Abstract
This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous conservation laws in the half space. It is proved that the solution converges to the corresponding degenerate stationary wave at the rate t- α / 4 as t → ∞, provided that the initial perturbation is in the weighted space Lα2 = L2 (R+ ; (1 + x)α) for α < αc (q) : = 3 + 2 / q, where q is the degeneracy exponent. This restriction on α is best possible in the sense that the corresponding linearized operator cannot be dissipative in Lα2 for α > αc (q). Our stability analysis is based on the space-time weighted energy method combined with a Hardy type inequality with the best possible constant.
| Original language | English |
|---|---|
| Pages (from-to) | 1-19 |
| Number of pages | 19 |
| Journal | Journal of Functional Analysis |
| Volume | 257 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 2009 Jul 1 |
| Externally published | Yes |
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Keywords
- Asymptotic stability
- Degenerate stationary waves
- Hardy inequality
- Viscous conservation laws
ASJC Scopus subject areas
- Analysis
Cite this
Hardy type inequality and application to the stability of degenerate stationary waves. / Kawashima, Shuichi; Kurata, Kazuhiro.
In: Journal of Functional Analysis, Vol. 257, No. 1, 01.07.2009, p. 1-19.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Hardy type inequality and application to the stability of degenerate stationary waves
AU - Kawashima, Shuichi
AU - Kurata, Kazuhiro
PY - 2009/7/1
Y1 - 2009/7/1
N2 - This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous conservation laws in the half space. It is proved that the solution converges to the corresponding degenerate stationary wave at the rate t- α / 4 as t → ∞, provided that the initial perturbation is in the weighted space Lα2 = L2 (R+ ; (1 + x)α) for α < αc (q) : = 3 + 2 / q, where q is the degeneracy exponent. This restriction on α is best possible in the sense that the corresponding linearized operator cannot be dissipative in Lα2 for α > αc (q). Our stability analysis is based on the space-time weighted energy method combined with a Hardy type inequality with the best possible constant.
AB - This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous conservation laws in the half space. It is proved that the solution converges to the corresponding degenerate stationary wave at the rate t- α / 4 as t → ∞, provided that the initial perturbation is in the weighted space Lα2 = L2 (R+ ; (1 + x)α) for α < αc (q) : = 3 + 2 / q, where q is the degeneracy exponent. This restriction on α is best possible in the sense that the corresponding linearized operator cannot be dissipative in Lα2 for α > αc (q). Our stability analysis is based on the space-time weighted energy method combined with a Hardy type inequality with the best possible constant.
KW - Asymptotic stability
KW - Degenerate stationary waves
KW - Hardy inequality
KW - Viscous conservation laws
UR - http://www.scopus.com/inward/record.url?scp=64849098118&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=64849098118&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2009.04.003
DO - 10.1016/j.jfa.2009.04.003
M3 - Article
AN - SCOPUS:64849098118
VL - 257
SP - 1
EP - 19
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
SN - 0022-1236
IS - 1
ER -