Contents 1. Introduction 1 2. Preliminaries 10 3. Outline of the proof 18 4. The algorithm 37 4.1. Approximate flux function 38 4.2. Approximate Hugoniot curves 40 4.3. Interpolations 41 4.4. Approximate mixed-curves 46 4.5. Approximate elementary curves 48 5. Basic interaction estimates 51 5.1. Strength estimates for interaction between wave-fronts of the same family 52 5.2. Speed estimates for interaction between wave-fronts of the same family 53 5.3. Interaction between waves of different families 55 5.4. Strength estimates for two coupled Riemann problems 56 6. Bounds on the total variation and on the interaction potential 60 6.1. Wave-front basic notations 63 6.2. Main estimates 72 6.3. Estimates on the increase of the interaction potential 86 6.4. Proof of Proposition 3.1 95 7. Estimates on the number of discontinuities 97 7.1. Proof of Proposition 3.2 98 7.2. Proof of Proposition 3.3 100 7.3. Proof of Proposition 3.4 108 8. Estimates on shift differentials 112 8.1. Estimates on shifting interactions 114 8.2. Basic notations 119 8.3. Main estimates 125 8.4. Estimates on the increase of the shift-interaction functional 135 8.5. Proof of Propositions 3.5-3.6 151
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