几类非线性系统的动力学研究

发布时间：2018-06-06 13:05

本文选题：非线性 + 黏弹性　；参考：《吉林大学》2017年博士论文

【摘要】：本文主要研究几类常见非线性系统的动力学性质.全文内容共分四章,第一章为绪论,第二章至第四章为论文主体部分.在第二章中,我们考虑一类含有非线性耗散项的黏弹性波动方程的初边值问题.问题的具体形式如下:其中Ω(?)Rn为具有光滑边界(?)Ω的有界区域,R+ = {t|0t∞},△表示Rn中的Laplace算子,'表示(?).对该方程中的h与g,我们有以下假设:(H.h)h ∈ × R)关于第二个变量单调递增,并且存在一个严格单调递增的奇函数kk∈ C1(R+)使得对某个正常数r01,有kk(r0)1以及K(v)=(?)((?))为[0,r02]上的严格凸函数.同时满足其中a(x)∈ L∞为一正函数,c1,c2,c3,c4均为正常数,k-1表示kk的反函数;(H.g)g(t)≥ 0是一个单调不增函数,且满足同时,存在一个非负的凸函数G ∈ C1(R+),满足G(0)= G'(0)= 0并且使得和成立.在这些假设条件下,我们首先给出该问题解的存在性定理.定理1.假设h,g满足(H.h),(H.g),且有u0∈H01(Ω)∩H2(Ω),u1∈H01(Ω).那么,问题(1)存在唯一弱解满足然后,通过构造适当的乘子并运用积分形式的Gronwall不等式,我们得到了关于这类非线性系统的能量衰减率估计.定理2.假设(H.g),(H.h)成立,并且u0,u1满足定理1中条件.那么存在某个正常数C0以及与E(0)有关的常数ω使得其中能量泛函E(t)的定义为上式中||·||表示L2范数.同时,η是一个当t→+∞时衰减到0的单调递减函数,其具体形式如下:式中L,v以及常数C11将分别在(2.47),(2.50)和(2.49)中给出.上述定理给出了同时具有弱耗散项以及黏弹性项的波动方程的能量估计公式.众所周知,耗散项和黏弹性项都会造成波动方程能量的衰减,但是关于两者同时存在时方程能量衰减问题的研究目前还处于刚起步阶段,许多关于这方面的研究所处理的耗散项以及黏弹性项的形式相对简单.在这一章中,我们利用凸函数的性质,对于复杂的耗散项和黏弹性项进行了讨论,并在一定程度上对比了这两者在方程能量衰减中起到作用的大小.定理中的公式可应用于多种具非线性耗散项的黏弹性波动方程的能量衰减率估计.在第二章的最后,我们将会通过两个例子来展示定理2的应用.在第三章中,我们考虑一类具非线性耗散项的变系数梁方程的初边值问题.问题的具体形式如下:其中Q(?)Rn为具有C2边界(?)Ω如的非空有界区域,R+ = {t|0t+O∞},‖·‖为L2范数,%接搿鞣直鹞猂n中的梯度算子与Laplace算子,'表示(?),v是边界(?)Ω上的单位外法向量,(?)表示法方向导数.变系数梁方程可以用于描述由非均匀材料所构成的梁的振动.相对于一般的常系数梁方程,它可以更好的描述实际情形,但变系数的出现也为方程的求解过程带来了困难.除此之外,非线性耗散项的存在也使得方程的求解变得复杂.在这一章中,我们定义了两个特殊的能量泛函,通过对于这两个泛函的估计并运用Aubin-Lions定理解决了这些问题.对于方程中的变系数a(x,t),b(x t)以及非线性项M(x,t,λ),h(s),我们有以下假设:(H.a)函数 a 与 a'满足a,a'∈ a(Ω × R+),函数 a"与 a"'满足 a",a"' ∈L1(Ω ×R+),并且存在常数 a00,a1 ∈R,A00,0,A20,A30使得(H.b)对于所有的t ∈ R+,有b(.,t)∈H02(Ω);对于所有的x ∈ Ω,有b(x,·)∈C1(R+).此外,b'(x,t)∈L1(Ω× R+)并且存在常数b00,Bo0,B10,B20使得(H.M)函数M满足M(x,t,λ)∈C1 × R+ × R+)并且存在常数M0 ≥ 0,M10,M2 ≥ 0,M3 ≥ 0 使得(H.h)函数h∈C1(R)为一非减函数且满足并且存在常数使得首先,我们运用Feado-Galerkin方法以及能量方法对问题强解的存在唯一性进行了研究并得出了以下定理.定理3.(存在性)假设(H.a),(H.b),(H.M)以及(H.h)成立,并且初值满足同时,存在常数b0使得且满足其中函数H(t)以及常数C2的具体形式将在(3.12)和(3.17)中给出,C为Poincare常数使得对所有的u ∈H02(Ω),有‖u‖≤C‖%絬‖,‖%絬‖≤‖%絬‖,‖u‖≤C‖△u‖.那么,上述初边值问题的强解存在.定理4.(唯一性)如果条件(H.a),(H.b),(H.M)以及(H.h)成立,并且M(x,t,λ)满足M ∈C1(Ω ×[0,+∞)×[0,∞)),△M,%組 ∈ L∞(Ω ×[0,+∞)×[0,+∞)).那么,上述问题的解是唯一的.在此基础上,我们还研究了该方程解的长时间渐近行为,证明了能量泛函的指数衰减,得到以下定理.定理5.在定理4的假设下,存在常数0,γ0使得上述问题的能量泛函满足E(t)≤ζe-γt,(?)t≥0,其中能量泛函E(t)的定义如下:E(t)=1/2∫Ω(a(x,t)|u'|2 + b(x,t)|△u|2 + M(x,1t,‖%絬(t)‖2)%絬|2)dx.在第四章中,我们研究了稳态Poisson-Nernst-Plank(PNP)系统.系统具体形式如下:其中x∈(0,1),k =1,2,...,n.其边值条件为:Φ(0)= V,ck(0)=lk≥ 0;Φ(1)= 0,ck(1)= rk≥0.上式中,Φ表示电压,ck,Jk分别表示第kk种离子的离子浓度以及流量密度,区间[0,1]表示整个离子通道,ε2《1为一与维度无关的参数,h(x)表示x位置的横截面积,Q(x)表示离子通道内壁上的永久电荷分布,αk≠0表示第kk种离子的带电量,lk和rk分别表示离子通道两端的离子浓度.由于ε很小,我们可以把上述边值问题看成奇异连接问题.然后,运用问题的特殊结构,我们将寻找奇异轨的问题转化为求与系统相对应的某个矩阵的特ξ征值问题.再通过进一步的推导,我们又将该求解特征值问题转化为求某个方程h(z)=0的根的问题,其中h(z)的具体形式为为叙述方便,我们首先定义了以下变量.之后,我们考虑了 n = 3时上述问题所对应矩阵的特征值λ1,λ2,λ3的分布情况,由于边界条件满足电中和的性质,因此我们有= ∑i=1αili=∑i=13αiri=0,进而可以推出λ1 = 0恒成立,于是我们只需要求得另外两个特征值A2,A3的分布即可求出以上PNP系统的奇异轨.关于λ2,λ3的分布情况我们得到了以下结论.命题1.当mr = ml时,我们有此时,λ2,λ3具有以下形式当mrml时,A2,A3分别为h(z)的两个实零点,更进一步,我们有λ3ml且命题2.当mrml且pmrml 时,λ2,λ3分别为h(z)=0的两个不等实根,且满足0ml;当mrm 且pmr=ml 时,A2,A3分别为h(z)=0的两个不等实根,且满足对于mrm 且pmrml的情况,A2,A3的分布情况比较复杂,为研究它们的分布我们首先定义了两个辅助函数:应用这两个函数的性质并通过进一步计算,我们得到了如下定理.定理6.的零点且其有以下几种分布情况:为方便理解,我们在文中针对上述定理中的每种情况都给出了相应的特征值分布的直观示意图.以上是在相关领域中第一次从具体的特征值分布情况入手研究PNP系统所得到的结果,在这些结论的基础上,我们又对该系统的电流L进行了研究,给出了不同特征值分布下电流L的公式.更进一步,我们根据所求电流L的公式对电流与电压的关系进行了细致的分析.这些分析对于科学实验以及数值模拟都有着指导性作用。
[Abstract]:This paper mainly studies the dynamic properties of several kinds of common nonlinear systems. The full text is divided into four chapters. The first chapter is the introduction, the second to the fourth chapter is the main part of the paper. In the second chapter, we consider a class of initial boundary value questions of the viscoelastic wave equation with nonlinear dissipative term. The concrete form of the problem is as follows: omega (?) Rn is a tool. The bounded region with smooth boundary (?) Omega, R+ = {t|0t infinity}, the Laplace operator in Rn, '(?). For the H and G in the equation, we have the following hypothesis: (H.h) h h * R) on the monotonically increasing second variables, and there is a strictly monotonically increasing odd function KK, C1 (R+). (?)) ((?)) a strict convex function on [0, r02], and a (x) L infinity as a positive function, C1, C2, C3, C4 are all normal, k-1 is a KK inverse function; (H.g) (H.g) > 0 is a monotonous non increment function, and satisfies the existence of a non negative convex function (0) = 0 and is made and established. We first give the existence theorem of the solution of this problem. Theorem 1. suppose h, G satisfies (H.h), (H.g), and there is U0 H01 (omega) H2 (omega), U1 H01 (omega). Then, the problem (1) exists the unique weak solution, and then we get the energy of this kind of nonlinear system by constructing the proper multiplier and using the Gronwall inequality in the integral form. Attenuation rate estimation. Theorem 2. hypothesis (H.g), (H.h) is established, and U0, U1 satisfies the conditions in Theorem 1. Then there is a normal number C0 and the constant Omega related to E (0) so that the energy functional E (T) is defined as the upper formula L2 norm. At the same time, the ETA is a monotone decreasing function that attenuates to 0 when t to + infinity, and its specific form is as follows: L, V and constant C11 will be given in (2.47), (2.50) and (2.49) respectively. The above theorem gives the energy estimation formula for the wave equation with both weak dissipation and viscoelastic terms. It is known that both the dissipative term and the viscoelastic term will cause the attenuation of the energy of the wave equation, but the energy attenuation of the equation at the same time is asked. In this chapter, we use the properties of the convex function to discuss the complex dissipative term and the viscoelastic term, and compare the two in the energy decay of the equation at a certain degree. The formula in the theorem can be applied to the energy attenuation estimation of a variety of viscoelastic wave equations with nonlinear dissipation terms. At the end of the second chapter, we will show the application of theorem 2 through two examples. In the third chapter, we consider a class of initial boundary value questions of a class of variable coefficient Beam Equations with nonlinear dissipation terms. The specific forms of the problem are as follows: Q (?) Rn is a non empty bounded region with a C2 boundary (?) Omega, R+ = {t|0t+O infinity}, a L2 norm, a gradient operator and a Laplace operator in the n of the tanning harrier n, '(?) ", V is a single external vector on the boundary (?) Omega, (?) to represent the derivative of the direction. The variable coefficient beam equation can be used to describe the reason. The vibration of a beam composed of nonuniform materials can describe the actual situation better than the ordinary constant coefficient beam equation, but the appearance of variable coefficients also brings difficulties to the solving process of the equation. In addition, the existence of the nonlinear dissipative term makes the solution of the equation complicated. In this chapter, we define two special cases. The energy functional is solved by the estimation of these two functionals and using the Aubin-Lions theorem. For the variable coefficients a (x, t), B (x T) and the nonlinear term M (x, t, lambda), H (s), we have the following hypothesis: Constant A00, A1 R, A00,0, A20, A30 make (H.b) for all t R+, B (, t) H02 (omega). H.h) function H C1 (R) is a non subtraction function and satisfies and exists constant. First, we use the Feado-Galerkin method and the energy method to study the existence and uniqueness of the strong solution of the problem and get the following theorem. Theorem 3. (existence) hypothesis (H.a), (H.b), (H.M) and (H.h), and the initial value satisfies the existence of a constant. B0 makes and satisfies the specific forms of the function H (T) and constant C2 in (3.12) and (3.17), and C is a Poincare constant for all u H02 (omega). (H.M) and (H.h) are established, and M (x, t, lambda) satisfies M C1 (omega * [0, + infinity) * [0, infinity), Delta M,% group, L infinity (omega * [0, + infinity) x, + infinity). Then, the solution of the above problem is unique. On this basis, we also study the long time asymptotic behavior of the solution of the equation, prove the exponential decay of the energy functional, and get the following theorem. Theorem 5. in the theorem. Under the assumption of Theorem 4, the existence constant 0, gamma 0 make the energy functional of the above problem satisfy E (T) < e- gamma T, (?) t > 0, and the energy functional E (T) is defined as follows: E (T) =1/2 (a) (a) in the fourth chapter, we have studied the steady state system. The formula is as follows: X (0,1), K =1,2,..., the boundary value conditions of N. are: (0) = V, CK (0) =lk > 0; diameter (1) = 0, CK (1) = rk > = 0., voltage, CK, Jk, respectively representing the ion concentration and the flow density, respectively. Q (x) represents the permanent charge distribution on the inner wall of the ion channel, and the alpha K 0 represents the charged amount of the KK ions. LK and rk represent the ionic concentration at both ends of the ion channels. By further derivation, we turn the solution of the eigenvalue problem into the root of an equation H (z) =0 by further derivation. In which the specific form of H (z) is convenient for the description, we first define the following variables. After that, we consider the corresponding moments of the above problems when n = 3. The distribution of the eigenvalue of the matrix [lambda] 1, lambda 2, and lambda 3, because the boundary condition satisfies the property of the electric neutralization, we have = = i=1 alpha ili= Sigma i=13 alpha iri=0, and then we can introduce [1] 1 = 0 constant, so we only need to obtain the other two eigenvalues A2, the distribution of the A3 system can be obtained by the singular rails of the upper PNP system. The distribution of [lambda] 2, [lambda] 3 We get the following conclusion. When Mr = ml, when Mr = ml, we have this time, when lambda 2, lambda 3 have the following form as mrml, A2, A3 are two real zeros of Z respectively, and further, we have a 3ml and propositional 2. when mrml and pmrml, lambda 2, and lambda 3 are two unequal real roots of H (z), respectively. In the case of MRM and pmrml, the distribution of A2 and A3 is more complex. In order to study their distribution, we first define two auxiliary functions: using the properties of the two functions and by further calculation, we get the following theorem. The zero point of theorem 6. and it has the following distribution: it is convenient to understand. In this paper, we give an intuitive diagram of the corresponding eigenvalue distribution for each of the above theorems. The above is the first time to study the results obtained by the PNP system from the specific eigenvalue distribution in the related fields. On the basis of these conclusions, we have also studied the current L of the system and gave the results. The formula of current L under the distribution of different eigenvalues is given. Further, we make a detailed analysis of the relationship between current and voltage according to the formula of current L. These analyses have a guiding role for both scientific experiments and numerical simulation.
【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175

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