Regularity Properties of Degenerate Diffusion Equations with Drifts

Kim, In-Won; Zhang, Yuming Paul

doi:10.1137/17m1159749

Cited by 17 publications

(15 citation statements)

References 16 publications

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“…The proof is based on the method of intrinsic scaling introduced by DiBenedetto for the porous medium equation [ DiB79 , Urb08 ]. It is also similar in spirit to the proof in [ KZ18 ] where regularity was proved for a degenerate diffusion equation posed on with a potentially singular drift term. We also direct the readers to [ HZ19 ] where Hölder regularity was proven for drift-diffusion equations with sharp conditions on the drift term using a different strategy of proof.…”

Section: Existence and Regularity Of Solutionssupporting

confidence: 63%

Phase Transitions for Nonlinear Nonlocal Aggregation-Diffusion Equations

Carrillo

Gvalani

2021

Commun. Math. Phys.

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We are interested in studying the stationary solutions and phase transitions of aggregation equations with degenerate diffusion of porous medium-type, with exponent $$1< m < \infty $$ 1 < m < ∞ . We first prove the existence of possibly infinitely many bifurcations from the spatially homogeneous steady state. We then focus our attention on the associated free energy, proving existence of minimisers and even uniqueness for sufficiently weak interactions. In the absence of uniqueness, we show that the system exhibits phase transitions: we classify values of m and interaction potentials W for which these phase transitions are continuous or discontinuous. Finally, we comment on the limit $$m \rightarrow \infty $$ m → ∞ and the influence that the presence of a phase transition has on this limit.

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Section: Existence and Regularity Of Solutionssupporting

confidence: 63%

Phase Transitions for Nonlinear Nonlocal Aggregation-Diffusion Equations

Carrillo

Gvalani

2021

Commun. Math. Phys.

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“…which can be done by means of several results in the literature for nonlinear-diffusion and transport equations, see [5,48]. In the next step we use the densities ρ i obtained above and evaluated at time t 1 as initial data for the the following hyperbolic equations…”

Section: Null Results: What We Tried and Did Not Workmentioning

confidence: 99%

Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts

Alasio¹,

Bruna²,

Fagioli³

et al. 2021

Preprint

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We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate-parabolic PDEs with self-and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of species with quadratic porous-medium interactions in a bounded domain Ω in any spatial dimension. The cross interactions between different species are scaled by a parameter δ < 1, with the δ = 0 case corresponding to no interactions across species. A smallness condition on δ ensures existence of solutions up to an arbitrary time T > 0 in a subspace of L 2 (0, T ; H 1 (Ω)). This is shown via a Schauder fixed point argument for a regularised system combined with a vanishing diffusivity approach. The behaviour of solutions for extreme values of δ is studied numerically.

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“…[8,16,35], etc). Concerning free boundary regularity of (1.1), there are recent developments of C 1,α -regularity assuming enough smoothness on V and ρ 0 ∈ L 1 (R d ) ∩ L ∞ (R d ) in [33,34] and references therein for relavant works when V = 0. The large-time asymptotics and regularity of nonlinear diffusion equations with drifts are also analyzed as well: for example, the rate of decay to equilibrium of self-similar solutions in [12], viscosity solutions, equivalent to weak solutions, and their aymptotic convergence of the free boundary to the equilibrium in [32], and Hölder continuous solutions for diffusion-aggregation equations with singular interaction kernels in [51].…”

Section: Introductionmentioning

confidence: 99%

“…The drift may affect the well-posedness of diffusive parabolic equations. There are works showing counterexamples of lack of regularities under certain conditions on the drift, for example, [43,46,52] for (fractional) linear diffusion and [27,33] for porous medium equations. From the perspective of critical conditions on V providing continuity of (1.1) (for example, [14,27,33]), we recall…”

Section: Introductionmentioning

confidence: 99%

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Existence of weak solutions for Porous medium equation with a divergence type of drift term

Hwang¹,

Kang²,

Kim³

2021

Preprint

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We consider degenerate porous medium equations with a divergence type of drift terms. We establish the existence of L q -weak solutions (satisfying energy estimates or even further with moment and speed estimates in Wasserstein spaces), in case the drift term belongs to a sub-scaling (including scaling invariant) class depending on q and m caused by nonlinear structure of diffusion, which is a major difference compared to that of a linear case. It is noticeable that the classes of drift terms become wider if the drift term is divergence-free. Similar conditions of gradients of drift terms is also provided to ensure the existence of such weak solutions. Uniqueness results follow under an additional condition on the gradients of the drift terms with the aid of methods developed in Wasserstein spaces. One of our main tools is so called the splitting method to construct a sequence of approximated solutions, which implies, by passing to the limit, the existence of weak solutions satisfying not only an energy inequality but also moment and speed estimates. One of crucial points in the construction is uniform Hölder continuity up to initial time for homogeneous porous medium equations, which seems to be of independent interest. As an application, we improve a regularity result for solutions of a repulsive Keller-Segel system of porous medium type.

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Regularity Properties of Degenerate Diffusion Equations with Drifts

Cited by 17 publications

References 16 publications

Phase Transitions for Nonlinear Nonlocal Aggregation-Diffusion Equations

Phase Transitions for Nonlinear Nonlocal Aggregation-Diffusion Equations

Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts

Existence of weak solutions for Porous medium equation with a divergence type of drift term

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