Special Session 136: Analysis and Applications of the Boltzmann equation

Thermalization rate for solutions to the Landau-Fermi-Dirac equation

Ricardo Alonso Texas A&M USA Co-Author(s):    V. Bagland, L. Desvillettes, B. Lods

Abstract: In this talk we discuss the essential properties of the LFD equation with focus on the thermal convergence of solutions to the Fermi Dirac distribution for moderately soft potentials. The key point of the analysis, valid only under a weak saturation regime, is to obtain estimates free of the Planck`s quantum parameter. Using such estimates, algebraic rate of convergence is obtained after applying a recent entropy - entropy dissipation inequality for the equation.

Short- and long-time behavior in evolution equations: the role of the hypocoercivity index

Anton Arnold Vienna University of Technology Austria Co-Author(s):

Abstract: The index of hypocoercivity is defined via a coercivity-type estimate for the self-adjoint/skew-adjoint parts of the generator, and it quantifies `how degenerate` a hypocoercive evolution equation is, both for ODEs and for evolutions equations in a Hilbert space. We show that this index characterizes the polynomial decay of the propagator norm for short time and illustrate these concepts for the Lorentz kinetic equation on a torus. This talk is based on joint work with F. Achleitner, E. Carlen, E. Nigsch, and V. Mehrmann.

Well/Ill-posedness separation of the Boltzmann equation with cut-off

Xuwen Chen University of Rochester USA Co-Author(s):    Justin Holmer, Shunlin Shen, Zhifei Zhang

Abstract: We report the finding of the sharp separation of well/ill-posedness for the Boltzmann equation with cut-off using dispersive PDE techniques. The separation is unexpectly 1/2-derivative above scaling and the illposedness is represented by forward in time norm deflation.

The Non-cutoff Boltzmann Equation in Bounded Domains

Dingqun Deng Pohang University of Science and Technology Korea Co-Author(s):

Abstract: In this talk, we will investigate the existence of the non-cutoff Boltzmann equation near a global Maxwellian in a general $C^3$ bounded domain $\Omega$. This includes convex and non-convex cases with inflow or Maxwell reflection boundary conditions. We obtain global-in-time existence, which has an exponential decay rate for both hard and soft potentials. The crucial method is to extend the boundary problem in a bounded domain to the whole space without regular velocity dissipation and to construct an extra damping from the advection operator, followed by the De Giorgi iteration and the $L^2$--$L^\infty$ method.

Polynomial tail solutions for Boltzmann equation in the whole space

Zongguang Li The Hong Kong Polytechic University Hong Kong Co-Author(s):

Abstract: We are concerned with the Cauchy problem on the Boltzmann equation in the whole space. The goal is to study solutions near Maxwellians with the perturbation admitting a polynomial tail in large velocities. We obtain global-in-time bounded mild solutions near global Maxwellians and construct solutions up to any finite time around local Maxwellians whose fluid quantities are classical solutions to the corresponding compressible Euler system around constant states. The main difficulty to be overcome in case of the whole space is the polynomial time decay of solutions which is much slower than the exponential rate in contrast with the torus case.

Analysis and numerical methods for the Boltzmann equation with uncertainties

Liu Liu Chinese University of Hong Kong Peoples Rep of China Co-Author(s):    Kunlun Qi, Xueyu Zhu, Shi Jin

Abstract: In this talk, we will first discuss the hypocoercivity analysis for general space-inhomogeous collisional kinetic problems with uncertainties, on the regularity and long-time behavior of the solution in the random space, proved spectral accuracy and long time (exponential decay) error estimates for the stochastic Galerkin method. For the spatially homogenous Boltzmann equation, we will show the spectral convergence for the numerical system with discrete velocity and uncertainty variables. Regarding numerical simulations for kinetic models with uncertainties, we will introduce the multi-fidelity method and asymptotic-preserving neural network approach, then discuss about how to apply the above analyses to obtain convergence and error estimates for the numerical methods.

The spatially inhomogeneous Vlasov-Nordstr\{o}m-Fokker-Planck system in the intrinsic weak diffusion regime

Shuangqian Liu Central China Normal University Peoples Rep of China Co-Author(s):

Abstract: The spatially homogeneous Vlasov-Nordstr\{o}m-Fokker-Planck system is known to exhibit nontrivial large time behavior, naturally leading to weak diffusion of the Fokker-Planck operator. This weak diffusion, combined with the singularity of relativistic velocity, present a significant challenge for the spatially inhomogeneous counterpart. In this talk, we demonstrate that the Cauchy problem for the spatially inhomogeneous Vlasov-Nordstr\{o}m-Fokker-Planck system, without friction, maintains dynamic stability relative to the corresponding spatially homogeneous system. Our results are twofold: (1) we establish the existence of a unique global classical solution and characterize the asymptotic behavior of the spatially inhomogeneous system using a refined weighted energy method; (2) we directly verify the dynamic stability of the spatially inhomogeneous system within the framework of self-similar solutions.

Precise boundary behavior of the kinetic Fokker-Planck equation

Giacomo Lucertini University of Bologna Italy Co-Author(s):    Christopher Henderson, Weinan Wang

Abstract: We present a study on the boundary behavior of a kinetic Fokker-Planck equation in the half-space with absorbing boundary condition. This equation can be seen as a base model for the study of Boltzmann equations: although the kinetic equation that we are taking into account is linear and present a local diffusion, it shares with the Boltzmann equation an analogue hypoelliptic structure. Our main result is the sharp regularity of the solution at the absorbing boundary and grazing set. The technique is based on a new kinetic Nash-type inequality. This talk is based on a joint work with Christopher Henderson (University of Arizona) and Weinan Wang (University of Oklahoma).

Global regularity for the Rayleigh-Taylor unstable Muskat bubble problem with critical regularity

Robert Strain University of Pennsylvania USA Co-Author(s):

Abstract: This talk concerns the Muskat problem with surface tension, modeling the filtration of two incompressible immiscible fluids in porous media. This non-local and non-linear partial differential equation is a basic mathematical model in petroleum engineering; it was formulated by the petroleum engineer M. Muskat in 1934 to describe the mixture of water into an oil sand. Given its origins and its equivalence with Hele-Shaw flows, the Muskat problem has received a lot of attention from the physics community. We consider the case in which the fluids have different constant densities together with different constant viscosities. The Rayleigh-Taylor condition cannot hold for a closed curve, which makes this situation unstable. In this case the equations are non-local, not only in the evolution system, but also in the implicit relation between the amplitude of the vorticity and the free interface. Among other extra difficulties, no maximum principles are available to bound the amplitude and the slopes of the interface. We prove global in time existence and uniqueness results for medium size initial stable data in critical functional spaces. In particular we prove for the first time the global in time stability of star shaped bubbles influenced by Gravity. This is joint work with Gancedo, Garcia-Juarez, and Patel.

Interplay of inertia and rarefaction in weakly nonlinear rarefied gas flow

Satoshi Taguchi Kyoto University Japan Co-Author(s):

Abstract: In this talk, we discuss the asymptotic behavior of a rarefied flow, governed by the Boltzmann equation, in the weakly nonlinear regime where both the Reynolds number and the Knudsen number are small. Specifically, we first address boundary-value problems of the Boltzmann equation and apply the Hilbert expansion for small Knudsen numbers (i.e., scaled mean free path) to derive a fluid-dynamic system in the case where the Reynolds number is of the same order as the Knudsen number. Using the matched asymptotic expansion, this system is then applied to analyze the flow past a sphere, providing insights into gas behavior around the sphere. In particular, we derive a drag formula that accounts for both rarefaction and inertia effects as well as their coupling. The work is in collaboration with Yuki Tatsudani and Tetsuro Tsuji.

Convergence to self-similar solution of the Boltzmann equation with shear flow

Shuaikun WANG Shandong University Peoples Rep of China Co-Author(s):

Abstract: In this report, we consider the wellposeness of the measure valued solution to the solutions to the Boltzmann equation with shear flow under the non-cutoff collision kernel and discusse the long time behavior of the solution. We show that the strong convergence rate to the self-similar solution is exponentially for the asymmetrical initial data.

Classical limit of the relativistic Cucker-Smale model

Qinghua Xiao Innovation Academy for Precision Measurement Science and Technology, CAS Peoples Rep of China Co-Author(s):    Seung-Yeal Ha; Tommaso Ruggeri

Abstract: We study quantitative estimates for the flocking and uniform-in-time classical limit to a relativistic Cucker-Smale model (in short RCS) with non-zero pressure, which is a subsystem of the relativistic thermomechanical Cucker-Smale model. For this RCS model, we provide its uniform-in-time classical limit with optimal convergence rate which is the same as in finite-in-time classical limit. As its direct application, for the corresponding mean-field limit equation, relativistic kinetic Cucker-Smale equation (in short RKCS), the uniform-in-time classical limit is also obtained.

The stability of the Boltzmann equation with deformation

Anita Yang The Chinese University of Hong Kong Hong Kong Co-Author(s):    Shuangqian Liu, Yating Wang, Xueying Zhang

Abstract: In this talk, we will present the existence and long-time behavior of the Boltzmann equation with a deformation matrix $A$, which describes the shear flow. Assuming the shear rate is small, we will study the corresponding Cauchy problem under the cutoff assumption. We will consider two cases. In the case where $tr A < 0$, we will prove the well-posedness in the case of hard potentials. In the case where $tr A \geq 0$, we will specifically consider the planar shear flow case and establish the stability of the stationary solution when the collision kernel is limited to the Maxwell molecules. This is a joint work with Prof. Shuangqian Liu, Yating Wang and Xueying Zhang.

KdV limit for the Vlasov-Poisson-Landau system

Dongcheng Yang South China University of Technology Peoples Rep of China Co-Author(s):

Abstract: In this talk, we are concerned with the fluid limit to KdV equations for the one-dimensional Vlasov-Poisson-Landau system which describes the dynamics of ions in plasma with the electron density determined by the self-consistent electric potential through the so-called Boltzmann relation. Formally, it is well known that as the Knudsen number $\varepsilon\to 0$ the Vlasov-Poisson-Landau system in the compressible scaling converges to the Euler-Poisson equations which further under the Gardner-Morikawa transformation $$ (t,x)\to (\delta^{\frac{3}{2}}t,\delta^{\frac{1}{2}}(x-\sqrt{\frac{8}{3}}t)) $$ converge to the KdV equations as the parameter $\delta\to 0$. Our goal of this paper is to construct smooth solutions of the correspondingly rescaled Vlasov-Poisson-Landau system over an arbitrary finite time interval that can converge uniformly to smooth solutions of the KdV equations as $\varepsilon\to 0$ and $\delta\to 0$ simultaneously under an extra condition $ \varepsilon^{\frac{2}{3}}\leq \delta\leq \varepsilon^{\frac{2}{5}}$. Moreover, the explicit rate of convergence in $\delta$ is also obtained. The proof is established by an appropriately chosen scaling and an intricate weighted energy method through the macro-micro decomposition around local Maxwellians. We design a $\varepsilon$-$\delta$-dependent high order energy functional to capture the singularity of such fluid limit problem.

The solution of the steady Boltzmann equation

Hongjun Yu School of Mathematical Sciences, South China Normal University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we will talk about some existence and stability results of the solution to the one-dimension steady Boltzmann equation.

Diffusive limit of one-species VPB and VMB with angular cutoff

Fujun Zhou South China University of Technology Peoples Rep of China Co-Author(s):

Abstract: In this talk, we give recent progress on diffusive limit of one-species VPB and VMB with angular cutoff. For one-species VPB, by a newly H^2-W^{2,\infty} framework, the time decay estimate and the weighted energy estimate, we justify the incompressible Navier-Stokes-Fourier Poisson (NSFP) limit for all potentials \gamma>-3. For one-species VMB, by employing the weighted energy method with two newly introduced weight functions, we justify the incompressible Navier-Stokes-Fourier-Maxwell (NSFM) limit for hard potentials \gamma \geq 0.