| Abstract |
Abstract: A central challenge in constructing entropy solutions of hyperbolic conservation laws is passing to the limit in nonlinear approximation schemes. Uniform (L^\infty) bounds provide weak compactness, but this alone is insufficient to identify the limit of the nonlinear flux. This talk introduces compensated compactness as a method for overcoming this difficulty by exploiting the entropy structure of the equations. We explain how compactness of entropy production yields constraints on oscillations in approximate solutions and how, under suitable structural assumptions, these constraints lead to strong convergence. We then discuss how this approach can be used to establish the existence of entropy solutions through vanishing viscosity and relaxation approximations.
References
[1] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4th ed., Springer, 2016.
[2] R. J. DiPerna, Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc. 292 (1985), 383–420. |
