Undercompressive shocks and Riemann problems for scalar conservation laws with non-convex fluxes

1999 ◽  
Vol 129 (4) ◽  
pp. 733-754 ◽  
Author(s):  
Brian Hayes ◽  
Michael Shearer
Keyword(s):  
Initial Data ◽  
Conservation Laws ◽  
Riemann Problem ◽  
Initial Value Problem ◽  
Inflection Point ◽  
Travelling Wave ◽  
Problem Solution ◽  
Scalar Conservation Laws ◽  
Initial Value ◽  
Scalar Conservation

The Riemann initial value problem is studied for scalar conservation laws whose fluxes have a single inflection point. For a regularization consisting of balanced diffusive and dispersive terms, the travelling wave criterion is used to select admissible shocks. In some cases, the Riemann problem solution contains an undercompressive shock. The analysis is illustrated by exploring parameter space for the Buckley–Leverett flux. The boundary of the set of parameters for which there is a physical solution of the Riemann problem for all data is computed. Within the region of acceptable parameters, the solution hasseveral different forms, depending on the initial data; the different forms are illustrated by numerical computations. Qualitatively similar behaviour is observed in Lax–Wendroff approximations of solutions of the Buckley–Leverett equation with no dissipation or dispersion.

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.

Non-classical global solutions for a class of scalar conservation laws

2004 ◽  
Vol 134 (2) ◽  
pp. 225-240
Author(s):  
Paolo Baiti
Keyword(s):  
Conservation Laws ◽  
Global Solutions ◽  
Entropy Inequality ◽  
Travelling Wave ◽  
Scalar Conservation Laws ◽  
Kinetic Relation ◽  
Bounded Variation Functions ◽  
The Cauchy Problem ◽  
Scalar Conservation ◽  
The One

We consider the Cauchy problem for a class of scalar conservation laws with flux having a single inflection point. We prove existence of global weak solutions satisfying a single entropy inequality together with a kinetic relation, in a class of bounded variation functions. The kinetic relation is obtained by the travelling-wave criterion for a regularization consisting of balanced diffusive and dispersive terms. The result is applied to the one-dimensional Buckley-Leverett equation.

HIGH ORDER GEOMETRIC SMOOTHNESS FOR CONSERVATION LAWS

2005 ◽  
Vol 02 (01) ◽  
pp. 39-59
Author(s):  
MARTIN CAMPOS PINTO ◽  
ALBERT COHEN ◽  
PENCHO PETRUSHEV
Keyword(s):  
Conservation Laws ◽  
Hausdorff Distance ◽  
Besov Spaces ◽  
Important Implication ◽  
Adaptive Strategy ◽  
Uniform Norm ◽  
Scalar Conservation Laws ◽  
Initial Value ◽  
Piecewise Polynomials ◽  
Scalar Conservation

The smoothness of the solutions of 1D scalar conservation laws is investigated and it is shown that if the initial value has smoothness of order α in Lq with α > 1 and q = 1/α, this smoothness is preserved at any time t > 0 for the graph of the solution viewed as a function in a suitably rotated coordinate system. The precise notion of smoothness is expressed in terms of a scale of Besov spaces which also characterizes the functions that are approximated at rate N-α in the uniform norm by piecewise polynomials on N adaptive intervals. An important implication of this result is that a properly designed adaptive strategy should approximate the solution at the same rate N-α in the Hausdorff distance between the graphs.

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.

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