arXiv subjects

Ling-Bing He

Publications and source records attributed to Ling-Bing He.

26 records · Page 2Linked to original sources

arXiv2021

Asymptotic analysis of the linearized Boltzmann collision operator from angular cutoff to non-cutoff

We give quantitative estimates on the asymptotics of the linearized Boltzmann collision operator and its associated equation from angular cutoff to non cutoff. On one hand, the results disclose the link between the hyperbolic property resulting from the Grad's cutoff assumption and the smoothing property due to the long-range interaction. On the other hand, with the help of the localization techniques in the phase space, we observe some new phenomenon in the asymptotic limit process. As a consequence, we give the affirmative answer to the question that there is no jump for the property that the collision operator with cutoff does not have the spectrum gap but the operator without cutoff does have for the moderate soft potentials.

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arXiv2021

On semi-classical limit of spatially homogeneous quantum Boltzmann equation: weak convergence

It is expected in physics that the homogeneous quantum Boltzmann equation with Fermi-Dirac or Bose-Einstein statistics and with Maxwell-Boltzmann operator (neglecting effect of the statistics) for the weak coupled gases will converge to the homogeneous Fokker-Planck-Landau equation as the Planck constant $\hbar$ tends to zero. In this paper and the upcoming work \cite{HLP2}, we will provide a mathematical justification on this semi-classical limit. Key ingredients into the proofs are the new framework to catch the {\it weak projection gradient}, which is motivated by Villani \cite{V1} to identify the $H$-solution for Fokker-Planck-Landau equation, and the symmetric structure inside the cubic terms of the collision operators.

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arXiv2020

A new monotonicity formula for the spatially homogeneous Landau equation with Coulomb potential and its applications

We describe a time-dependent functional involving the relative entropy and the $\dot{H}^1$ seminorm, which decreases along solutions to the spatially homogeneous Landau equation with Coulomb potential. The study of this monotone functionial sheds light on the competition between the dissipation and the nonlinearity for this equation. It enables to obtain new results concerning regularity/blowup issues for the Landau equation with Coulomb potential.

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arXiv2020

Boltzmann equation with cutoff Rutherford scattering cross section near Maxwellian

The well-known Rutherford differential cross section, denoted by $ dΩ/dσ$, corresponds to a two body interaction with Coulomb potential. It leads to the logarithmically divergence of the momentum transfer (or the transport cross section) which is described by $$\int_{{\mathbb S}^2} (1-\cosθ) \frac{dΩ}{dσ} dσ\sim \int_0^π θ^{-1}dθ. $$ Here $θ$ is the deviation angle in the scattering event. Due to screening effect, physically one can assume that $θ_{\min}$ is the order of magnitude of the smallest angles for which the scattering can still be regarded as Coulomb scattering. Under ad hoc cutoff $θ\geq θ_{\min}$ on the deviation angle, L. D. Landau derived a new equation in \cite{landau1936transport} for the weakly interacting gas which is now referred to as the Fokker-Planck-Landau or Landau equation. In the present work, we establish a unified framework to justify Landau's formal derivation in \cite{landau1936transport} and the so-called Landau approximation problem proposed in \cite{alexandre2004landau} in the close-to-equilibrium regime. Precisely, (i). we prove global well-posedness of the Boltzmann equation with cutoff Rutherford cross section which is perhaps the most singular kernel both in relative velocity and deviation angle. (ii). we prove a global-in-time error estimate between solutions to Boltzmann and Landau equations with logarithm accuracy, which is consistent with the famous Coulomb logarithm. Key ingredients into the proofs of these results include a complete coercivity estimate of the linearized Boltzmann collision operator, a uniform spectral gap estimate and a novel linear-quasilinear method.

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arXiv2017

Sharp bounds for Boltzmann and Landau collision operators

The aim of the work is to provide a stable method to get sharp bounds for Boltzmann and Landau operators in weighted Sobolev spaces and in anisotropic spaces. All the sharp bounds are given for the original Boltzmann and Landau operators. The sharpness means the lower and upper bounds for the operators are consistent with the behavior of the linearized operators. Moreover, we make clear the difference between the bounds for the original operators and those for the linearized ones. According to the Bobylev's formula, we introduce two types of dyadic decompositions performed in both phase and frequency spaces to make full use of the interaction and the cancellation. It allows us to see clearly which part of the operator behaves like a Laplace type operator and which part is dominated by the anisotropic structure. It is the key point to get the sharp bounds in weighted Sobolev spaces and in anisotropic spaces. Based on the geometric structure of the elastic collision, we make a geometric decomposition to capture the anisotropic structure of the collision operator. More precisely, we make it explicit that the fractional Laplace-Beltrami operator really exists in the structure of the collision operator. It enables us to derive the sharp bounds in anisotropic spaces and then complete the entropy dissipation estimates. The structures mentioned above are so stable that we can apply them to the rescaled Boltzmann collision operator in the process of the grazing collisions limit. Then we get the sharp bounds for the Landau collision operator by passing to the limit. We remark that our analysis used here will shed light on the investigation of the asymptotics from Boltzmann equation to Landau equation.

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arXiv2017

On the global dynamics of the inhomogeneous Boltzmann equations without angular cutoff: Hard potentials and Maxwellian molecules

This is the first one of two papers on the global dynamics of the original Boltzmann equations without angular cutoff on the torus. We address the problem for the hard potentials and Maxwellian molecules in the present paper. The case of soft potentials is left to a forthcoming paper. The key to solve the problem is the energy-entropy method which characterizes the propagation of the regularity, $H$-theorem and the interplay between the energy and the entropy. Our main results are as follows: (i) We present a unified framework to prove the well-posedness for the original Boltzmann equation for both angular cutoff and without cutoff in weighted Sobolev spaces with polynomial weights. As a consequence, we obtain an explicit formula for the asymptotics of the equation from angular cutoff to non-cutoff. (ii) We describe the global dynamics of the equation under the almost optimal assumption on the solution which makes sure that the Boltzmann collision operator behaves like a fractional Laplace operator for the velocity variable. More precisely, we obtain the propagation of the regularity for the solution and a new mechanism for the convergence of the solution to its equilibrium with quantitative estimates. (iii) We prove that any global and smooth solution to the equation is stable, i.e., any perturbed solution will remain close to the reference solution if initially they are close to each other.

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arXiv2017

High order approximation for the Boltzmann equation without angular cutoff

In order to solve the Boltzmann equation numerically, in the present work, we propose a new model equation to approximate the Boltzmann equation without angular cutoff. Here the approximate equation incorporates Boltzmann collision operator with angular cut-off and the Landau collision operator. As a first step, we prove the well-posedness theory for our approximate equation. Then in the next step we show the error estimate between the solutions to the approximate equation and the original equation. Compared to the standard angular cut-off approximation method, our method results in higher order of accuracy.

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arXiv2016

On global dynamics of three dimensional magnetohydrodynamics: nonlinear stability of Alfvén waves

We construct and study global solutions for the 3-dimensional incompressible MHD systems with arbitrary small viscosity. In particular, we provide a rigorous justification for the following dynamical phenomenon observed in many contexts: the solution initially behaves like non-dispersive waves and the shape of the solution persists for a very long time (proportional to the Reynolds number), thereafter, the solution will be damped due to the long-time accumulation of the diffusive effects, eventually, the total energy of the system becomes extremely small compared to the viscosity so that the diffusion takes over and the solution afterwards decays fast in time. We do not assume any condition on the symmetry or on the vorticity. The size of data and the a priori estimates do not depend on viscosity. The proof is builded upon a novel use of the basic energy identity and a geometric study of the characteristic hypersurfaces. The approach is partly inspired by Christodoulou-Klainerman's proof of the nonlinear stability of Minkowski space in general relativity.

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