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Lack of uniqueness for weak solutions of the incompressible porous media equation. (English) Zbl 1241.35156
The authors study existence of weak solutions for incompressible two-dimensional porous media system \[ \partial_t\rho + \nabla \cdot (v\rho ) = 0, \;\;\nabla \cdot v = 0, \;\;v=-\nabla p-(0,\rho ), \] where \(\rho (x,t)\) is the density, \(v\) the velocity, and \(p\) the pressure of the fluid. The weak formulation of this system is considered. Some non-trivial solutions with \(\rho ,v \in L^{\infty }(\mathbb{T}^2\times [0,T])\) under the initial condition \(\rho (x,0)=0\) are constructed. Here \(\mathbb{T}^2\) is the two-dimensional flat torus. The weak solutions satisfy \[ \lim\limits_{t\to0^{+}}\sup \|\rho \|_{H^s(t)}=+\infty \] for any \(s>0\). From Darcy’s law and the incompressibility of the fluid it follows that the velocity can be represented as a singular integral operator w.r.t. the density with a kernel of Calderon-Zygmund type. By using the De Lellis-Székelyhidi method it is shown the existence of non-unique \(L^{\infty }\)-solutions (w.r.t. space and time). The analytic perturbation property of the solutions is discussed as well.

MSC:
35Q35 PDEs in connection with fluid mechanics
76S05 Flows in porous media; filtration; seepage
35D30 Weak solutions to PDEs
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