Shock Waves Increase of Entropy And Loss of Information
Shock Waves Increase of Entropy And Loss of Information
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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.
What to read after Shock Waves Increase of Entropy And Loss of Information?
You can find similar books in the "Read Also" column, or choose other free books by Peter D Lax to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
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.
What to read after Shock Waves Increase of Entropy And Loss of Information?
You can find similar books in the "Read Also" column, or choose other free books by Peter D Lax to read onlineMoreLess
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