scholarly journals A note on front tracking and the equivalence between viscosity solutions of Hamilton–Jacobi equations and entropy solutions of scalar conservation laws

2002 ◽  
Vol 50 (4) ◽  
pp. 455-469 ◽  
Author(s):  
Kenneth Hvistendahl Karlsen ◽  
Nils Henrik Risebro
Keyword(s):  
Conservation Laws ◽  
Viscosity Solutions ◽  
Front Tracking ◽  
Entropy Solutions ◽  
Scalar Conservation Laws ◽  
Scalar Conservation ◽  
Jacobi Equations

scholarly journals A NOTE ON ADMISSIBLE SOLUTIONS OF 1D SCALAR CONSERVATION LAWS AND 2D HAMILTON–JACOBI EQUATIONS

2004 ◽  
Vol 01 (04) ◽  
pp. 813-826 ◽  
Author(s):  
LUIGI AMBROSIO ◽  
CAMILLO DE LELLIS
Keyword(s):  
Conservation Laws ◽  
Viscosity Solutions ◽  
Entropy Solutions ◽  
Uniformly Convex ◽  
Scalar Conservation Laws ◽  
Open Set ◽  
Admissible Solutions ◽  
Scalar Conservation ◽  
Jacobi Equations

Let Ω⊂ℝ2 be an open set and f∈C2(ℝ) with f" > 0. In this note we prove that entropy solutions of Dtu+Dxf(u) = 0 belong to SBV loc (Ω). As a corollary we prove the same property for gradients of viscosity solutions of planar Hamilton–Jacobi PDEs with uniformly convex Hamiltonians.

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

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.

STRUCTURE OF ENTROPY SOLUTIONS TO SCALAR CONSERVATION LAWS WITH STRICTLY CONVEX FLUX

2012 ◽  
Vol 09 (04) ◽  
pp. 571-611 ◽  
Author(s):  
ADIMURTHI ◽  
SHYAM SUNDAR GHOSHAL ◽  
G. D. VEERAPPA GOWDA
Keyword(s):  
Shock Wave ◽  
Initial Data ◽  
Conservation Laws ◽  
Compact Support ◽  
Space Dimension ◽  
Structure Theorem ◽  
Regions Of Interest ◽  
Entropy Solutions ◽  
Scalar Conservation Laws ◽  
Scalar Conservation

We consider scalar conservation laws in one space dimension with convex flux and we establish a new structure theorem for entropy solutions by identifying certain shock regions of interest, each of them representing a single shock wave at infinity. Using this theorem, we construct a smooth initial data with compact support for which the solution exhibits infinitely many shock waves asymptotically in time. Our proof relies on Lax–Oleinik explicit formula and the notion of generalized characteristics introduced by Dafermos.

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