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The Green’s function of the Navier-Stokes equations for gas dynamics in \(\mathbb R^3\). (English) Zbl 1075.76053
Summary: We derive pointwise estimates for the Green’s function of the Navier-Stokes equations for the compressible fluid. Our analysis shows that the short-time behavior of the Green’s function is dominated by the high-frequency waves but the large-time behavior is dictated by low-frequency waves. Furthermore, the low-frequency waves consist of entropy and acoustic waves that demonstrate a weaker form of Huygens’ principle.

76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q30 Navier-Stokes equations
Full Text: DOI
[1] Alfors, L.V.: Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable. New York: McGraw-Hill, second edition, 1966
[2] Evans, L.C.: Paritial Differential Equations. Volume 19 of Graduate Studies in Mathematics. Providence, Rhode Island: American Mathematical Society, 1998
[3] Hoff, D., Zumbrun, K.: Pointwise decay estimates for multidimensional Navier-Stokes diffusion Waves. Z. angew. Math. Phys. 48, 597–614 (1997) · Zbl 0882.76074 · doi:10.1007/s000330050049
[4] Kawashima, S.: Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics. PhD thesis, Kyoto University, 1983
[5] Liu, T.-P., Wang, W.: The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions. Commun. Math. Phys. 196, 145–173 (1998) · Zbl 0912.35122 · doi:10.1007/s002200050418
[6] Liu, T.-P., Zeng, Y.: Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws. Memoirs of the American Mathematical Society, 125, January 1997, pp. 599
[7] Zeng, Y.: L1 Asymptotic Behavior of Compressible Isentropic Viscous 1-D Flow. Commun. Pure Appl. Math. 47, 1053–1082 (1994) · Zbl 0807.35110 · doi:10.1002/cpa.3160470804
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