eBook ISBN:  9781470403997 
Product Code:  MEMO/169/801.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 
eBook ISBN:  9781470403997 
Product Code:  MEMO/169/801.E 
List Price:  $71.00 
MAA Member Price:  $63.90 
AMS Member Price:  $42.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 169; 2004; 170 ppMSC: Primary 35;
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 nonlinear. 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, entropyadmissible solutions of the Cauchy problem (1) with small initial data, for systems satisfying the above assumption.ReadershipGraduate students and research mathematicians interested in partial differential equations.

Table of Contents

Chapters

1. Introduction

2. Preliminaries

3. Outline of the proof

4. The algorithm

5. Basic interaction estimates

6. Bounds on the total variation and on the interaction potential

7. Estimates on the number of discontinuities

8. Estimates on shift differentials

9. Completion of the proof

10. Conclusion


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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 nonlinear. 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, entropyadmissible solutions of the Cauchy problem (1) with small initial data, for systems satisfying the above assumption.
Graduate students and research mathematicians interested in partial differential equations.

Chapters

1. Introduction

2. Preliminaries

3. Outline of the proof

4. The algorithm

5. Basic interaction estimates

6. Bounds on the total variation and on the interaction potential

7. Estimates on the number of discontinuities

8. Estimates on shift differentials

9. Completion of the proof

10. Conclusion