scholarly journals Regularity and Lyapunov Stabilization of Weak Entropy Solutions to Scalar Conservation Laws

2017 ◽  
Vol 62 (4) ◽  
pp. 1620-1635 ◽  
Author(s):  
Sebastien Blandin ◽  
Xavier Litrico ◽  
Maria Laura Delle Monache ◽  
Benedetto Piccoli ◽  
Alexandre Bayen
Keyword(s):  
Conservation Laws ◽  
Entropy Solutions ◽  
Scalar Conservation Laws ◽  
Lyapunov Stabilization ◽  
Scalar Conservation

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.

scholarly journals Optimal regularity for all time for entropy solutions of conservation laws in $$BV^s$$

2020 ◽  
Vol 27 (5) ◽  
Author(s):  
Shyam Sundar Ghoshal ◽  
Billel Guelmame ◽  
Animesh Jana ◽  
Stéphane Junca
Keyword(s):  
Conservation Laws ◽  
Entropy Solutions ◽  
Optimal Regularity

scholarly journals Fractional BV spaces and applications to scalar conservation laws

2014 ◽  
Vol 11 (04) ◽  
pp. 655-677 ◽  
Author(s):  
C. Bourdarias ◽  
M. Gisclon ◽  
S. Junca
Keyword(s):  
Conservation Laws ◽  
Stability Result ◽  
Entropy Solutions ◽  
Smoothing Effect ◽  
Scalar Conservation Laws ◽  
One Dimensional ◽  
Regular Functions ◽  
Bv Functions ◽  
Scalar Conservation ◽  
First Time

We obtain new fine properties of entropy solutions to scalar nonlinear conservation laws. For this purpose, we study the "fractional BV spaces" denoted by BVs(ℝ) (for 0 < s ≤ 1), which were introduced by Love and Young in 1937 and closely related to the critical Sobolev space Ws,1/s(ℝ). We investigate these spaces in connection with one-dimensional scalar conservation laws. The BVs spaces allow one to work with less regular functions than BV functions and appear to be more natural in this context. We obtain a stability result for entropy solutions with BVs initial data. Furthermore, for the first time, we get the maximal Ws,p smoothing effect conjectured by Lions, Perthame and Tadmor for all nonlinear (possibly degenerate) convex fluxes.

scholarly journals Conservation laws driven by Lévy white noise

2015 ◽  
Vol 12 (03) ◽  
pp. 581-654 ◽  
Author(s):  
Imran H. Biswas ◽  
Kenneth H. Karlsen ◽  
Ananta K. Majee
Keyword(s):  
Conservation Laws ◽  
Entropy Solution ◽  
Young Measure ◽  
Entropy Solutions ◽  
Contraction Principle ◽  
Viscosity Method ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Lévy White Noise ◽  
Entropy Inequalities

We consider multidimensional conservation laws perturbed by multiplicative Lévy noise. We establish existence and uniqueness results for entropy solutions. The entropy inequalities are formally obtained by the Itó–Lévy chain rule. The multidimensionality requires a generalized interpretation of the entropy inequalities to accommodate Young measure-valued solutions. We first prove the existence of entropy solutions in the generalized sense via the vanishing viscosity method, and then establish the L1-contraction principle. Finally, the L1 contraction principle is used to argue that the generalized entropy solution is indeed the classical entropy solution.

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