scholarly journals The generalized nonlinear initial–boundary Riemann problem for linearly degenerate quasilinear hyperbolic systems of conservation laws

2011 ◽  
Vol 379 (2) ◽  
pp. 589-615 ◽  
Author(s):  
Zhi-Qiang Shao
Keyword(s):  
Conservation Laws ◽  
Riemann Problem ◽  
Hyperbolic Systems ◽  
Initial Boundary ◽  
Quasilinear Hyperbolic Systems ◽  
Systems Of Conservation Laws ◽  
Linearly Degenerate

Local exact boundary controllability of entropy solutions to linearly degenerate quasilinear hyperbolic systems of conservation laws

2018 ◽  
Vol 24 (2) ◽  
pp. 793-810
Author(s):  
Tatsien Li ◽  
Lei Yu
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Systems ◽  
Front Tracking ◽  
Entropy Solutions ◽  
Exact Boundary Controllability ◽  
Boundary Controllability ◽  
Quasilinear Hyperbolic Systems ◽  
Exact Boundary ◽  
Systems Of Conservation Laws ◽  
Linearly Degenerate

In this paper, we study the local exact boundary controllability of entropy solutions to linearly degenerate quasilinear hyperbolic systems of conservation laws with characteristics of constant multiplicity. We prove the two-sided boundary controllability, the one-sided boundary controllability and the two-sided boundary controllability with fewer controls, by applying the strategy used in [T. Li and L. Yu, J. Math. Pures et Appl. 107 (2017) 1–40; L. Yu, Chinese Ann. Math., Ser. B (To appear)]. Our constructive method is based on the well-posedness of semi-global solutions constructed by the limit of ε-approximate front tracking solutions to the mixed initial-boundary value problem with general nonlinear boundary conditions, and on some further properties of both ε-approximate front tracking solutions and limit solutions.

GLOBAL EXISTENCE OF PIECEWISE C1 SOLUTIONS TO THE GENERALIZED RIEMANN PROBLEM

2004 ◽  
Vol 01 (02) ◽  
pp. 329-350 ◽  
Author(s):  
TATSIEN LI ◽  
LIBIN WANG
Keyword(s):  
Global Existence ◽  
Riemann Problem ◽  
Hyperbolic Systems ◽  
Quasilinear Hyperbolic Systems ◽  
Generalized Riemann Problem ◽  
Contact Discontinuities ◽  
Systems Of Conservation Laws ◽  
Global Existence And Uniqueness ◽  
Linearly Degenerate ◽  
Characteristic Field

Assuming that each characteristic field is either genuinely nonlinear or linearly degenerate in the sense of Lax, we prove the global existence and uniqueness of piecewise C1 solution containing n small amplitude waves (shocks corresponding to genuinely nonlinear fields and contact discontinuities corresponding to linearly degenerate fields) for the generalized Riemann problem associated with quasilinear hyperbolic systems of conservation laws with small, decaying, piecewise C1 initial data. The solution possesses a global structure similar to that of the similarity solution to the corresponding Riemann problem.

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