%0 Journal Article
%D 2014
%T Global Structure of Admissible BV Solutions to Piecewise Genuinely Nonlinear, Strictly Hyperbolic Conservation Laws in One Space Dimension
%A Stefano Bianchini
%A Lei Yu
%X The paper describes the qualitative structure of an admissible BV solution to a strictly hyperbolic system of conservation laws whose characteristic families are piecewise genuinely nonlinear. More precisely, we prove that there are a countable set of points Θ and a countable family of Lipschitz curves T{script} such that outside T{script} ∪ Θ the solution is continuous, and for all points in T{script}{set minus}Θ the solution has left and right limit. This extends the corresponding structural result in [7] for genuinely nonlinear systems. An application of this result is the stability of the wave structure of solution w.r.t. -convergence. The proof is based on the introduction of subdiscontinuities of a shock, whose behavior is qualitatively analogous to the discontinuities of the solution to genuinely nonlinear systems.
%I Taylor & Francis
%G en
%U http://urania.sissa.it/xmlui/handle/1963/34694
%1 34908
%2 Mathematics
%4 1
%# MAT/05
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2015-10-22T09:34:23Z
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%R 10.1080/03605302.2013.775153
%0 Report
%D 2014
%T Structure of entropy solutions to general scalar conservation laws in one space dimension
%A Stefano Bianchini
%A Lei Yu
%I SISSA
%G en
%U http://hdl.handle.net/1963/7259
%1 7305
%2 Mathematics
%4 -1
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2014-03-10T11:39:18Z
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Global structure of entropy solutions to scalar conservation laws.ref.pdf: 464993 bytes, checksum: 9b83b6f3f845eda8eb202f0748617756 (MD5)
%0 Thesis
%D 2013
%T The structure and regularity of admissible BV solutions to hyperbolic conservation laws in one space dimension
%A Lei Yu
%X 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.
%I SISSA
%G en
%1 7210
%2 Mathematics
%4 1
%# MAT/05 ANALISI MATEMATICA
%$ Submitted by Lei Yu (yulei@sissa.it) on 2013-10-23T15:42:25Z
No. of bitstreams: 1
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%] 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
%0 Report
%D 2012
%T Global structure of admissible BV solutions to piecewise genuinely nonlinear, strictly hyperbolic conservation laws in one space dimension
%A Stefano Bianchini
%A Lei Yu
%K Hyperbolic conservation laws, Wave-front tracking, Global structure of solution.
%X The paper gives an accurate description of the qualitative structure of an admissible BV solution to a strictly hyperbolic, piecewise genuinely nonlinear system of conservation laws. We prove that there are a countable set $\\\\Theta$ which contains all interaction points and a family of countably many Lipschitz curves $\\\\T$ such that outside $\\\\T\\\\cup \\\\Theta$ $u$ is continuous, and along the curves in $\\\\T$, u has left and right limit except for points in $\\\\Theta$. This extends the corresponding structural result in \\\\cite{BL,Liu1} for admissible solutions.\\r\\n\\r\\nThe proof is based on approximate wave-front tracking solutions and a proper selection of discontinuity curves in the approximate solutions, which converge to curves covering the discontinuities in the exact solution $u$.
%I SISSA
%G en
%U http://hdl.handle.net/1963/6316
%1 6225
%2 Mathematics
%3 Functional Analysis and Applications
%4 1
%# MAT/05 ANALISI MATEMATICA
%$ Submitted by Lei Yu (yulei@sissa.it) on 2012-11-15T08:50:10Z\\nNo. of bitstreams: 1\\nglobal structure of solutions to PWGN hyperbolic conservation laws.pdf: 452219 bytes, checksum: 85bd51fc08fa53a087cee8aec2b9544a (MD5)