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Continuity of the saturation in the flow of two immiscible fluids in a porous medium. (English) Zbl 1246.35110
The authors study the weakly coupled system \[ \begin{aligned} & v_t - \text{div }[A(v)\nabla v + \vec B(v)] = \vec V\cdot \nabla C(v) \quad \text{in } E_T = E\times (0,T],\\ &\text{div }\vec V = 0, \end{aligned} \] where \(E \subset \mathbb R^N\) is a bounded domain with boundary \(\partial E\) of class \(C^1\). This system is separated into a parabolic equation for \(v\), and an elliptic equation for \(u\), and it arises in the theory of flow of immiscible fluids in a porous medium. The unknown functions \(u\) and \(v\) and the equations above represent the pressure and the saturation respectively due to the Kruzkov-Sukorjanski transformation, subject to Darcy’s law and the Buckley-Leverett coupling.
Under suitable structural assumptions on \(A\), \(\vec B\) and \(C\), the authors prove that if \((u,v)\) satisfy the above-mentioned system locally in the weak sense, then the saturation \(v\) is a locally continuous function in \(E_T\), irrespective of the nature of the degeneracy of the principal part of the system. Moreover, for every compact subset \(K_T \subset E_T\), there exists a continuous function \(\omega(\cdot)\) that can be determined a priori only in terms of the data and the distance from \(K_T\) to the parabolic boundary of \(E_T\), such that \(\omega(0) = 0\), and \[ |v(x_1,t_1) - v(x_2,t_2)| \leq \omega(|x_1 - x_2| + |t_1 - t_2|^{1/2}) \] for \((x_i,t_i) \in K_T\).

35K65 Degenerate parabolic equations
35B65 Smoothness and regularity of solutions to PDEs
35B45 A priori estimates in context of PDEs
35Q35 PDEs in connection with fluid mechanics
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