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Weakly nonlinear geometric optics for hyperbolic systems of conservation laws (双曲守恒律方程组的弱非线性几何光学方法)
张永前教授(复旦大学数学科学学院)
2018-01-01 12:13  华东师范大学

学术报告

报告内容简介:We present a new approach to analyze the validation of weakly
nonlinear geometric optics for entropy solutions of nonlinear
hyperbolic systems of conservation laws whose eigenvalues are allowed
to have constant multiplicity and corresponding characteristic fields
to be linearly degenerate. The approach is based on our careful
construction of more accurate auxiliary approximation to weakly
nonlinear geometric optics, the properties of wave front-tracking
approximate solutions, the behavior of solutions to the approximate
asymptotic equations, and the standard semigroup estimates. To
illustrate this approach more clearly, we focus first on the Cauchy
problem for the hyperbolic systems with compact support initial data
of small bounded variation and establish that the $L^1-$estimate
between the entropy solution and the geometric optics expansion
function is bounded by $O(\varepsilon^2)$, independent of the time
variable. This implies that the simpler geometric optics expansion
functions can be employed to study the behavior of general entropy
solutions to hyperbolic systems of conservation laws. Finally, we
extend the results to the case with non-compact support initial data
of bounded variation. (joint work with G. Chen and W. Chen)