We are concerned with the compactness in L^1_loc of the semigroup (St)_{t>0} of entropy weak solutions generated by hyperbolic conservation laws in one space dimension. This note provides a survey of recent results establishing upper and lower estimates for the Kolmogorov "-entropy of the image through the mapping S_t of bounded sets in L^1 \cap L^\infty, both in the case of scalar and of systems of conservation laws. As suggested by Lax [16], these quantitative compactness estimates could provide a measure of the order of "resolution" of the numerical methods implemented for these equations.
On quantitative compactness estimates for hyperbolic conservation laws
ANCONA, FABIO;
2014
Abstract
We are concerned with the compactness in L^1_loc of the semigroup (St)_{t>0} of entropy weak solutions generated by hyperbolic conservation laws in one space dimension. This note provides a survey of recent results establishing upper and lower estimates for the Kolmogorov "-entropy of the image through the mapping S_t of bounded sets in L^1 \cap L^\infty, both in the case of scalar and of systems of conservation laws. As suggested by Lax [16], these quantitative compactness estimates could provide a measure of the order of "resolution" of the numerical methods implemented for these equations.
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