Mathematics

This thesis is devoted to the study of the qualitative properties of admissible BV solutions to the strictly hyperbolic conservation laws in one space dimension by using wave-front tracking approximation. This thesis consists of two parts:
• SBV-like regularity of vanishing viscosity BV solutions to strict hyperbolic systems of conservation laws.
• Global structure of admissible BV solutions to strict hyperbolic conservation laws.

Mathematics

MAT/05 ANALISI MATEMATICA

Contents
0.1 HyperbolicConservationLaws. .......................... 1
0.2 SBV and SBV-like regularity ........................... 3
0.3 Global structure of BV solutions ......................... 6
0.4 Main notations ................................... 9
1 Preliminary results 11
1.1 BV and SBV functions............................... 11
1.2 Coarea formula for BV function.......................... 15
1.3 The singular conservation law........................... 16
1.3.1 The Riemann problem........................... 17
1.3.2 Front tracking algorithm.......................... 18
1.3.3 Uniform boundedness estimates on the speed of wave fronts . . . . . . 19
1.4 The Cauchy problem for systems ......................... 21
1.4.1 Solution of Riemann problem....................... 22
1.4.2 Construction of solution by wave-front tracking approximation . . . . 26
2 SBV-like regularity for strictly hyperbolic systems of conservation laws 33
2.1 Overview of the chapter .............................. 33
2.2 The scalar case ................................... 34
2.3 Notations and settings for general systems.................... 37
2.3.1 Preliminary notation............................ 37
2.3.2 Construction of solutions to the Riemann problem . . . . . . . . . . . 38
2.3.3 Cantor part of the derivative of characteristic for i-th waves . . . . . 39
2.4 Main SBV regularity argument .......................... 40
2.5 Review of wave-front tracking approximation for general system . . . . . . . . 41
2.5.1 Description of the wave-front tracking approximation . . . . . . . . . . 42
2.5.2 Jump part of i-th waves.......................... 43
2.6 Proof of Theorem2.4.1............................... 46
2.6.1 Decay estimate for positive waves..................... 46
2.6.2 Decay estimate for negative waves .................... 47
2.7 SBV regularity for the i-th component of the i-th eigenvalue . . . . . . . . . 54
i
CONTENTS
3 Global structure of admissible BV solutions to the piecewise genuinely nonlinear system 57 3.1 Description of wave-front tracking approximation . . . . . . . . . . . . . . . . 62
3.2 Construction of subdiscontinuity curves ..................... 63
3.3 Proof of the main theorems ............................ 67
3.4 A counterexample on general strict hyperbolic systems . . . . . . . . . . . . . 71
4 Global structure of entropy solutions to general scalar conservation law 75
4.1 Overview ...................................... 75
4.2 Estimates on the level sets of the front tracking approximations . . . . . . . . 76
4.2.1 Bounds on the initial points of the boundary curves of level sets . . . 77
4.2.2 Bound estimates on the derivative of the boundary curves of level sets 77
4.3 Level sets in the exact solutions.......................... 78
4.4 Lagrangian representative for the entropy solution . . . . . . . . . . . . . . . 84
4.5 Pointwise structure................................. 88