Boundary effect on a stationary viscous shock wave for scalar viscous conservation laws
In: Journal of mathematical analysis and applications, Jg. 255 (2001), Heft 2, S. 535-550
Online
academicJournal
- print, 5 ref
The initial-boundary value problem on the negative half-line R― {ut + f(u)x = uxx, (x,t) ∈ R―×(0,∞) u(0,t) = u+, u(x,0) = u0(x) = {→ u-, x→-∞ {=u+, x=0 is considered, subsequently to T.-P. Liu and K. Nishihara (1997, J. Differential Equations 133, 296-320). Here, the flux f is a smooth function satisfying f(u±) = 0 and the Oleinik shock condition f(Φ) < 0 for u+ < ø < u― if u+ < u― or f(Φ) > 0 for u+ > Φ > u if u+ < u-. In this situation the corresponding Cauchy problem on the whole line R = (-∞,∞) to (*) has a stationary viscous shock wave Φ(x + x0) for any fixed x0. Our aim in this paper is to show that the solution u(x,t) to (*) behaves as ø(x + d(t)) with d(t) = O(ln t) as t → ∞ under the suitable smallness conditions, When f = u2/2, the fact was shown by T.-P. Liu and S.-H. Yu (1997, Arch. Rational Mech, Anal. 139, 57-82), based on the Hopf-Cole transformation. Our proof is based on the weighted energy method.
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Boundary effect on a stationary viscous shock wave for scalar viscous conservation laws
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Autor/in / Beteiligte Person: | NISHIHARA, Kenii |
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Zeitschrift: | Journal of mathematical analysis and applications, Jg. 255 (2001), Heft 2, S. 535-550 |
Veröffentlichung: | San Diego, CA: Elsevier, 2001 |
Medientyp: | academicJournal |
Umfang: | print, 5 ref |
ISSN: | 0022-247X (print) |
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