上海交通大学 硕士学位论文 拟线性对称双曲方程组非线性初边值问题的适定性 姓名：梁俊辉 申请学位级别：硕士 专业：应用数学 指导教师：王亚光 20080201 ?a? ?555VVV?| ?555? ?fl flfl KKK?555 ? u?5V|?fl K?fi kL? ?u?fl K?a?a?A?.?/ ,?aA?.?/u?A?.?/? l?V8?c?mu?5?)?fl K? 5.?/uA?.?/g?Vl?c?5?k Nk?A?/e9?5?. ?u?afl K?5?fl K?8?y?w ,kd? ?O?kA?.!?A?.?5V|? ?fl K?.?5?5V? e?!?v?N5?5z?fl K ?v4?K.?5?fl K)?3 ?5?d?kA?.?A?.?/e?5 zfl KO3?5?5?a Sobolev ?m?U ?O,?L?5?1 Picard S?.?1 Newton S?5fl K?E?Cq)S?L|c ?5zfl K)?U?O?Cq)S?5l? ?5fl K?) ?c?5V?5.?4?K.? 5 -i- ?a? WELL-POSEDNESS OF NONLINEAR IBVP FOR QUASILINEAR SYMMETRIC HYPERBOLIC SYSTEMS ABSTRACT There already have been a lot of results on Initial Boundary Value Problems (IBVP) for quasilinear hyperbolic systems. These works can be mainly divided into two cases, the case of noncharacteristic boundaries and of characteristic boundaries. The theory on noncharacteristic problems had been developing since 1960s, while people started to study the IBVP with characteristic boundaries in 1980s. However, most of these results were restricted to the problems with linear boudary conditions. Obviously, it is important and interesting to study the IBVP of quasilinear hyperbolic equations with nonlinear boundary conditions. In this thesis,we consider the IBVP of quasilinear hyperbolic systems with nonlinear boundary conditions.Both of the characteristic boundary case and noncharacteristic boundary case are studied. Under the assumption that the initial data and boundary data satisfy the compatibility conditions, and the linearized problem satisfi es the maximal dissipative condition, we establish the existence and uniqueness of solutions to the quasilinear hyperbolic equations with nonlinear boundary conditions. First, certain energy estimates are obtained in isentropic and non-isentropic sobolev spaces for the linearized problems with noncharacteristic boudary and characteristic boudary respectively.Then, we construct approximate solutions to nonlinear problems by using the Picard iteration for the equations and the Newton iteration for the nonlinear boundary condition. Finally, by employing the energy estimates of linearized problems for the iteration scheme, we obtain the -ii- ?a? boundedness and convergence of the approximate solutions, which leads to the limit being the unique solution to the original nonlinear problem. KEYWORDS: quasilinear hyperbolic systems,nonlinear boundary condition, maximal dissipative boundary condition, well-posedness -iii- ?a? 1?X? 1.1 ?fl K ?5V?|3?+?X?n!?6N?!? ?N?n?|X?N? | Maxwell ?|?z?V|u?|?8 ?K1986cII? National Research Council ?uJ?8?:?Kk?=?“?5V? |”? 16 u?5V|?c?fl K?a?a?A?.?/,?aA?. ?/u?A?.?/3?V8?c?fi k? 678101821 ?)?fl K?5?.? uA?.?/3?Vl?c?kN?A? /e9?5?.? 59111517192022 Lax ? Phillips )?.?e3.?S?e)3 L2 e?55uA?.fl K)u.? K5? 2224 d?e?A?.?/? ?)k?K5=?y?)3? Sobolev ?m Hm ?u,?A?Xu5?6N? Euler ?)?k ?K5? 1234 ?,3?v?r?e ?)?k?K5? 915 ?k?fl K3 Hm(),m 3 ?m?kk?K5? 23 uA?.?A?.?fl K?.?5 .?u?5.?fl K3SA?2 -1- ?a? ?u?5V?|?5.?k7?k? u?fl K?k? A. Majda 3?p?-?5 ? 1213 ? 12 ?)?A?.fl K ouA?.?fl K9?5.?A?vk? ? ?k?Xefl K: - Rn,(n 2) ?k.m8 3?. ?- QT= (0,T) ,T = (0,T) ?e?fl K? L(t,x,u,t,x)u = F(t,x,u)in QT,(1-1) M(t,x,u) = gon T(1-2) u|t=0= fin (1-3) ? L(t,x,u,t,x) = A0(t,x,u)t+ n j=1 Aj(t,x,u)j t= /t,j= /xj? A0,A1, ,Anu (t,x,u) ? N N ? A0? u = (u1,u2, ,uN)T f(x) N? F(t,x,u) u (t,x,u) ?N?P N0, v Rn|v f| 0 ? 0, ?u A0?3 a0 0 ? u N0 a0I A0(t,x,u) 1 a0I ? (t,x) 0,T ?- = (1, ,n) . ? A? Xe A= n j=1 Ajj K A?.? 1.1? 1 ? ? N N ? e RN?5f?m ?g. u u ?K? ?U*?f?m? u u ?KK ? ?4?Kf?m ufl K (1-1)(1-2)(1-3) ?e? -2- ?a? ( A1 )M(t,x,u) 3 ? u (t,x,u) 1 w ? d ? ? ? ? ? uM(t,x,u) 3 0,T N0? d ( A2 )u? u N0,keruM(t,x,u) A?4?Kf?m ( A3 )u? u N0 A3 0,+) ?_ ( A3 0 )u? (t,x,u) 0,+) N0 rank(A) = k . ? .? (A3) ?A?.?.? (A3 0) ?A? .? 1.2?(J ?fl K (1 1),(1 2),(1 3) u?A?.?/XJ? (A1),(A2) ? f Hm+ 1 2() ? g Hm(T) ?v?