Hardy type inequality and application to the stability of degenerate stationary waves
Hardy type inequality and application to the stability of degenerate stationary waves
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DOI：
10.1016/j.jfa.2009.04.003
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发表时间：
2009-07
影响因子：
1.7
通讯作者：
S. Kawashima;K. Kurata
中科院分区：
文献类型：
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作者：
S. Kawashima;K. Kurata
研究了半空间中具有粘性守恒律的简并平稳波的渐近稳定性。证明了当初始扰动在加权空间Lα2=L2（R+;(1+x)α）中，当α<αc(q):=3+2/q，其中q为简并指数时，解收敛于相应的简并平稳波，速率为t−α/4as t→∞。对于α>αc(q)，相应的线性化算子在l α2中不可能是耗散的，因此对α的这种限制是最好的。我们的稳定性分析基于时空加权能量法，并结合具有最佳可能常数的Hardy型不等式。
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/4as 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 Lα2for 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.