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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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