zbMATH — the first resource for mathematics

Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical \(L^{p}\) framework. (English) Zbl 1366.35126
This work is devoted to the derivation of the (optimal) time decay rates of solutions for the isentropic compressible Navier-Stokes equations in the norms of critical spaces. The existence of global in time solutions has been discussed previously in \(L^2\) [R. Danchin, Invent. Math. 141, No. 3, 579–614 (2000; Zbl 0958.35100)] and \(L^p\) frameworks [F. Charve and R. Danchin, Arch. Ration. Mech. Anal. 198, No. 1, 233–271 (2010; Zbl 1229.35167)], [Q. Chen et al., Commun. Pure Appl. Math. 63, No. 9, 1173–1224 (2010; Zbl 1202.35002)], [B. Haspot, Arch. Ration. Mech. Anal. 202, No. 2, 427–460 (2011; Zbl 1427.76230)]. The method of proof relies on refined time weighted inequalities in the Fourier space.

35Q35 PDEs in connection with fluid mechanics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35B40 Asymptotic behavior of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
Full Text: DOI arXiv
[1] Bahouri, H., Chemin, J.-Y., Danchinm, R.: Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, Vol. 343, Springer, Berlin, 2011 · Zbl 0921.35092
[2] Charve, F.; Danchin, R., A global existence result for the compressible Navier-Stokes equations in the critical \(L\)\^{\(p\)} framework, Arch. Ration. Mech. Anal., 198, 233-271, (2010) · Zbl 1229.35167
[3] Chemin, J.-Y., Théorèmes d’unicité pour le système de Navier-Stokes tridimensionnel, Journal d’Analyse Mathématique, 77, 27-50, (1999) · Zbl 0938.35125
[4] Chemin, J.-Y.; Lerner, N., Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differ. Equ., 121, 314-328, (1995) · Zbl 0878.35089
[5] Chen, Q.; Miao, C.; Zhang, Z., Global well-posedness for the compressible Navier-Stokes equations with the highly oscillating initial velocity, Commun. Pure App. Math., 63, 1173-1224, (2010) · Zbl 1202.35002
[6] Danchin, R., Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141, 579-614, (2000) · Zbl 0958.35100
[7] Danchin, R., On the well-posedness of the incompressible density-dependent Euler equations in the \(L\)\^{\(p\)} framework, J. Differ. Equ., 248, 2130-2170, (2010) · Zbl 1192.35137
[8] Danchin, R., A Lagrangian approach for the compressible Navier-Stokes equations, Annales de l’Institut Fourier, 64, 753-791, (2014) · Zbl 1311.35214
[9] Danchin, R.: Fourier analysis methods for the compressible Navier-Stokes equations. Handbook of Mathematical Analysis in Mechanics of Viscous Fluids (to appear) · Zbl 1098.76062
[10] Danchin, R.; He, L., The incompressible limit in \(L\)\^{\(p\)} type critical spaces, Math. Ann.,, 366, 1365-1402, (2016) · Zbl 1354.35096
[11] Guo, Y.; Wang, Y.J., Decay of dissipative equations and negative Sobolev spaces, Commun. Part. Differ. Equ., 37, 2165-2208, (2012) · Zbl 1258.35157
[12] Haspot, B., Well-posedness in critical spaces for the system of compressible Navier-Stokes in larger spaces, J. Differ. Equ., 251, 2262-2295, (2011) · Zbl 1229.35182
[13] Haspot, B., Existence of global strong solutions in critical spaces for barotropic viscous fluids, Arch. Ration. Mech. Anal., 202, 427-460, (2011) · Zbl 1427.76230
[14] Hoff, D., Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differ. Equ.,, 120, 215-254, (1995) · Zbl 0836.35120
[15] Hoff, D.; Zumbrun, K., Multidimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44, 604-676, (1995) · Zbl 0842.35076
[16] Hoff, D., Zumbrun, K.: Multidimensional diffusion waves for the Navier-Stokes diffusion waves. Z. Angew. Math. Phys. 48, 597-614, 1997 · Zbl 0882.76074
[17] Kagei, Y.; Kobayashi, T., On large time behavior of solutions to the compressible Navier-Stokes equations in the half space in \({\mathbb{R}^3}\), Arch. Ration. Mech. Anal., 165, 89-159, (2002) · Zbl 1016.35055
[18] Kagei, Y.; Kobayashi, T., Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space, Arch. Ration. Mech. Anal., 177, 231-330, (2005) · Zbl 1098.76062
[19] Kobayashi, T., Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in \({\mathbb{R}^3}\). J. differ, Equ., 184, 587-619, (2002) · Zbl 1069.35051
[20] Kobayashi, T.; Shibata, Y., Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain of \({\mathbb{R}^3}\), Comm. Math. Phys., 200, 621-659, (1999) · Zbl 0921.35092
[21] Liu, T.-P.; Wang, W.-K., The pointwise estimates of diffusion waves for the Navier-Stokes equations in odd multi-dimensions, Commun. Math. Phys., 196, 145-173, (1998) · Zbl 0912.35122
[22] Matsumura, A.; Nishida, T., The initial value problem for the equation of motion of compressible viscous and heat-conductive fluids, Proc. Jpn. Acad. Ser. A, 55, 337-342, (1979) · Zbl 0447.76053
[23] Matsumura, A.; Nishida, T., The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20, 67-104, (1980) · Zbl 0429.76040
[24] Okita, M., Optimal decay rate for strong solutions in critical spaces to the compressible Navier-Stokes equations, J. Differ. Equ., 257, 3850-3867, (2014) · Zbl 1300.35080
[25] Ponce, G., Global existence of small solution to a class of nonlinear evolution equations, Nonlinear Anal. TMA, 9, 339-418, (1985) · Zbl 0576.35023
[26] Sohinger, V.; Strain, R.M., The Boltzmann equation, Besov spaces, and optimal time decay rates in \({\mathbb{R}^{n}_{x}}\), Adv. Math., 261, 274-332, (2014) · Zbl 1293.35195
[27] Xu, J.; Kawashima, S., The optimal decay estimates on the framework of Besov spaces for generally dissipative systems, Arch. Ration. Mech. Anal., 218, 275-315, (2015) · Zbl 1323.35141
[28] Zeng, Y., \(L\)\^{1} asymptotic behavior of compressible isentropic viscous 1-D flow, Commun. Pure Appl. Math., 47, 1053-1082, (1994) · Zbl 0807.35110
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.