Consider the Landau equation with Coulomb potential in a periodic box. We develop
a new L2 to L∞ framework to construct global unique solutions near Maxwellian with
small L∞ norm. The first step is to establish global L2 estimates with strong velocity
weight and time decay, under the assumption of L∞ bound, which is further controlled
by such L2 estimates via De Giorgi’s method (Golse et al. in Ann. Sc. Norm. Super. Pisa
Cl. Sci. (5) 19(1), 253–295 (2019), Imbert and Mouhot in arXiv :1505.04608 (2015)).
The second step is to employ estimates in Sp spaces to control velocity derivatives to
ensure uniqueness, which is based on Hölder estimates via De Giorgi’s method (Golse
et al. in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19(1), 253–295 (2019), Golse and Vasseur
in arXiv :1506.01908 (2015), Imbert and Mouhot in arXiv :1505.04608 (2015)).
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Global Hilbert Expansion for the Relativistic Vlasov–Maxwell–Boltzmann System
Consider the relativistic Vlasov–Maxwell–Boltzmann system describing the
dynamics of an electron gas in the presence of a fixed ion background. Thanks to recent
works Germain and Masmoudi (Ann Sci Éc Norm Supér 47(3):469–503, 2014), Guo
et al. (J Math Phys 55(12):123102, 2014) and Deng et al. (Arch Ration Mech Anal
225(2):771–871, 2017), we establish the global-in-time validity of its Hilbert expansion
and derive the limiting relativistic Euler–Maxwell system as the mean free path goes
to zero. Our method is based on the L2 − L∞ framework and the Glassey–Strauss
Representation of the electromagnetic field, with auxiliary H1 estimates and W1,∞
estimates to control the characteristic curves and corresponding L∞ norm.
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- Award ID(s):
- 1810868
- NSF-PAR ID:
- 10330710
- Date Published:
- Journal Name:
- Communications in mathematical physics
- Volume:
- 384
- ISSN:
- 1432-0916
- Page Range / eLocation ID:
- 341–401
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00220-021-04079-x
