Title: Global solutions to the Boltzmann equation without angular cutoff and the Landau equation with Coulomb potential
This report succinctly summarizes results proved in the authors' recent work (2019) where the unique existence of solutions to the Boltzmann equation without angular cut-off and the Landau equation with Coulomb potential are studied in a perturbation framework. A major feature is the use of the Wiener space $$A(\Omega)$$, which can be expected to play a similar role to $$L^\infty$$. Compared to the $L^2$-based solution spaces that were employed for prior known results, this function space enables us to establish a new global existence theory. One further feature is that, not only an initial value problem, but also an initial boundary value problem whose boundary conditions can be regarded as physical boundaries in some simple situation, are considered for both equations. In addition to unique existence, large-time behavior of the solutions and propagation of spatial regularity are also proved. In the end of report, key ideas of the proof will be explained in a concise way. more »« less
S. Berhanu
(, Communications in analysis and geometry)
Kefeng Liu
(Ed.)
We establish results on unique continuation at the boundary for the solutions of ∆u = f, f harmonic, and the biharmonic equation ∆^2u = 0. The work is motivated by analogous results proved for harmonic functions by X. Huang et al in [HK1], [HK2], and [HKMP] and by M. S. Baouendi and L. P. Rothschild in [BR1] and [BR2].
Berhanu, S.
(, Communications in analysis and geometry)
Kefeng Liu
(Ed.)
We establish results on unique continuation at the boundary for the solutions of ∆u = f, f harmonic, and the biharmonic equation ∆^2u = 0. The work is motivated by analogous results proved for harmonic functions by X. Huang et al in [HK1], [HK2], and [HKMP] and by M. S. Baouendi and L. P. Rothschild in [BR1] and [BR2].
Guo, Yan; Hwang, Hyung Ju; Jang, Jin Woo; Ouyang, Zhimeng
(, Archive for Rational Mechanics and Analysis)
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 (Kim et al. in Peking Math J, 2020). Our methods consist of the generalization of the wellposedness theory for the Fokker–Planck equation (Hwang et al. SIAM J Math Anal 50(2):2194–2232, 2018; Hwang et al. Arch Ration Mech Anal 214(1):183–233, 2014) 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 (Golse et al. Ann Sc Norm Super Pisa Cl Sci 19(1):253–295, 2019) and the Morrey estimates (Bramanti et al. J Math Anal Appl 200(2):332–354, 1996) 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.
Abstract We establish local well‐posedness in the sense of Hadamard for a certain third‐order nonlinear Schrödinger equation with a multiterm linear part and a general power nonlinearity, known as higher‐order nonlinear Schrödinger equation, formulated on the half‐line . We consider the scenario of associated coefficients such that only one boundary condition is required and hence assume a general nonhomogeneous boundary datum of Dirichlet type at . Our functional framework centers around fractional Sobolev spaces with respect to the spatial variable. We treat both high regularity () and low regularity () solutions: in the former setting, the relevant nonlinearity can be handled via the Banach algebra property; in the latter setting, however, this is no longer the case and, instead, delicate Strichartz estimates must be established. This task is especially challenging in the framework of nonhomogeneous initial‐boundary value problems, as it involves proving boundary‐type Strichartz estimates that are not common in the study of Cauchy (initial value) problems. The linear analysis, which forms the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method (also known as the unified transform) for the associated forced linear problem. In this connection, we note that the higher‐order Schrödinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivatives in the linear part of the equation. This feature manifests itself via several complications throughout the analysis, including (i) analyticity issues related to complex square roots, which require careful treatment of branch cuts and deformations of integration contours; (ii) singularities that emerge upon changes of variables in the Fourier analysis arguments; and (iii) complicated oscillatory kernels in the weak solution formula for the linear initial‐boundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data. The present work provides a first, complete treatment via the Fokas method of a nonhomogeneous initial‐boundary value problem for a partial differential equation associated with a multiterm linear differential operator.
Hu, G.; Li, P.; Zhao, Y.
(, Journal of differential equations)
Consider the elastic scattering of a plane or point incident wave by an unbounded and rigid rough surface. The angular spectrum representation (ASR) for the time-harmonic Navier equation is derived in three dimensions. The ASR is utilized as a radiation condition to the elastic rough surface scattering problem. The uniqueness is proved through a Rellich-type identity for surfaces given by uniformly Lipschitz functions. In the case of flat surfaces with local perturbations, an equivalent variational formulation is deduced in a truncated bounded domain and the existence of solutions are shown for general incoming waves. The main ingredient of the proof is the radiating behavior of the Green tensor to the first boundary value problem of the Navier equation in a half-space.
Duan, Renjun, Liu, Shuangqian, Sakamoto, Shota, and Strain, Robert M. Global solutions to the Boltzmann equation without angular cutoff and the Landau equation with Coulomb potential. Retrieved from https://par.nsf.gov/biblio/10349693. RIMS kokyuroku bessatsu B82.Jun-2020
Duan, Renjun, Liu, Shuangqian, Sakamoto, Shota, & Strain, Robert M. Global solutions to the Boltzmann equation without angular cutoff and the Landau equation with Coulomb potential. RIMS kokyuroku bessatsu, B82 (Jun-2020). Retrieved from https://par.nsf.gov/biblio/10349693.
Duan, Renjun, Liu, Shuangqian, Sakamoto, Shota, and Strain, Robert M.
"Global solutions to the Boltzmann equation without angular cutoff and the Landau equation with Coulomb potential". RIMS kokyuroku bessatsu B82 (Jun-2020). Country unknown/Code not available. https://par.nsf.gov/biblio/10349693.
@article{osti_10349693,
place = {Country unknown/Code not available},
title = {Global solutions to the Boltzmann equation without angular cutoff and the Landau equation with Coulomb potential},
url = {https://par.nsf.gov/biblio/10349693},
abstractNote = {This report succinctly summarizes results proved in the authors' recent work (2019) where the unique existence of solutions to the Boltzmann equation without angular cut-off and the Landau equation with Coulomb potential are studied in a perturbation framework. A major feature is the use of the Wiener space $A(\Omega)$, which can be expected to play a similar role to $L^\infty$. Compared to the $L^2$-based solution spaces that were employed for prior known results, this function space enables us to establish a new global existence theory. One further feature is that, not only an initial value problem, but also an initial boundary value problem whose boundary conditions can be regarded as physical boundaries in some simple situation, are considered for both equations. In addition to unique existence, large-time behavior of the solutions and propagation of spatial regularity are also proved. In the end of report, key ideas of the proof will be explained in a concise way.},
journal = {RIMS kokyuroku bessatsu},
volume = {B82},
number = {Jun-2020},
author = {Duan, Renjun and Liu, Shuangqian and Sakamoto, Shota and Strain, Robert M.},
editor = {Takayoshi Ogawa and Keiichi Kato and Mishio Kawashita}
}
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