Cauchy Problem and Exponential Stability for the Inhomogeneous Landau Equation

Carrapatoso, Kleber; Tristani, Isabelle; Wu, Kaidi

doi:10.1007/s00205-015-0963-x

Cited by 57 publications

(69 citation statements)

References 27 publications

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“…The methods of Guo have moreover inspired many subsequent perturbative results for various kinetic models [39,40,57,58,59,42,60], including the remarkable works of the global nonlinear stability of Maxwellians for the non-cutoff Boltzmann equation [36,3,4,2]. See also [19,20,21] for more recent results on near-Maxwellian solutions.…”

Section: Regularity Theory For Landau Equationmentioning

confidence: 99%

Stability of Vacuum for the Landau Equation with Moderately Soft Potentials

Luk

2019

Ann. PDE

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Consider the spatially inhomogeneous Landau equation with moderately soft potentials (i.e. with γ ∈ (−2, 0)) on the whole space R 3 . We prove that if the initial data fin are close to the vacuum solution fvac ≡ 0 in an appropriate norm, then the solution f remains regular globally in time. This is the first stability of vacuum result for a binary collisional model featuring a long-range interaction.Moreover, we prove that the solutions in the near-vacuum regime approach solutions to the linear transport equation as t → +∞. Furthermore, in general, solutions do not approach a traveling global Maxwellian as t → +∞.Our proof relies on robust decay estimates captured using weighted energy estimates and the maximum principle for weighted quantities. Importantly, we also make use of a null structure in the nonlinearity of the Landau equation which suppresses the most slowly-decaying interactions. * jluk@stanford.edu 1 Each of these terms depends also on (t, x). For brevity, we have suppressed these dependence in (1.2).

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Section: Regularity Theory For Landau Equationmentioning

confidence: 99%

Stability of Vacuum for the Landau Equation with Moderately Soft Potentials

Luk

2019

Ann. PDE

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“…Theorem 13. Let f be the weak solution of (60) with initial-boundary value conditions (61), which satisfies the conservation laws (7), and (8) if Ω has a rotational symmetry. Suppose that g ∞ < for some > 0.…”

Section: Technical Lemmasmentioning

confidence: 99%

“…Assume g ∞ < for some > 0. Let f be a weak solution of (60) and (14) with (7) and (8). Then there exist a constant 0 < δ ≤ 1/4 and a function 0 ≤ η(t) ≤ C f (t) 2 2 , such that…”

Section: Technical Lemmasmentioning

confidence: 99%

The Landau Equation with the Specular Reflection Boundary Condition

Guo

Hwang

Jang

et al. 2020

Arch Rational Mech Anal

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The existence and stability of the Landau equation (1936) in a general bounded domain with a physical boundary condition is a long-outstanding open problem. This work proves the global stability of the Landau equation with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. The highlight of this work also comes from the low-regularity assumptions made for the initial distribution. This work generalizes the recent global stability result for the Landau equation in a periodic box [42]. Our methods consist of the generalization of the wellposedness theory for the Fokker-Planck equation [37, 38] and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations [27] and the Morrey estimates [56] to further control the velocity derivatives, which ensures the uniqueness. Our methods provide a new understanding of the grazing collisions in the Landau theory for an initial-boundary value problem. 2010 Mathematics Subject Classification. Primary: 35Q84, 35Q20, 82C40, 35B45, 34C29, 35B65.

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“…From all the estimates obtained in Section 3.1.1, following [19,8] we can then obtain the corresponding nonlinear estimates (that is, in the same spaces) of Section 3.1.2 and 3.1.3.…”

Section: Functional Spaces and Main Estimatesmentioning

confidence: 99%