Shock Waves Increase of Entropy And Loss of Information
Shock Waves Increase of Entropy And Loss of Information
Peter D Lax
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It follows then, as shown at the beginning of this section, that u = lim u satisfies (2. 1) in the sense of distributions. The argument outlined above shows that for every sequence of €-. 0 we can select a subsequence such that the solutions u ' of (2. 17) with prescribed initial value Uq tend in the L sense to a distribution solution u of (2. 1) with initial value Uq. To prove that lim n exist, we have to show that any two subsequences have the €-►0 same limjt. For this we need the following c...haracterization of such limits, see [11]: Theorem 3. 4 : Let u be the L limit of a subsequence u ' of solutions of (2. 27). Let /? be any convex function, and V related to tj by (3. 22). Then (3. 32) ;?(u)^ + SO(u)jj $ in the sense of distribution. The proof follows from (3. 23); for when ;? is convex. N" ^ 0, and so (3. 23) implies /5^ - ^^V ^ €, (€) XX (3. 32) is the limit in the distribution sense of this relation as cO. Condition (3. 32) is called an entropy condition ; this notion will be elaborated in Sections 5 and 6.
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