# Well-Posedness for General 2 X 2 Systems of Conservation Laws (Memoirs of the American Mathematical Society Series #801)

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$60.00 Seller since 2007 Feedback rating: (466) Condition: New — never opened or used in original packaging. Like New — packaging may have been opened. A "Like New" item is suitable to give as a gift. Very Good — may have minor signs of wear on packaging but item works perfectly and has no damage. Good — item is in good condition but packaging may have signs of shelf wear/aging or torn packaging. All specific defects should be noted in the Comments section associated with each item. Acceptable — item is in working order but may show signs of wear such as scratches or torn packaging. All specific defects should be noted in the Comments section associated with each item. Used — An item that has been opened and may show signs of wear. All specific defects should be noted in the Comments section associated with each item. Refurbished — A used item that has been renewed or updated and verified to be in proper working condition. Not necessarily completed by the original manufacturer. New Brand new. We distribute directly for the publisher. 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 (1) with small initial data, for systems satysfying the above assumption. Read more Show Less Ships from: Boonsboro, MD Usually ships in 1-2 business days • Canadian • International • Standard, 48 States • Standard (AK, HI) • Express, 48 States • Express (AK, HI) Page 1 of 1 Showing All Close Sort by ## More About This Textbook ### Overview 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). Vice versa, 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 (1) with small initial data, for systems satisfying the above assumption.

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### Product Details

• ISBN-13: 9780821834350
• Publisher: American Mathematical Society
• Publication date: 5/1/2004
• Series: Memoirs of the American Mathematical Society Series , #801
• Edition description: 2 of 4 Numbers, May 2004 Edition
• Pages: 170
• Product dimensions: 6.96 (w) x 9.98 (h) x 0.38 (d)

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