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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 $$(x_i,t_i) \in K_T$$.

##### MSC:
 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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