几类高阶非线性抛物方程解的存在性与渐近性
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摘要
本文主要研究几类高阶非线性抛物方程(组)解的存在性和长时间渐近行为.本质性困难是作为通常工具所使用的二阶抛物方程的最大值原理和比较原理在高阶情形不再有效.所讨论的问题包括广义薄膜方程解的长时间性态,具有梯度项主部的非线性四阶抛物方程解的存在性和长时间渐近极限,以及粘性量子流体动力学模型解、粘性流体动力学模型解的指数衰减性.首先考虑一个具有零边界流的非线性四阶抛物方程-广义薄膜方程,利用熵泛函的方法证明了解在时间趋于无穷大时趋近于对应的定态问题解的结论,并讨论了带二阶扩散项的薄膜方程解的存在性和正性.然后关注一类具三阶退化项的四阶抛物方程解的存在性与渐近极限.最后研究粘性双极量子流体动力学模型,粘性项对能量耗散速率的影响是:耗散随粘性增大而增大.
第1章概述本文所研究问题的物理背景和国内外发展状况,并简要介绍本文的主要工作.
第2章首先关注一类四阶抛物型方程:u_t=-▽·(u~n▽△u+αu~(n-1)△u▽u+βu~(n-2)|▽u|~2▽u)解的长时间行为.此方程可以理解为薄膜方程u_t+(u~nu_(xxx))_x=0的推广.方程由润滑近似法推导而来,描述粘性薄膜的演化及蔓延传播情况,函数u表示薄膜的高度.对于Neumann初边值问题,我们分别得到一维问题解在L~∞范数意义下以代数速率衰减到其初值的均值,高维问题解在L~1范数意义下以指数衰减速率收敛到其初值的均值.主要技术思想是构造相应的熵泛函,对熵进行两方面的运算,一是建立熵泛函微分的(与熵有关的)负上界,二是证明熵泛函具有L~p范数的下界(0n时,得到弱意义下解的存在性,以及参数n的值对解的正性的影响.最后证明,问题的古典解在t→∞时、以指数速率收敛于其初值的均值.
第3章致力于一类非线性四阶抛物方程u_t+▽·(|▽△u|~(p-2)▽△u)=f(u)第一初边值问题解的存在性和渐近极限的研究.通过不动点理论结合半离散方法,分别得到定态情形和发展方程解的存在性.与一般以往文章方法不同之处是:我们构造两类不同的逼近解、分别处理关于时间变量和空间变量的一致估计,再由必要的先验估计和紧性讨论进一步证明这两类逼近解收敛于同一个函数.也就是所求问题的解.另外.利用熵泛函方法还得到解在时间趋于无穷大时、收敛于其一个常定态解的结论.最后,我们指出当参数p→∞时,解u恰好趋于初值函数u_0.
第4章考虑粘性量子流体动力学模型解的长时间行为.共振穿隧等量子效应可由微观模型表示,例如Wigner方程和Schr(o|¨)dinger系统.这些微观模型又可以由宏观变量,如流密度、电子密度、温度等来描述.这个模型是经典的流体动力学模型经量子作用的修正,由Wigner-Boltzmann方程利用矩方法或者由Schr(o|¨)dinger方程导出.这里,我们用熵泛函方法和一系列先验估计,得到一维和高维两种情形下粘性量子流体动力学模型解的指数衰减,且在t→∞时,解收敛于一个定态解.此外,还得到粘性流体动力学模型解的衰减性。
This thesis deals with existence and asymptotic behavior of solutions for multinonlinear high-order parabolic equations(systems).The substantial difficulty is that the general maximum principle does not hold any more for the high order cases.The topics include the large time behavior and the existence of solutions for a generalized thin film equation and a nonlinear fourth-order parabolic equation with gradient principal part,and the exponential decays for a viscous bipolar quantum hydrodynamic model with special third-order terms.Firstly,we consider a generalized thin film equation with zero-boundary fluxes.The decay of solution towards its mean is given by entropy functional method.Existence and positivity of solutions are studied for a thin film equation with a second-order diffusion term.Secondly,we concern the existence and asymptotic behavior of solutions for a nonlinear fourth-order parabolic equation with degenerate third-order derivative terms.Finally,we study the large-time behavior of solutions for a viscous bipolar quantum hydrodynamic model.The viscosity affects the speed of energy change,and specifically the bigger one yields the faster dissipation.
Chapter 1 is to summarize the background of the related issues and to briefly introduce the main results of the present thesis.
Chapter 2 is firstly concerned with the long time behavior of solutions to a class of fourth order nonlinear parabolic equation:u_t=-▽·(u~n▽△u +αu~(n-1)△u▽u+βu~(n-2)｜▽u｜~2▽u).The equation can be regarded as a generalization of the thin film equation u_t +(u~nu_(xxx))_x = 0 which is derived from a lubrication approximation and models the evolution of thin viscous films and spreading droplets.The function u represents thickness of the film.For the Neumann problem,we prove the algebra decay of solution towards its mean in L~∞-norm for the one-dimensional problem,and the exponential decay of solution to its average in L~1-norm for the multi-dimensional case,respectively.The main technical idea is to construct required dissipative entropies.We show that the derivative of entropy has a negative bound related to itself and for another the entropy has a L~p-norm low bound(0＜p≤∞).Thus,some long-time decay results can be obtained in the sense of L~p-norm.Secondly,we investigate nonnegative solutions of the thin film equation with a second-order diffusion term:u_t+(u~nu_(xxx))_x-(u~m)_(xx)= 0.For m＞n,existence of solutions is obtained in a weak sense.Positivity of solutions is collected,which is depending on n. Finally,we show that the classical solutions also converge to their mean at an exponential rate as the time t→∞.
Chapter 3 is devoted to studying the existence and asymptotic behavior of solutions to a nonlinear fourth-order parabolic equation:u_t+▽·(｜▽△u｜~(P-2)▽△u) = f(u) inΩ(?)R~N with boundary condition u =△u = 0 and initial data u_0.The solutions are obtained for both the steady-state case and the developing case by the fixed point theorem and the semi-discretization method.Unlike the general procedures used in the previous papers on the subject,we introduce two families of approximate solutions with determining the uniform bounds of derivatives with respect to the time and space variables,respectively.By a compactness argument with necessary estimates,we show that the two approximation sequences converge to the same limit,i.e.,the solution to be determined.In addition, the decays of solutions towards the constant steady states are established via the entropy method.Finally,it is interesting to observe that the solutions just tend to the initial data u_0 as p→∞.
Chapter 4 deals with large-time behavior of solutions for a viscous bipolar quantum hydrodynamic model.Generally,quantum effects,for example,resonant tunneling,are presented via microscopic models,such as the Wigner equation and the Schr(o｜¨)dinger system. These microscopic models can be described by macroscopic variables like current densities,electron densities and temperatures.QHD model,being as the extension of the classical hydrodynamic equations with quantum corrections,can be obtained from the Wigner-Boltzmann equation by employing the moment method or the Schr(o｜¨)dinger system.By applying the entropy method,we prove the exponential decays of solutions towards the constant steady states for the one-dimensional and the multi-dimensional cases.The argument is based on a series of a priori estimates.As a byproduct,the decay of solutions for the viscous hydrodynamic model is obtained as well.
第1章概述本文所研究问题的物理背景和国内外发展状况,并简要介绍本文的主要工作.
第2章首先关注一类四阶抛物型方程:u_t=-▽·(u~n▽△u+αu~(n-1)△u▽u+βu~(n-2)|▽u|~2▽u)解的长时间行为.此方程可以理解为薄膜方程u_t+(u~nu_(xxx))_x=0的推广.方程由润滑近似法推导而来,描述粘性薄膜的演化及蔓延传播情况,函数u表示薄膜的高度.对于Neumann初边值问题,我们分别得到一维问题解在L~∞范数意义下以代数速率衰减到其初值的均值,高维问题解在L~1范数意义下以指数衰减速率收敛到其初值的均值.主要技术思想是构造相应的熵泛函,对熵进行两方面的运算,一是建立熵泛函微分的(与熵有关的)负上界,二是证明熵泛函具有L~p范数的下界(0
第3章致力于一类非线性四阶抛物方程u_t+▽·(|▽△u|~(p-2)▽△u)=f(u)第一初边值问题解的存在性和渐近极限的研究.通过不动点理论结合半离散方法,分别得到定态情形和发展方程解的存在性.与一般以往文章方法不同之处是:我们构造两类不同的逼近解、分别处理关于时间变量和空间变量的一致估计,再由必要的先验估计和紧性讨论进一步证明这两类逼近解收敛于同一个函数.也就是所求问题的解.另外.利用熵泛函方法还得到解在时间趋于无穷大时、收敛于其一个常定态解的结论.最后,我们指出当参数p→∞时,解u恰好趋于初值函数u_0.
第4章考虑粘性量子流体动力学模型解的长时间行为.共振穿隧等量子效应可由微观模型表示,例如Wigner方程和Schr(o|¨)dinger系统.这些微观模型又可以由宏观变量,如流密度、电子密度、温度等来描述.这个模型是经典的流体动力学模型经量子作用的修正,由Wigner-Boltzmann方程利用矩方法或者由Schr(o|¨)dinger方程导出.这里,我们用熵泛函方法和一系列先验估计,得到一维和高维两种情形下粘性量子流体动力学模型解的指数衰减,且在t→∞时,解收敛于一个定态解.此外,还得到粘性流体动力学模型解的衰减性。
This thesis deals with existence and asymptotic behavior of solutions for multinonlinear high-order parabolic equations(systems).The substantial difficulty is that the general maximum principle does not hold any more for the high order cases.The topics include the large time behavior and the existence of solutions for a generalized thin film equation and a nonlinear fourth-order parabolic equation with gradient principal part,and the exponential decays for a viscous bipolar quantum hydrodynamic model with special third-order terms.Firstly,we consider a generalized thin film equation with zero-boundary fluxes.The decay of solution towards its mean is given by entropy functional method.Existence and positivity of solutions are studied for a thin film equation with a second-order diffusion term.Secondly,we concern the existence and asymptotic behavior of solutions for a nonlinear fourth-order parabolic equation with degenerate third-order derivative terms.Finally,we study the large-time behavior of solutions for a viscous bipolar quantum hydrodynamic model.The viscosity affects the speed of energy change,and specifically the bigger one yields the faster dissipation.
Chapter 1 is to summarize the background of the related issues and to briefly introduce the main results of the present thesis.
Chapter 2 is firstly concerned with the long time behavior of solutions to a class of fourth order nonlinear parabolic equation:u_t=-▽·(u~n▽△u +αu~(n-1)△u▽u+βu~(n-2)｜▽u｜~2▽u).The equation can be regarded as a generalization of the thin film equation u_t +(u~nu_(xxx))_x = 0 which is derived from a lubrication approximation and models the evolution of thin viscous films and spreading droplets.The function u represents thickness of the film.For the Neumann problem,we prove the algebra decay of solution towards its mean in L~∞-norm for the one-dimensional problem,and the exponential decay of solution to its average in L~1-norm for the multi-dimensional case,respectively.The main technical idea is to construct required dissipative entropies.We show that the derivative of entropy has a negative bound related to itself and for another the entropy has a L~p-norm low bound(0＜p≤∞).Thus,some long-time decay results can be obtained in the sense of L~p-norm.Secondly,we investigate nonnegative solutions of the thin film equation with a second-order diffusion term:u_t+(u~nu_(xxx))_x-(u~m)_(xx)= 0.For m＞n,existence of solutions is obtained in a weak sense.Positivity of solutions is collected,which is depending on n. Finally,we show that the classical solutions also converge to their mean at an exponential rate as the time t→∞.
Chapter 3 is devoted to studying the existence and asymptotic behavior of solutions to a nonlinear fourth-order parabolic equation:u_t+▽·(｜▽△u｜~(P-2)▽△u) = f(u) inΩ(?)R~N with boundary condition u =△u = 0 and initial data u_0.The solutions are obtained for both the steady-state case and the developing case by the fixed point theorem and the semi-discretization method.Unlike the general procedures used in the previous papers on the subject,we introduce two families of approximate solutions with determining the uniform bounds of derivatives with respect to the time and space variables,respectively.By a compactness argument with necessary estimates,we show that the two approximation sequences converge to the same limit,i.e.,the solution to be determined.In addition, the decays of solutions towards the constant steady states are established via the entropy method.Finally,it is interesting to observe that the solutions just tend to the initial data u_0 as p→∞.
Chapter 4 deals with large-time behavior of solutions for a viscous bipolar quantum hydrodynamic model.Generally,quantum effects,for example,resonant tunneling,are presented via microscopic models,such as the Wigner equation and the Schr(o｜¨)dinger system. These microscopic models can be described by macroscopic variables like current densities,electron densities and temperatures.QHD model,being as the extension of the classical hydrodynamic equations with quantum corrections,can be obtained from the Wigner-Boltzmann equation by employing the moment method or the Schr(o｜¨)dinger system.By applying the entropy method,we prove the exponential decays of solutions towards the constant steady states for the one-dimensional and the multi-dimensional cases.The argument is based on a series of a priori estimates.As a byproduct,the decay of solutions for the viscous hydrodynamic model is obtained as well.
引文
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