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Existence of traveling waves associated with Lax shocks which violate Oleinik’s entropy criterion

http://repository.vnu.edu.vn/handle/VNU_123/28808

This paper answers to the question whether a shock wave in conservation laws satisfying the Lax shock inequalities but not Oleinik’s entropy criterion is admissible under the vanishing viscosity-capillarity effects.

Such a shock appears in van der Waals fluids when a secant line meets the graph of the flux function at four distinct points, and the shock jumps between the two farthest points.

The existence of the corresponding traveling waves would justify the admissibility of the shock.

For this purpose, we will first show that the corresponding traveling waves satisfy a system of differential equations with two saddle points and two asymptotically stable points.

Second, we estimate the domains of attractionof the asymptotically stable equilibrium points, relying on Lyapunov’s stability theory.

Third, we investigate the circumstances when an unstable trajectory leaving the saddle point corresponding to the left-hand state of the shock will ever enter the domain of attraction of each of the two asymptotically stable equilibrium points.

Finally, we establish the existence of traveling waves associatedwith a Lax shock but violating the Oleinik’s entropy criterion.

 

Title: Existence of traveling waves associated with Lax shocks which violate Oleinik’s entropy criterion
Authors: Nguyen Huu Hiep
Mai Duc Thanh
Nguyen Dinh Huy
Keywords: Shock wave
traveling wave
Lax shock inequalities
Oleinik’s entropy criterion
viscosity
capillarity
Lyapunov stability
Issue Date: 2017
Publisher: TAYLOR & FRANCIS LTD, 2-4 PARK SQUARE, MILTON PARK, ABINGDON OR14 4RN, OXON, ENGLAND
Citation: ISIKNOWLEDGE
Abstract: This paper answers to the question whether a shock wave in conservation laws satisfying the Lax shock inequalities but not Oleinik’s entropy criterion is admissible under the vanishing viscosity-capillarity effects. Such a shock appears in van der Waals fluids when a secant line meets the graph of the flux function at four distinct points, and the shock jumps between the two farthest points. The existence of the corresponding traveling waves would justify the admissibility of the shock. For this purpose, we will first show that the corresponding traveling waves satisfy a system of differential equationswith two saddle points and two asymptotically stable points. Second, we estimate the domains of attractionof the asymptotically stable equilibrium points, relying on Lyapunov’s stability theory. Third, we investigate the circumstances when an unstable trajectory leaving the saddle point corresponding to the left-hand state of the shock will ever enter the domain of attraction of each of the two asymptotically stable equilibrium points. Finally, we establish the existence of traveling waves associatedwith a Lax shock but violating the Oleinik’s entropy criterion.
Description: TNS07010 ; APPLICABLE ANALYSIS Volume: 96 Issue: 5 Pages: 810-826 Published: 2017
URI: http://repository.vnu.edu.vn/handle/VNU_123/28808
ISSN: 0003-6811
1563-504X
Appears in Collections: Bài báo của ĐHQGHN trong Web of Science

 

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