## On distributional solutions of local and nonlocal problems of porous medium type. (Sur des solutions distributionnelles de problèmes locaux et non locaux de type milieux poreux.)(English. Abridged French version)Zbl 1382.35243

The authors develop well-posedness and a priori estimates ($$L^1$$ contraction, comparison, $$L^p$$ estimates, energy estimate, time regularity and conservation of mass) for bounded distributional solutions of the nonlinear and nonlocal evolution equation $$u_t-{\mathcal L}^{\sigma,\mu}\varphi(u)=g(x,t)$$ where $$\varphi$$ is merely continuous and nondecreasing, and $${\mathcal L}^{\sigma,\mu}$$ is the generator of a general symmetric Lévy process. New uniqueness results for bounded distributional solutions to this problem and the corresponding elliptic equation are shown. A key role is played by a new Liouville type property for $${\mathcal L}^{\sigma,\mu}$$. Existence and a priori estimates are deduced from a numerical approximation, and energy-type estimates are also obtained.

### MSC:

 35Q35 PDEs in connection with fluid mechanics 35K65 Degenerate parabolic equations 76S05 Flows in porous media; filtration; seepage 35B45 A priori estimates in context of PDEs 60G51 Processes with independent increments; Lévy processes
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### References:

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