ISBN: 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]<

2004, 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.<

2004, ISBN: 9780821834350

Softcover, Buch, [PU: American Mathematical Society]

2004, ISBN: 9780821834350

Softcover, Buch, [PU: American Mathematical Society]