Breakdown of smoothness for the Muskat problem. (English) Zbl 1293.35234
The authors study the Muskat equation, which desribes the evolution of the interface separating two incompressible fluids in a porous medium. They show the existence of a solution, defined for \(t \in [t_0, t_2]\), with the following properties:
“A. At time \(t_0\), the interface is a graph of a function. B. At some time \(t_1 \in (t_0, t_2)\), the interface is no longer a graph. C. For each time \(t \in [t_0, t_2)\), the interface is real analytic. D. At time \(t_2\), the interface is \(C^3\) smooth but not \(C^4\) smooth.” (From the introduction)
In order to prove the main result, the authors construct Muskat solutions analytic on a specially chosen domain \(\Omega(t) = \{|\operatorname{Im} x| < h(\operatorname{Re} x, t) \}\). A small perturbation of the real-analytic “turnover” solution, constructed in [A. Castro et al., Ann. Math. (2) 175, No. 2, 909–948 (2012; Zbl 1267.76033)], is taken as the initial datum.
“A. At time \(t_0\), the interface is a graph of a function. B. At some time \(t_1 \in (t_0, t_2)\), the interface is no longer a graph. C. For each time \(t \in [t_0, t_2)\), the interface is real analytic. D. At time \(t_2\), the interface is \(C^3\) smooth but not \(C^4\) smooth.” (From the introduction)
In order to prove the main result, the authors construct Muskat solutions analytic on a specially chosen domain \(\Omega(t) = \{|\operatorname{Im} x| < h(\operatorname{Re} x, t) \}\). A small perturbation of the real-analytic “turnover” solution, constructed in [A. Castro et al., Ann. Math. (2) 175, No. 2, 909–948 (2012; Zbl 1267.76033)], is taken as the initial datum.
Reviewer: Natalia Bondarenko (Saratov)
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76S05 | Flows in porous media; filtration; seepage |
| 76D27 | Other free boundary flows; Hele-Shaw flows |
| 35A20 | Analyticity in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
Citations:
Zbl 1267.76033References:
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