Asymptotic behavior of solutions to a model system of a radiating gas

Liu, Yongqin; Kawashima, Shuichi

doi:10.3934/cpaa.2011.10.209

Cited by 21 publications

(24 citation statements)

References 27 publications

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“…The next lemma will serve us for estimating the function ψ(x, t) (see [4]). The next lemma will serve us for estimating the function ψ(x, t) (see [4]).…”

Section: It Is Well Known That the Burgers Equation Can Be Linearizedmentioning

confidence: 99%

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Asymptotic behavior of solutions of the Hamer equation

Соловьев¹

2015

St. Petersburg Math. J.

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In a preceding paper the leading term was found for the asymptotics as t → +∞ of the solution of the initial problem for the Hamer equation, which is a simplest model for the motion of a radiating gas. Here, the second asymptotic term is constructed. It is proved that this term is proportional to the second term of the asymptotics of the solution of the initial problem for the Burgers equation.2010 Mathematics Subject Classification. Primary 35G55.

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“…The next lemma will serve us for estimating the function ψ(x, t) (see [4]). The next lemma will serve us for estimating the function ψ(x, t) (see [4]).…”

Section: It Is Well Known That the Burgers Equation Can Be Linearizedmentioning

confidence: 99%

“…The behavior of the solution (u(x, t), q(x, t)) as t → ∞ of system (1.1) with initial data belonging to H s (R) ∩ L 1 (R) was studied in [4].…”

Section: §1 Introductionmentioning

confidence: 99%

Asymptotic behavior of solutions of the Hamer equation

Соловьев¹

2015

St. Petersburg Math. J.

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“…Our study is based on the following global existence theorem (cf. ). Theorem Assume that

u_{0} (x) MathClass-rel\in H^{N} ((), {double-struckR}^{n}) (n MathClass-rel\geq 1)

for an integer

N MathClass-rel\geq ([], \frac{n}{2}) MathClass-bin+ 2

.…”

Section: Introductionmentioning

confidence: 97%

“…For the study of decay rates to the Cauchy problem , pointwise estimates of solutions were obtained in . Under the assumption that

MathClass-rel∥ u_{0} {MathClass-rel∥}_{L^{1}}

is small, Liu and Kawashima in studied large time behavior of solutions to the problem by using a time‐weighted energy method. Under a weaker assumption that

MathClass-rel∥ u_{0} {MathClass-rel∥}_{L^{1}}

is bounded, Duan et al in showed that the optimal L 2 ‐norm time‐decay rate of solutions is

{(1 MathClass-bin+ t)}^{MathClass-bin- \frac{n}{4}}

, whereas time‐decay estimates of the derivatives of the solutions have not been considered.…”

Section: Introductionmentioning

confidence: 99%

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Decay of a model system of radiating gas

Wang

2013

Math. Meth. Appl. Sci.

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This paper is concerned with optimal time-decay estimates of solutions of the Cauchy problem to a model system of the radiating gas in $\mathbb{R}^n$. Compared to Liu and Kawashima (2011) \cite{Liu1} and Wang and Wang (2009) \cite{Wang}, without smallness assumption of initial perturbation in $L^1$-norm, we study large time behavior of small amplitude classical solutions to the Cauchy problem. The optimal $H^N$-norm time-decay rates of the solutions in $\mathbb{R}^n$ with $1\leq n\leq4$ are obtained by applying the Fourier splitting method introduced in Schonbek (1980) \cite{Schonbek1} with a slight modification and an energy method. Furthermore, basing on a refined pure energy method introduced in Guo and Wang \cite{Guo} (2011), we give optimal $L^p$-$L^2(\mathbb{R}^3)$ decay estimates of the derivatives of solutions when initial perturbation is bounded in $L^p$-norm with some $p\in(1,2]$.Comment: 15 page

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Global existence and optimal decay rates of solutions to conservation laws with diffusion‐type terms

Wang

2016

Math Methods in App Sciences

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We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term. Based on a low-frequency and high-frequency decomposition, Green's function method and the classical energy method, we not only obtain L 2 time-decay estimates but also establish the global existence of solutions to Cauchy problem when the initial data u 0 .x/ satisfies the smallness condition on kr u 0 k H s , but not on ku 0 k L 2 . Furthermore, by taking a time-frequency decomposition, we obtain the optimal decay estimates of solutions.

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Asymptotic behavior of solutions to a model system of a radiating gas

Cited by 21 publications

References 27 publications

Asymptotic behavior of solutions of the Hamer equation

Asymptotic behavior of solutions of the Hamer equation

Decay of a model system of radiating gas

Global existence and optimal decay rates of solutions to conservation laws with diffusion‐type terms

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