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On A(p) weights and the Landau equation
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). George Washington Univ, Dept Math, Washington, DC 20052 USA.
Univ Massachusetts, Dept Math, Amherst, MA 01003 USA..
2019 (English)In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 58, no 1, article id 17Article in journal (Refereed) Published
Abstract [en]

In this manuscript we investigate the regularization of solutions for the spatially homogeneous Landau equation. For moderately soft potentials, it is shown that weak solutions become smooth instantaneously and stay so over all times, and the estimates depend only on the initial mass, energy, and entropy. For very soft potentials we obtain a conditional regularity result, hinging on what may be described as a nonlinear Morrey space bound, assumed to hold uniformly over time. This bound always holds in the case of very soft potentials, and nearly holds for general potentials, including Coulomb. This latter phenomenon captures the intuition that for very soft potentials, the dissipative term in the equation is of the same order as the quadratic term driving the growth (and potentially, singularities). In particular, for the Coulomb case, the conditional regularity result shows a rate of regularization much stronger than what is usually expected for regular parabolic equations. The main feature of our proofs is the analysis of the linearized Landau operator around an arbitrary and possibly irregular distribution. This linear operator is shown to be a degenerate elliptic Schrodinger operator whose coefficients are controlled by A(p)-weights.

Place, publisher, year, edition, pages
Springer Berlin/Heidelberg, 2019. Vol. 58, no 1, article id 17
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-240692DOI: 10.1007/s00526-018-1451-6ISI: 000452764900002Scopus ID: 2-s2.0-85058027788OAI: oai:DiVA.org:kth-240692DiVA, id: diva2:1277660
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QC 20190111

Available from: 2019-01-11 Created: 2019-01-11 Last updated: 2019-01-11Bibliographically approved

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Gualdani, Maria

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