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Shen, Rong; Wang, Yong

Optimal \(L^p\)-\(L^q\)-type decay rates of solutions to the three-dimensional nonisentropic compressible Euler equations with relaxation. (English) Zbl 1481.35069

Adv. Math. Phys. 2021, Article ID 8636092, 15 p. (2021).
Summary: In this paper, we consider the three-dimensional Cauchy problem of the nonisentropic compressible Euler equations with relaxation. Following the method of Y. Wu et al. [ibid. 2021, Article ID 5512285, 13 p. (2021; Zbl 1490.35290)], we show the existence and uniqueness of the global small \(H^k\) (\(k \geqslant 3\)) solution only under the condition of smallness of the \(H^3\) norm of the initial data. Moreover, we use a pure energy method with a time-weighted argument to prove the optimal \(L^p\)-\(L^q\) (\(1 \leqslant p \leqslant 2\), \(2 \leqslant q \leqslant \infty\))-type decay rates of the solution and its higher-order derivatives.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs
35Q31 Euler equations

Keywords:

energy method; Cauchy problem; small initial data

Citations:

Zbl 1490.35290
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\textit{R. Shen} and \textit{Y. Wang}, Adv. Math. Phys. 2021, Article ID 8636092, 15 p. (2021; Zbl 1481.35069)
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