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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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[2] de Pablo, A.; Quirós, F.; Rodríguez, A.; Vázquez, J. L., A general fractional porous medium equation, Commun. Pure Appl. Math., 65, 9, 1242-1284, (2012) · Zbl 1248.35220
[3] F. del Teso, J. Endal, E.R. Jakobsen, Numerical methods and analysis for nonlocal (and local) equations of porous medium type, preprint, 2017. · Zbl 1349.35311
[4] del Teso, F.; Endal, J.; Jakobsen, E. R., On the well-posedness of solutions with finite energy for nonlocal equations of porous medium type, (Partial Differential Equations, Mathematical Physics, and Stochastic Analysis. A Volume in Honor of Helge Holden’s 60th Birthday, EMS Ser. Congr. Rep., (2017)), in press
[5] del Teso, F.; Endal, J.; Jakobsen, E. R., Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type, Adv. Math., 305, 78-143, (2017) · Zbl 1349.35311
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