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Item Details
| Title:
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WELL-POSEDNESS FOR GENERAL 2 X 2 SYSTEMS OF CONSERVATION LAWS
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| By: |
Fabio Ancona (Editor), Andrea Marson (Editor) |
| Format: |
Paperback |
| List price:
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£70.50 |
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We currently do not stock this item, please contact the publisher directly for
further information.
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| ISBN 10: |
0821834355 |
| ISBN 13: |
9780821834350 |
| Publisher: |
AMERICAN MATHEMATICAL SOCIETY |
| Pub. date: |
15 March, 2004 |
| Series: |
Memoirs of the American Mathematical Society No. 169 |
| Pages: |
170 |
| Description: |
Considers the Cauchy problem for a strictly hyperbolic $2\times 2$ system of conservation laws in one space dimension $u_t+[F(u)]_x=0, u(0,x)=\bar u(x),$ which is neither linearly degenerate nor genuinely non-linear. |
| Synopsis: |
We consider the Cauchy problem for a strictly hyperbolic $2\times 2$ system of conservation laws in one space dimension $u_t+[F(u)]_x=0, u(0,x)=\bar u(x),$ which is neither linearly degenerate nor genuinely non-linear. We make the following assumption on the characteristic fields. If $r_i(u), \i=1,2,$ denotes the $i$-th right eigenvector of $DF(u)$ and $\lambda_i(u)$ the corresponding eigenvalue, then the set $\{u: \nabla \lambda_i \cdot r_i (u) = 0\}$ is a smooth curve in the $u$-plane that is transversal to the vector field $r_i(u)$. Systems of conservation laws that fulfill such assumptions arise in studying elastodynamics or rigid heat conductors at low temperature.For such systems we prove the existence of a closed domain $\mathcal{D} \subset L^1,$ containing all functions with sufficiently small total variation, and of a uniformly Lipschitz continuous semigroup $S:\mathcal{D} \times [0,+\infty)\rightarrow \mathcal{D}$ with the following properties. Each trajectory $t \mapsto S_t \bar u$ of $S$ is a weak solution of (1). Viceversa, if a piecewise Lipschitz, entropic solution $u= u(t,x)$ of (1) exists for $t \in [0,T],$ then it coincides with the trajectory of $S$, i.e.$u(t,\cdot) = S_t \bar u. This result yields the uniqueness and continuous dependence of weak, entropy-admissible solutions of the Cauchy problem with small initial data, for systems satysfying the above assumption. |
| Publication: |
US |
| Imprint: |
American Mathematical Society |
| Returns: |
Returnable |
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