Local solutions of the Landau equation with rough, slowly decaying initial data. (English. French summary) Zbl 1455.35214
Summary: We consider the Cauchy problem for the spatially inhomogeneous Landau equation with soft potentials in the case of large (i.e. non-perturbative) initial data. We construct a solution for any bounded, measurable initial data with uniform polynomial decay in the velocity variable, and that satisfies a technical lower bound assumption (but can have vacuum regions). For uniqueness in this weak class, we have to make the additional assumption that the initial data is Hölder continuous. Our hypotheses are much weaker, in terms of regularity and decay, than previous large-data well-posedness results in the literature. We also derive a continuation criterion for our solutions that is, for the case of very soft potentials, an improvement over the previous state of the art.
MSC:
| 35Q49 | Transport equations |
| 35Q20 | Boltzmann equations |
| 35A09 | Classical solutions to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
Keywords:
Landau equation; large data well-posedness; vacuum regions; classical solutions; kinetic equationsReferences:
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