Glimm type functional and one dimensional systems of hyperbolic conservation laws
Student thesis: Doctoral Thesis
Related Research Unit(s)
Proposed by Glimm in the 1960’s , the Glimm scheme is widely used in one dimensional systems of hyperbolic conservation laws. In this scheme, one of the most important ideas is the construction of the Glimm functional, which plays an important role not only in proving the existence of the weak solution as Glimm did but also in many other problems. The original Glimm functional is applied to systems of hyperbolic conservation laws with each characteristic field genuinely nonlinear or linear degenerate. The uniqueness of the solution is proved in[6, 8, 11, 15]. To study the systems when the characteristic field may have finite transversal codimension one linear degeneracy manifolds, a new Glimm type functional is constructed in . The new functional introduces the concept of effective angle which measures the possibility of wave interaction between same family waves. Hence, this Glimm type functional is cubic order rather than quadratic, which causes difficulties in proving the consistency of the Glimm scheme. The convergence rate of the Glimm scheme in this case is shown as in , which is slower than the optimal one in the case of each characteristic field genuinely nonlinear or linear degenerate. To overcome this, in a joint work with Jiang Zaihong and Prof. Yang Tong, we have constructed another new Glimm type functional, which can be reduced to the classical one when the characteristic fields are genuinely nonlinear. With this new functional, the consistency and the same optimal convergence rate can be easily deduced. Hence, the functional may be regarded as optimal, which will be useful in other problems of general systems of hyperbolic conservation laws. The Glimm type functional is also essential in the well-posedness theory of systems of hyperbolic conservation laws with a moving source. In [32, 52], the BV and L1 stability of the system of hyperbolic conservation laws with a nonresonant source are studied. Here nonresonant source means that the source moves with a constant speed different from all characteristic speeds. In these works, the Glimm type functional with source term taken into account is the key issue. When the source is resonant with the system, the problem is more complicated. The BV and L1 stability of a single transonic shock wave solution are studied in [33, 48]. A criterion is given to test whether the transonic shock is stable or not, however, this condition is restricted to the case of weak shock. One can consider the case where the transonic i-th shock is relatively strong and stable in the sense of Majda. A new criterion of time asymptotic stability is obtained, which is an extension of the previous one. This is achieved by combining the Glimm type functional in [44, 45] and in  with some new fine estimates. By constructing the Liu-Yang functional, one can prove the L1 stability of the shock under the stability condition.
- Numerical solutions, Conservation laws (Mathematics), Cauchy problem