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Well-Posedness For General 2 X 2 Systems Of Conservation Laws
- new bookISBN: 9780821834350
We consider the Cauchy problem for a strictly hyperbolic $2 imes 2$ system of conservation laws in one space dimension $u t+[F(u)] x=0, u(0,x)= ar u(x),$ which is neither linearly degener… More...
We consider the Cauchy problem for a strictly hyperbolic $2 imes 2$ system of conservation laws in one space dimension $u t+[F(u)] x=0, u(0,x)= ar u(x),$ 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. We consider the Cauchy problem for a strictly hyperbolic $2 imes 2$ system of conservation laws in one space dimension $u t+[F(u)] x=0, u(0,x)= ar u(x),$ 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 ${u: ; abla lambda i cdot r i (u) = 0}$ 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 $S:mathcal{D} imes [0,+infty); ightarrow mathcal{D}$ with the following properties. Each trajectory $t mapsto S t ar u$ 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 ar 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. Books, Science and Geography~~Mathematics~~Calculus & Mathematical Analysis, Well-Posedness For General 2 X 2 Systems Of Conservation Laws~~Book~~9780821834350, , , , , , , , , ,, [PU: American Mathematical Society]<
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Ancona, Fabio:Well-Posedness for General 2 X 2 Systems of Conservation Laws
- used book ISBN: 9780821834350
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 degen… More...
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. Well-Posedness for General 2 X 2 Systems of Conservation Laws Ancona, Fabio, American Mathematical Society<
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Well-posedness for General 2 X 2 Systems of Conservation Laws (Paperback)
- Paperback2004, ISBN: 0821834355
[EAN: 9780821834350], Neubuch, [PU: American Mathematical Society, United States], Brand New Book. We consider the Cauchy problem for a strictly hyperbolic $2 times 2$ system of conservat… More...
[EAN: 9780821834350], Neubuch, [PU: American Mathematical Society, United States], Brand New Book. We consider the Cauchy problem for a strictly hyperbolic $2 times 2$ system of conservation laws in one space dimension $u t+[F(u)] x=0, u(0,x)= ar u(x),$ 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 $t mapsto S t ar u$ 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 ar 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.<
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Fabio Ancona; Andrea Marson:Well-posedness for General 2 X 2 Systems of Conservation Laws
- Paperback 2004, ISBN: 9780821834350
Softcover, Buch, [PU: American Mathematical Society]
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SAMPLE
Well-posedness for General 2 X 2 Systems of Conservation Laws
- Paperback2004, ISBN: 9780821834350
Softcover, Buch, [PU: American Mathematical Society]
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(*) Book out-of-stock means that the book is currently not available at any of the associated platforms we search.