Global Classical Solutions For 1-D Saint-Venant Systems

In the paper, we study the following 1-D Saint-Venant systemswhere A = A(t,x) is the cross-sectional area at time t and position x, v(t,x),g,h(A), are, respectively, the mean velocity over-slipping water-carrying section, the gravity acceleration and the water depth of the corresponding cross-sectional area A, and F(A, v)is the frictional resistance. For the Cauchy problem of Saint-Venant systems with different frictional resistanceF(A,v) (zero frictional resistance,linear dissipative frictional resistance,higher order dissipative frictional resistance and relaxation frictional resistance)and the initial-boundary value problem with zero frictional resistance, under certain hypotheses, we show the existenceof global classical solutions for the above problems, and give a sufficient conditions of blow up of solutions,also the estimates for life span of solutions is obtained. We also study the Saint-Venant systems with relaxation and prove the existence of weak solutions.The main results include the following three aspects:In chapter 2,we study the Cauchy problem for 1-D Saint-Venant systems. By introducing Riemann invariants, we consider the Cauchy problem for reducible Saint-Venant systems with frictional resistance F(A,v)≡0using the method of characteristics, we give a necessary and sufficient conditions of the existence for global classical solutions to Cauchy problem(0.0.2), and the life span of solutions is obtained. And then, we also discuss Cauchy problem with linear dissipative frictional resistanceF(A, v) = 2αv and higher order dissipative frictional resistance F(A, v) =α|v|^p-1v (whereα＞0 and p＞2 are constants), respectively. For the systems with linear dissipative frictional resistance, we show that if initial data is sufficiently small, the Cauchy problem admits a unique global classical solution on t≥0. At the same time, we verify that lower dissipation can guarantee existence of global classical solution to the first order quasilinear hyperbolic systems, but for higher order dissipation, generally speaking, the classical solution maybe blow up in a finite time, our results show that higher order dissipation will generate singulatity, and the life span of solution is obtained, too. Lastly,we discuss Cauchy problemof the systems with relaxation F(A,v) =(v-v_*(A))/δ:whereδ＞0 is the relaxation time, and v_*(A) is the equilirium state of the systems. The result indicates the dissipative effect of relaxation, i.e., relaxation can guarantee existence of global classical solution.It is well-konwn that, generally speaking, the initial-boundary value problem for the first order quasilinear hyperbolic systems there is no global classical solutions. In chapter 3, we study the initial-boundary value problem of Saint-Venant systems with zero frictional resistance, under certain assumptions, the global smooth resolvabillity is obtained.In chapter 4, we study the convergence of approximate solution to Saint-Venant systems with relaxation, by using of the normalized parabolic systems of(0.0.3):whereε＞0 is the coefficient of viscosity. With the method of compensated compactness we prove that there exists a sequence {A_δ^ε,v_δ^ε} which is subsequence of {A^ε,v^ε}, such thatasε→0+,and (A,v) is the weak solution to the Cauchy problem (0.0.3),where {A^ε,v^ε}is the solution sequence of Cauchy problem (0.0.4).In the last, we give the practical explaination of the thesis.