GLOBAL SOLUTION FOR THE SPATIALLY INHOMOGENEOUS NON-CUTOFF KAC EQUATION

Tong YANG; Hongjun YU

doi:10.1137/17M1135955

GLOBAL SOLUTION FOR THE SPATIALLY INHOMOGENEOUS NON-CUTOFF KAC EQUATION

Tong YANG, Hongjun YU

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

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Abstract

This paper is concerned with the Cauchy problem on the one-dimensional inhomogeneous non-cutoff Kac equation. Based on the analysis on the linearized operator obtained in [N. Lerner, Y. Morimoto, K. Pravda-Starov, and C.-J. Xu, J. Funct. Anal., 269 (2015), pp. 459-535], we first prove the existence of a global solution to the equation around a global Maxwellian by combining two sets of macro-micro decomposition. Then by using the dissipative norm of the linearized operator in the fractional Hermite-Sobolev space and using the perturbation theory, the spectrum structure of the linearized Kac equation is given. Based on this, the optimal time decay estimate for the nonlinear Kac equation is obtained.

Original language	English
Pages (from-to)	4503-4562
Journal	SIAM Journal on Mathematical Analysis
Volume	50
Issue number	4
Online published	30 Aug 2018
DOIs	https://doi.org/10.1137/17M1135955
Publication status	Published - 2018

Research Keywords

non-cutoff Kac equation
global existence
large time behavior
LARGE-TIME BEHAVIOR
BOLTZMANN-EQUATION
WHOLE SPACE
ANGULAR CUTOFF
STRICT POSITIVITY
MAXWELLIAN GAS
EQUILIBRIUM
REGULARITY
PROPAGATION
INEQUALITY

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COPYRIGHT TERMS OF DEPOSITED FINAL PUBLISHED VERSION FILE: © 2018 Society for Industrial and Applied Mathematics.

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@article{626e2b60a3c14b649b2dce834625efe3,

title = "GLOBAL SOLUTION FOR THE SPATIALLY INHOMOGENEOUS NON-CUTOFF KAC EQUATION",

abstract = "This paper is concerned with the Cauchy problem on the one-dimensional inhomogeneous non-cutoff Kac equation. Based on the analysis on the linearized operator obtained in [N. Lerner, Y. Morimoto, K. Pravda-Starov, and C.-J. Xu, J. Funct. Anal., 269 (2015), pp. 459-535], we first prove the existence of a global solution to the equation around a global Maxwellian by combining two sets of macro-micro decomposition. Then by using the dissipative norm of the linearized operator in the fractional Hermite-Sobolev space and using the perturbation theory, the spectrum structure of the linearized Kac equation is given. Based on this, the optimal time decay estimate for the nonlinear Kac equation is obtained.",

keywords = "non-cutoff Kac equation, global existence, large time behavior, LARGE-TIME BEHAVIOR, BOLTZMANN-EQUATION, WHOLE SPACE, ANGULAR CUTOFF, STRICT POSITIVITY, MAXWELLIAN GAS, EQUILIBRIUM, REGULARITY, PROPAGATION, INEQUALITY",

author = "Tong YANG and Hongjun YU",

year = "2018",

doi = "10.1137/17M1135955",

language = "English",

volume = "50",

pages = "4503--4562",

journal = "SIAM Journal on Mathematical Analysis",

issn = "0036-1410",

publisher = "Society for Industrial and Applied Mathematics",

number = "4",

}

TY - JOUR

T1 - GLOBAL SOLUTION FOR THE SPATIALLY INHOMOGENEOUS NON-CUTOFF KAC EQUATION

AU - YANG, Tong

AU - YU, Hongjun

PY - 2018

Y1 - 2018

N2 - This paper is concerned with the Cauchy problem on the one-dimensional inhomogeneous non-cutoff Kac equation. Based on the analysis on the linearized operator obtained in [N. Lerner, Y. Morimoto, K. Pravda-Starov, and C.-J. Xu, J. Funct. Anal., 269 (2015), pp. 459-535], we first prove the existence of a global solution to the equation around a global Maxwellian by combining two sets of macro-micro decomposition. Then by using the dissipative norm of the linearized operator in the fractional Hermite-Sobolev space and using the perturbation theory, the spectrum structure of the linearized Kac equation is given. Based on this, the optimal time decay estimate for the nonlinear Kac equation is obtained.

AB - This paper is concerned with the Cauchy problem on the one-dimensional inhomogeneous non-cutoff Kac equation. Based on the analysis on the linearized operator obtained in [N. Lerner, Y. Morimoto, K. Pravda-Starov, and C.-J. Xu, J. Funct. Anal., 269 (2015), pp. 459-535], we first prove the existence of a global solution to the equation around a global Maxwellian by combining two sets of macro-micro decomposition. Then by using the dissipative norm of the linearized operator in the fractional Hermite-Sobolev space and using the perturbation theory, the spectrum structure of the linearized Kac equation is given. Based on this, the optimal time decay estimate for the nonlinear Kac equation is obtained.

KW - non-cutoff Kac equation

KW - global existence

KW - large time behavior

KW - LARGE-TIME BEHAVIOR

KW - BOLTZMANN-EQUATION

KW - WHOLE SPACE

KW - ANGULAR CUTOFF

KW - STRICT POSITIVITY

KW - MAXWELLIAN GAS

KW - EQUILIBRIUM

KW - REGULARITY

KW - PROPAGATION

KW - INEQUALITY

UR - http://www.scopus.com/inward/record.url?scp=85055284587&partnerID=8YFLogxK

UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-85055284587&origin=recordpage

U2 - 10.1137/17M1135955

DO - 10.1137/17M1135955

M3 - RGC 21 - Publication in refereed journal

SN - 0036-1410

VL - 50

SP - 4503

EP - 4562

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

IS - 4

ER -

GLOBAL SOLUTION FOR THE SPATIALLY INHOMOGENEOUS NON-CUTOFF KAC EQUATION

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NSFC: Mathematical Theories of Some Kinetic and Fluid Models

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GLOBAL SOLUTION FOR THE SPATIALLY INHOMOGENEOUS NON-CUTOFF KAC EQUATION

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Research Keywords

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NSFC: Mathematical Theories of Some Kinetic and Fluid Models

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