Abstract
Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space R n+(n≥2) is considered around a given constant equilibrium. A solution formula for the linearized problem is derived, and Lp estimates for solutions of the linearized problem are obtained for 2≤p≤∞. It is shown that, as in the case of the Cauchy problem, the leading part of the solution of the linearized problem is decomposed into two parts, one that behaves like diffusion waves and the other one purely diffusively. There, however, are some aspects different from the Cauchy problem, especially in considering spatial derivatives. It is also shown that the solution of the linearized problem approaches for large times the solution of the nonstationary Stokes problem in some Lp spaces; and, as a result, a solution formula for the nonstationary Stokes problem is obtained. Large-time behavior of solutions of the nonlinear problem is then investigated in Lp spaces for 2≤p≤∞ by applying the results on the linearized analysis and the weighted energy method. The results indicate that there may be some nonlinear interaction phenomena not appearing in the Cauchy problem.
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References
Deckelnick, K.: Decay estimates for the compressible Navier-Stokes equations in unbounded domain. Math. Z. 209, 115–130 (1992)
Deckelnick, K.: L2–decay for the compressible Navier-Stokes equations in unbounded domains. Comm. Partial Differential Equations 18, 1445–1476 (1993)
Friedman, A.: Partial Differential Equations. Holt, Rinehart and Winston, 1969
Galdi, G. P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. 1. Springer-Verlag, New York, 1994
Hoff, D., Zumbrun, K.: Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow. Indiana Univ. Math. J. 44, 604–676 (1995)
Hoff, D., Zumbrun, K.: Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves. Z. Angew. Math. Phys. 48, 597–614 (1997)
Kagei, Y., Kobayashi, T.: On large time behavior of solutions to the Compressible Navier-Stokes Equations in the half space in R3. Arch. Ration. Mech. Anal. 165, 89–159 (2002)
Kawashima, S.: Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics. Ph. D. Thesis, Kyoto University, 1983
Kawashima, S., Matsumura, A., Nishida, T.: On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation. Comm. Math. Phys. 70, 97–124 (1979)
Kawashita, M.: On global solutions of Cauchy problems for compressible Navier-Stokes equations. Nonlinear Analysis 48, 1087–1105 (2002)
Kobayashi, T.: Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in R3. J. Differential Equations 184, 587–619 (2002)
Kobayashi, T., Shibata, Y.: Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in R3. Comm. Math. Phys. 200, 621–659 (1999)
Kobayashi, T., Shibata, Y.: Remarks on the rate of decay of solutions to linearized compressible Navier-Stokes equations. Pacific J. Math. 207, 199–234 (2002)
Liu, Tai-P., Wang, W.: The pointwise estimates of diffusion wave for the Navier-Stokes Systems in odd multi-dimensions. Comm. Math. Phys. 196, 145–173 (1998)
Matsumura, A.: An energy method for the equations of motion of compressible viscous and heat-conductive fluids. University of Wisconsin-Madison, MRC Technical Summary Report # 2194 1–16 (1981)
Matsumura, A., Nishida, T.: The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Japan Acad. Ser. A 55, 337–342 (1979)
Matsumura, A., Nishida, T.: The initial value problems for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20, 67–104 (1980)
Matsumura, A., Nishida, T.: Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Comm. Math. Phys. 89, 445–464 (1983)
Miyachi, A.: Oscillatory Integral Operators. (In Japanese) Gakushuin University Lecture Note Series 1, 1997
Ponce, G.: Global existence of small solutions to a class of nonlinear evolution equations. Nonlinear Anal. 9, 399–418 (1985)
Shibata, Y.: On the rate of decay of solutions to linear viscoelastic equation. Math. Meth. Appl. Sci. 23, 203–226 (2000)
Stein, E. M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Univ. Press, Princeton, 1993
Ukai, S.: A solution formula for the Stokes equation in Rn+. Comm. Pure Appl. Math. XL, 611–621 (1987)
Wang, W.: Large time behavior of solutions for general Navier-Stokes Systems in Multi-dimension. Wuhan Univ. J. Nat. Sci. 2, 385–393 (1997)
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Kagei, Y., Kobayashi, T. Asymptotic Behavior of Solutions of the Compressible Navier-Stokes Equations on the Half Space. Arch. Rational Mech. Anal. 177, 231–330 (2005). https://doi.org/10.1007/s00205-005-0365-6
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DOI: https://doi.org/10.1007/s00205-005-0365-6