u\right)=-\nabla u\cdot\mathbf{q}$, where $\mathbf{q}=(-v_{x_2},v_{x_1}{)}^T$, $\mathbf{q}\otimes \mathbf{q}=\mathbf{q}{\mathbf{q}}^T$. This problem has been proposed as a model for a fluid flowing through a porous medium under the influence of gravity and hydrodynamic dispersion. For each $T>0$ we obtain a so-called strong solution $(v,u)$ in the function space $L^{\infty }(0,T;{\left(W^{1,\infty }(\Omega )\right)}^2)$, where $\Omega$ is a bounded domain in ${\mathbb{R}}^2$. The key ingredient in our approach is the decomposition $A^2=\mbox{tr} (A)A-\mbox{det}(A) I$ for any $2\times 2$ symmetric matrix $A$. By exploring this decomposition, we are able to derive an equation of parabolic type for the function ${\left(\left((a|\mathbf{q}|+m)I+(b-a)\frac{\mathbf{q}\otimes \mathbf{q}}{|\mathbf{q}|}\right)\nabla u\cdot \nabla u\right)}^j,j\ge 1$. With the aid of this equation we obtain a uniform bound for $\nabla u$.

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