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