辐射流体动力学方程的Cauchy问
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摘要
本文主要考虑辐射流体动力学方程整体解的存在性和唯一性,以及解的大时间行为.本文的主要内容如下:
第一章为绪言.在这里,我们回顾了辐射流体动力学方程的物理背景及研究历史,并交代了将要研究的方程和相关的主要结论.
第二章中,我们研究了多维辐射流体动力学方程的一个简化模型,即一个双曲-椭圆耦合方程组.在第一节中,我们研究模型的Cauchy问题经典解的整体存在性和逐点估计.首先,利用能量估计的方法,我们得到经典解的整体存在唯一性.进一步,通过对线性化问题Green函数的研究,我们得到Green函数的逐点估计.并利用Duhamel原理将非线性微分方程转换成非线性积分方程.然后,利用Green函数的逐点估计,我们获得当初始值在一个常状态附近扰动时非线性方程解的逐点收敛速度.研究发现,解随时间增加不断地耗散,与此同时解的主部向某个方向在平移.最后,我们得到了解的最优的Lp模衰减估计.在第二节中,我们研究前面模型的一种修改形式.这一形式,因其不满足Shizuta-Kawashima条件,一般认为这类耗散结构较差.通过对这一问题的研究,我们得到了多维的较差耗散结构方程组解的整体存在性.这里利用加权能量的方法,我们获得解的Lp模的衰减估计.进一步,我们研究了解的大时间渐近行为.在第三节中,利用不动点定理和Green函数方法,我们研究了一个相关方程解的整体存在性和逐点估计.
第三章中,我们考虑的是带粘性的辐射流体动力学方程的Cauchy问题.在第一节中,我们得到了Cauchy问题经典解的整体存在唯一性和衰减估计.这里我们主要利用[54]中的应用能量方法得到解的先验估计.在第二节中,在前面Cauchy问题解的整体存在性基础上,利用线性化问题Green函数的逐点估计,我们获得解的逐点估计.在第三节中,我们考虑多维等熵球对称流体动力学方程组弱解的局部存在性.
第四章中,我们研究多维辐射流体动力学方程简化模型大扰动解的整体存在性和衰减估计.首先,我们考虑Cauchy问题经典解的爆破.然后,利用能量方法,我们得到当初始值的梯度满足某种小性假设条件时问题的经典解的整体存在性和衰减估计.
In this thesis, we consider the global existence and uniqueness of solutions to the initialvalue problem for the equations of radiation hydrodynamics. Furthermore, we study thelarge time behaviors of the solutions. The thesis is arranged as follows:
In Chapter 1, we review the physics background of the equations of radiation hydrody-namics and the history on studying the equations. We also introduce the problems addressedin this study and summarize the main results.
In Chapter 2, we investigate a model system for the radiating gas, i.e. a hyperbolic-elliptic coupled system. In Section 1, we consider global existence and pointwise estimatesof solutions to the Cauchy problem for the model system of the radiating gas. First of all,by using the energy method, we obtain the global existence and uniqueness of the solutions.Then, we derive pointwise estimates of the Green function by studying the Green functionproblem. Moreover, by using the Duhamel principle and the pointwise estimates of theGreen function, we obtain pointwise estimates of the solutions when the initial perturbationcorresponding to a positive constant state is sufficiently small in Hs(Rn). Furthermore, weshow the optimal Lp-norm estimates of the solutions. In Section 2, we consider a modi-fied model system for the radiating gas. This model doesn’t satisfy the Shizuta-Kawashimacondition. Here, we study the global existence and asymptotic behaviors of solutions to aregularity-loss type for nonlinear hyperbolic-elliptic system in multi-dimensions. By usinga weighted energy method, we obtain Lp-norm decay estimates of solutions to the Cauchyproblem. Fianlly, we get large time asymptotic behaviors of the solutions. In Section 3, westudy the global existence and pointwise estimates of solutions to an equation which satisfiesthe Shizuta-Kawashima condition.
In Chapter 3, we consider the initial value problem for a radiation hydrodynamic model with viscosity in R3. In Section 1, we obtain the global existence and some decay estimatesof solutions to the problem. In order to prove a priori estimates of the solutions, we applyan energy method as that in [54]. In Section 2, under the proof of the global existenceof the solutions in Section 1, we obtain pointwise estimates of solutions. Here, the decayestimates of the Green function for the linearized system to the equations is used. In Section3, when omitting the radiation effect, we consider the local existence of weak solutions tothe isentropic spherically symmetric Navier-Stokes equations.
In Chapter 4, we investigate the global existence and decay estimates of solutions tothe Cauchy problem for a model system of the radiating gas with large initial data. Firstly,we consider the blow up of classical solutions. Secondly, when the derivative of initial datasatisfies some smallness conditions, we obtain the global existence and decay estimates ofthe solutions by using an energy method.
第一章为绪言.在这里,我们回顾了辐射流体动力学方程的物理背景及研究历史,并交代了将要研究的方程和相关的主要结论.
第二章中,我们研究了多维辐射流体动力学方程的一个简化模型,即一个双曲-椭圆耦合方程组.在第一节中,我们研究模型的Cauchy问题经典解的整体存在性和逐点估计.首先,利用能量估计的方法,我们得到经典解的整体存在唯一性.进一步,通过对线性化问题Green函数的研究,我们得到Green函数的逐点估计.并利用Duhamel原理将非线性微分方程转换成非线性积分方程.然后,利用Green函数的逐点估计,我们获得当初始值在一个常状态附近扰动时非线性方程解的逐点收敛速度.研究发现,解随时间增加不断地耗散,与此同时解的主部向某个方向在平移.最后,我们得到了解的最优的Lp模衰减估计.在第二节中,我们研究前面模型的一种修改形式.这一形式,因其不满足Shizuta-Kawashima条件,一般认为这类耗散结构较差.通过对这一问题的研究,我们得到了多维的较差耗散结构方程组解的整体存在性.这里利用加权能量的方法,我们获得解的Lp模的衰减估计.进一步,我们研究了解的大时间渐近行为.在第三节中,利用不动点定理和Green函数方法,我们研究了一个相关方程解的整体存在性和逐点估计.
第三章中,我们考虑的是带粘性的辐射流体动力学方程的Cauchy问题.在第一节中,我们得到了Cauchy问题经典解的整体存在唯一性和衰减估计.这里我们主要利用[54]中的应用能量方法得到解的先验估计.在第二节中,在前面Cauchy问题解的整体存在性基础上,利用线性化问题Green函数的逐点估计,我们获得解的逐点估计.在第三节中,我们考虑多维等熵球对称流体动力学方程组弱解的局部存在性.
第四章中,我们研究多维辐射流体动力学方程简化模型大扰动解的整体存在性和衰减估计.首先,我们考虑Cauchy问题经典解的爆破.然后,利用能量方法,我们得到当初始值的梯度满足某种小性假设条件时问题的经典解的整体存在性和衰减估计.
In this thesis, we consider the global existence and uniqueness of solutions to the initialvalue problem for the equations of radiation hydrodynamics. Furthermore, we study thelarge time behaviors of the solutions. The thesis is arranged as follows:
In Chapter 1, we review the physics background of the equations of radiation hydrody-namics and the history on studying the equations. We also introduce the problems addressedin this study and summarize the main results.
In Chapter 2, we investigate a model system for the radiating gas, i.e. a hyperbolic-elliptic coupled system. In Section 1, we consider global existence and pointwise estimatesof solutions to the Cauchy problem for the model system of the radiating gas. First of all,by using the energy method, we obtain the global existence and uniqueness of the solutions.Then, we derive pointwise estimates of the Green function by studying the Green functionproblem. Moreover, by using the Duhamel principle and the pointwise estimates of theGreen function, we obtain pointwise estimates of the solutions when the initial perturbationcorresponding to a positive constant state is sufficiently small in Hs(Rn). Furthermore, weshow the optimal Lp-norm estimates of the solutions. In Section 2, we consider a modi-fied model system for the radiating gas. This model doesn’t satisfy the Shizuta-Kawashimacondition. Here, we study the global existence and asymptotic behaviors of solutions to aregularity-loss type for nonlinear hyperbolic-elliptic system in multi-dimensions. By usinga weighted energy method, we obtain Lp-norm decay estimates of solutions to the Cauchyproblem. Fianlly, we get large time asymptotic behaviors of the solutions. In Section 3, westudy the global existence and pointwise estimates of solutions to an equation which satisfiesthe Shizuta-Kawashima condition.
In Chapter 3, we consider the initial value problem for a radiation hydrodynamic model with viscosity in R3. In Section 1, we obtain the global existence and some decay estimatesof solutions to the problem. In order to prove a priori estimates of the solutions, we applyan energy method as that in [54]. In Section 2, under the proof of the global existenceof the solutions in Section 1, we obtain pointwise estimates of solutions. Here, the decayestimates of the Green function for the linearized system to the equations is used. In Section3, when omitting the radiation effect, we consider the local existence of weak solutions tothe isentropic spherically symmetric Navier-Stokes equations.
In Chapter 4, we investigate the global existence and decay estimates of solutions tothe Cauchy problem for a model system of the radiating gas with large initial data. Firstly,we consider the blow up of classical solutions. Secondly, when the derivative of initial datasatisfies some smallness conditions, we obtain the global existence and decay estimates ofthe solutions by using an energy method.
引文
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[3] J. W. Bond Jr., K. M. Watson, and J. A. Welch Jr., Atomic Theory of Gas Dynamics,Addison-Wesley Publishing Company, Inc, 1965
[4] G. Q. Chen, M. Kratka, Global solutions to the Navier-Stokes equations for compress-ible heat-conducting ?ow with symmetry and free boundary, Comm. Partial Differen-tial Equations, 27 (2002), 907-943.
[5] G. Q. Chen, T. H. Li, Formation ofδ-shocks and vacuum states in the vanishingpressure limit of solutions to the Euler equations for isentropic ?uids, , SIAM J. Math.Anal., 34 (2003), 925-938.
[6] G. Q. Chen, B. H. Li and T. H. Li, Entropy solutions in L∞for the Euler equations innonlinear elastodynamics and related equations, Arch. Ration. Mech. Anal., 170 (2003),331-357.
[7] P. Chen, T. Zhang, A vacuum problem for multidimensional compressible Navier-Stokes equations with degenerate viscosity coefficients, Comm. Pure Appl. Anal., 7(2008), 987-1016.
[8] R. J. Duan, K. Fellner and C. J. Zhu, Energy method for multi-dimensional balancelaws with non-local dissipation, J. Math. Pures Appl., in press.
[9] B. Ducomet, E. Feireisl, On the dynamics of gaseous stars, Arch. Rational Mech.Anal., 174 (2004), 221-266.
[10] B. Ducomet, E. Feireisl, The equations of magnetohydrodynamics: on the interactionbetween matter and radiation in the evolution of gaseous stars, Commun. Math. Phys.,266 (2006), 595-629.
[11] B. Ducomet, A. Zlotnik, Lyapunov functional methods for 1D radiative and reactiveviscous gas dynamics, Arch. Ration. Mech. Anal., 177 (2005), 185-229.
[12] M. D. Francesco, Initial value problem and relaxation limits of the hamer model forradiating gases in several space variables, NoDEA, Nonl. Differential Equations Appl.,13 (2007), 531-562.
[13] K. O. Fredrichs, Symmetric hyperbolic linear differential equations, Comm. Pure Appl.Math., 7 (1954), 345-392.
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