scholarly journals ON THE PIECEWISE SMOOTHNESS OF ENTROPY SOLUTIONS TO SCALAR CONSERVATION LAWS FOR A LARGER CLASS OF INITIAL DATA

2007 ◽  
Vol 04 (03) ◽  
pp. 369-389 ◽  
Author(s):  
TAO TANG ◽  
JINGHUA WANG ◽  
YINCHUAN ZHAO
Keyword(s):  
Initial Data ◽  
Conservation Laws ◽  
Schwartz Space ◽  
Entropy Solutions ◽  
Scalar Conservation Laws ◽  
Smooth Initial Data ◽  
Compression Waves ◽  
First Category ◽  
Piecewise Smoothness ◽  
Scalar Conservation

We prove that if the initial data do not belong to a certain subset of Ck, then the solutions of scalar conservation laws are piecewise Cksmooth. In particular, our initial data allow centered compression waves, which was the case not covered by Dafermos (1974) and Schaeffer (1973). More precisely, we are concerned with the structure of the solutions in some neighborhood of the point at which only a Ck+1shock is generated. It is also shown that there are finitely many shocks for smooth initial data (in the Schwartz space) except for a certain subset of 𝒮(ℝ) of the first category. It should be pointed out that this subset is smaller than those used in previous works. We point out that Thom's theory of catastrophes, which plays a key role in Schaeffer (1973), cannot be used to analyze the larger class of initial data considered in this paper.

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.

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.

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