Abstract. In this paper, we study the compactness in L^1_{loc} of the semigroup (S_t){t\geq 0} of entropy weak solutions to strictly convex scalar conservation laws in one space dimension. The compactness of S_t for each t > 0 was established by P. D. Lax. Upper estimates for the Kolmogorov e-entropy of the image of bounded sets in L^1 \cap L^\infty through S_t were given by C. De Lellis and F. Golse. Here we provide lower estimates on this e-entropy of the same order as the one established by De Lellis and Golse, thus showing that such an e-entropy is of size approximate to 1/e. Moreover, we extend these estimates of compactness to the case of convex balance laws.
Lower compactness estimates for scalar balance laws
ANCONA, FABIO;
2012
Abstract
Abstract. In this paper, we study the compactness in L^1_{loc} of the semigroup (S_t){t\geq 0} of entropy weak solutions to strictly convex scalar conservation laws in one space dimension. The compactness of S_t for each t > 0 was established by P. D. Lax. Upper estimates for the Kolmogorov e-entropy of the image of bounded sets in L^1 \cap L^\infty through S_t were given by C. De Lellis and F. Golse. Here we provide lower estimates on this e-entropy of the same order as the one established by De Lellis and Golse, thus showing that such an e-entropy is of size approximate to 1/e. Moreover, we extend these estimates of compactness to the case of convex balance laws.
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