Analysis and mean-field derivation of a porous-medium equation with fractional diffusion

Chen, Li; Holzinger, Alexandra; Jüngel, Ansgar; Zamponi, Nicola

doi:10.1080/03605302.2022.2118608

Cited by 4 publications

(3 citation statements)

References 44 publications

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“…Several forms of nonlocal extensions of the porous medium equation (PME) have been discussed in the literature, see, for example, [7,11,19]. For derivations of nonlocal PMEs from microscopic dynamcis we refer to [12,15]. Many works have been devoted to the analysis of the asymptotic behavior and apriori estimates, see, for example [9,50] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Optimal regularity in time and space for the porous medium equation

2020

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A broad class of possibly non-unique generalized kinetic solutions to hyperbolicparabolic PDEs is introduced. Optimal regularity estimates in time and space for such solutions to nonlocal, and spatially inhomogeneous variants of the porous medium equation are shown in the scale of Sobolev spaces. The optimality of these results is shown by comparison to the non-local Barenblatt solution. The regularity results are used in order to obtain existence of generalized kinetic solutions.

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Section: Introductionmentioning

confidence: 99%

Optimal regularity in time and space for the porous medium equation

2020

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“…The rigorous derivation of Vlasov–Poisson–Fokker–Planck through mean field limit was recently shown in earlier studies 8,9 . In the last decades, there have been many notable works in the derivation of one particle effective equation through the mean field limit of first‐ (or second‐) order particle systems (in both deterministic or stochastic setting), for example, in earlier research 10–17 to name a few. However, due to the singular coupling of the particle system on the right‐hand side of the momentum equation, a rigorous derivation of the whole Vlasov–Poisson–Fokker–Planck–Navier–Stokes system is still missing.…”

Section: Introductionmentioning

confidence: 94%

Global weak solutions to the Vlasov–Poisson–Fokker–Planck–Navier–Stokes system

Chen

et al. 2022

Math Methods in App Sciences

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We consider the compressible Vlasov-Poisson-Fokker-Planck-Navier-Stokes system in a three-dimensional bounded domain with nonhomogeneous Dirichlet boundary conditions. The system describes the evolution of charged particles ensemble dispersed in an isentropic fluid. For the adiabatic coefficient 𝛾 > 3∕2, we establish the global existence of weak solutions to this system with arbitrary large initial and boundary data.

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“…For s ∈ (0, 1) and ε = 0, it is called fractional porous medium equation and was studied in [4] and [69]. For ε > 0 and s ∈ [0, 1], it was considered as the viscosity approximation of fractional porous medium equation in [64] and [20]. In particular, our results can be applied to the above equation for ε > 0.…”

Section: Introductionmentioning

confidence: 96%

Second order fractional mean-field SDEs with singular kernels and measure initial data

Hao¹,

Röckner²,

Zhang³

2023

Preprint

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In this paper we establish the local and global well-posedness of weak and strong solutions to second order fractional mean-field SDEs with singular/distribution interaction kernels and measure initial value, where the kernel can be Newton or Coulomb potential, Riesz potential, Biot-Savart law, etc. Moreover, we also show the stability, smoothness and the short time singularity and large time decay estimates of the distribution density. Our results reveal a phenomenon that for nonlinear mean-field equations, the regularity of the initial distribution could balance the singularity of the kernel. The precise relationship between the singularity of kernels and the regularity of initial value are calculated, which belongs to the subcritical regime in scaling sense. In particular, our results provide microscopic probability explanation and establish a unified treatment for many physical models such as fractional Vlasov-Poisson-Fokker-Planck system, the vorticity formulation of 2D-fractal Navier-Stokes equations, surface quasi-geostrophic models, fractional porous medium equation with viscosity, etc.

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Analysis and mean-field derivation of a porous-medium equation with fractional diffusion

Cited by 4 publications

References 44 publications

Optimal regularity in time and space for the porous medium equation

Optimal regularity in time and space for the porous medium equation

Global weak solutions to the Vlasov–Poisson–Fokker–Planck–Navier–Stokes system

Second order fractional mean-field SDEs with singular kernels and measure initial data

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