×

Optimal decay rates to conservation laws with diffusion-type terms of regularity-gain and regularity-loss. (English) Zbl 1241.35133

Summary: We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term related to an index \(s \in \mathbb R\) over the whole space \(\mathbb R^{n}\) for any spatial dimension \(n \geq 1\). Here, the diffusion-type source term behaves as the usual diffusion term over the low frequency domain while it admits on the high frequency part a feature of regularity-gain and regularity-loss for \(s < 1\) and \(s > 1\), respectively. For all \(s \in \mathbb R\), we not only obtain the \(L^{p}\)-\(L^{q}\) time-decay estimates on the linear solution semigroup but also establish the global existence and optimal time-decay rates of small-amplitude classical solutions to the nonlinear Cauchy problem. In the case of regularity-loss, the time-weighted energy method is introduced to overcome the weakly dissipative property of the equation. Moreover, the large-time behavior of solutions asymptotically tending to the heat diffusion waves is also studied. The current results have general applications to several concrete models arising from physics.

MSC:

35L65 Hyperbolic conservation laws
35B40 Asymptotic behavior of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] DOI: 10.1007/BF00376814 · Zbl 0851.35010
[2] DOI: 10.1007/3-540-29089-3
[3] DOI: 10.1142/S0219891611002421 · Zbl 1292.76080
[4] DOI: 10.1016/j.matpur.2009.10.007 · Zbl 1197.35050
[5] Duan R. J., Commun. Pure Appl. Math. 64 pp 1497–
[6] DOI: 10.1007/s00030-006-4023-y · Zbl 1132.35061
[7] DOI: 10.1142/S0218202508002760 · Zbl 1163.35324
[8] DOI: 10.1016/j.jde.2008.02.023 · Zbl 1154.35010
[9] DOI: 10.1093/qjmam/24.2.155 · Zbl 0219.76088
[10] DOI: 10.1142/S021820250600173X · Zbl 1108.35014
[11] DOI: 10.1142/S0218202508002802 · Zbl 1153.35013
[12] DOI: 10.1142/S0218202508002930 · Zbl 1160.35346
[13] Iguchi T., Hiroshima Math. J. 32 pp 229–
[14] DOI: 10.1142/S0218202596000109 · Zbl 0853.76076
[15] DOI: 10.1007/BF00280740 · Zbl 0343.35056
[16] DOI: 10.2206/kyushujm.63.139 · Zbl 1177.35034
[17] Kružkov S., Mat. Sb. 8 pp 228–
[18] DOI: 10.1016/S0022-0396(02)00158-4 · Zbl 1052.35126
[19] DOI: 10.1016/j.jde.2007.02.014 · Zbl 1120.35044
[20] DOI: 10.3934/cpaa.2011.10.209 · Zbl 1229.35016
[21] DOI: 10.1103/PhysRevA.40.7193
[22] DOI: 10.1016/j.jde.2010.07.029 · Zbl 1207.35231
[23] DOI: 10.1007/BF00375117 · Zbl 0793.76005
[24] DOI: 10.1007/978-1-4684-9320-7
[25] Vincenti W. G., Introduction to Physical Gas Dynamics (1965)
[26] DOI: 10.1016/j.na.2008.11.050 · Zbl 1178.35104
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.