Locally Lipschitz solutions of a single conservation law in several space variables |
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Authors: | David Hoff |
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Institution: | Department of Mathematics, Indiana University, Bloomington, Indiana 47401 USA |
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Abstract: | We consider the Cauchy problem for a single conservation law in several space variables. Letting u(x, t) denote the solution with initial data u0, we state necessary and sufficient conditions on u0 so that u(x, t) is locally Lipschitz continuous in the half space {t > 0}. These conditions allow for the preservation of smoothness of u0 as well as for the smooth resolution of discontinuities in u0. One consequence of our result is that u(x, t) cannot be locally Lipschitz unless u0 has locally bounded variation. Another is that solutions which are bounded and locally Lipschitz continuous in {t > 0} automatically have boundary values u0 at t = 0 in the sense that u(·, t) → u0 in Lloc1. Finally, we give an elementary proof that locally Lipschitz solutions satisfy Kruzkov's uniqueness condition. |
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