Glimm type functional and one dimensional systems of hyperbolic conservation laws
Glimm 類泛函與一維系統雙曲守恒律
Student thesis: Doctoral Thesis
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Award date  15 Jul 2009 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(1b616c6f7b724e3c9d55f0713699676c).html 

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Abstract
Proposed by Glimm in the 1960’s [30], 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 [60]. 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 [38], 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 wellposedness 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 ith 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 [52] with some new fine estimates. By constructing
the LiuYang functional, one can prove the L1 stability of the shock under
the stability condition.
 Numerical solutions, Conservation laws (Mathematics), Cauchy problem