Green’s function and pointwise convergence for compressible Navier-Stokes equations
Authors:
Shijin Deng and Shih-Hsien Yu
Journal:
Quart. Appl. Math. 75 (2017), 433-503
MSC (2010):
Primary 35Q30, 65M80
DOI:
https://doi.org/10.1090/qam/1461
Published electronically:
February 2, 2017
MathSciNet review:
3636165
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Abstract: In this paper we introduce a program to construct the Green’s function for the linearized compressible Navier-Stokes equations in several space dimensions. This program contains three components, a procedure to isolate global singularities in the Green’s function for a multi-spatial-dimensional problem, a long wave-short wave decomposition for the Green’s function and an energy method together with Sobolev inequalities. These three components together split the Green’s function into singular and regular parts with the singular part given explicitly and the regular part bounded by exponentially sharp pointwise estimates. The exponentially sharp singular-regular description of the Green’s function together with Duhamel’s principle and results of Matsumura-Nishida on $L^\infty$ decay yield through a bootstrap procedure an exponentially sharp space-time pointwise description of solutions of the full compressible Navier-Stokes equations in ${\mathbb R}^n(n=2,3)$.
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- Shijin Deng, Weike Wang, and Shih-Hsien Yu, Pointwise convergence to Knudsen layers of the Boltzmann equation, Comm. Math. Phys. 281 (2008), no. 2, 287–347. MR 2410897, DOI https://doi.org/10.1007/s00220-008-0496-3
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- B. Hanouzet and R. Natalini, Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Arch. Ration. Mech. Anal. 169 (2003), no. 2, 89–117. MR 2005637, DOI https://doi.org/10.1007/s00205-003-0257-6
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- David Hoff and Kevin Zumbrun, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. Angew. Math. Phys. 48 (1997), no. 4, 597–614. MR 1471469, DOI https://doi.org/10.1007/s000330050049
- David Hoff and Kevin Zumbrun, Asymptotic behavior of multidimensional scalar viscous shock fronts, Indiana Univ. Math. J. 49 (2000), no. 2, 427–474. MR 1793680, DOI https://doi.org/10.1512/iumj.2000.49.1942
- David Hoff and Kevin Zumbrun, Pointwise Green’s function bounds for multidimensional scalar viscous shock fronts, J. Differential Equations 183 (2002), no. 2, 368–408. MR 1919784, DOI https://doi.org/10.1006/jdeq.2001.4125
- Feimin Huang, Akitaka Matsumura, and Xiaoding Shi, Viscous shock wave and boundary layer solution to an inflow problem for compressible viscous gas, Comm. Math. Phys. 239 (2003), no. 1-2, 261–285. MR 1997442, DOI https://doi.org/10.1007/s00220-003-0874-9
- Yoshiyuki Kagei and Takayuki Kobayashi, Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space, Arch. Ration. Mech. Anal. 177 (2005), no. 2, 231–330. MR 2188049, DOI https://doi.org/10.1007/s00205-005-0365-6
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- Tai-Ping Liu and Weike Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Comm. Math. Phys. 196 (1998), no. 1, 145–173. MR 1643525, DOI https://doi.org/10.1007/s002200050418
- Tai-Ping Liu and Shih-Hsien Yu, The Green’s function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math. 57 (2004), no. 12, 1543–1608. MR 2082240, DOI https://doi.org/10.1002/cpa.20011
- Tai-Ping Liu and Shih-Hsien Yu, Green’s function of Boltzmann equation, 3-D waves, Bull. Inst. Math. Acad. Sin. (N.S.) 1 (2006), no. 1, 1–78. MR 2230121
- Tai-Ping Liu and Shih-Hsien Yu, Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation, Comm. Pure Appl. Math. 60 (2007), no. 3, 295–356. MR 2284213, DOI https://doi.org/10.1002/cpa.20172
- T.-P. Liu and S.-H. Yu, Multi-Dimensional Wave Propagation Over Burgers Shock Profile , preprint.
- Tai-Ping Liu and Yanni Zeng, Shock waves in conservation laws with physical viscosity, Mem. Amer. Math. Soc. 234 (2015), no. 1105, vi+168. MR 3244333, DOI https://doi.org/10.1090/memo/1105
- Akitaka Matsumura and Takaaki Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 9, 337–342. MR 555060
- Akitaka Matsumura and Takaaki Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ. 20 (1980), no. 1, 67–104. MR 564670, DOI https://doi.org/10.1215/kjm/1250522322
- Akitaka Matsumura and Takaaki Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys. 89 (1983), no. 4, 445–464. MR 713680
- Akitaka Matsumura and Kenji Nishihara, Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas, Comm. Math. Phys. 222 (2001), no. 3, 449–474. MR 1888084, DOI https://doi.org/10.1007/s002200100517
- O. Guès, G. Métivier, M. Williams, and K. Zumbrun, Boundary layer and long time stability for multidimensional viscous shocks, Discrete Contin. Dyn. Syst. 11 (2004), no. 1, 131–160. MR 2073950, DOI https://doi.org/10.3934/dcds.2004.11.131
- Olivier Guès, Guy Métivier, Mark Williams, and Kevin Zumbrun, Multidimensional viscous shocks. I. Degenerate symmetrizers and long time stability, J. Amer. Math. Soc. 18 (2005), no. 1, 61–120. MR 2114817, DOI https://doi.org/10.1090/S0894-0347-04-00470-9
- Gustavo Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal. 9 (1985), no. 5, 399–418. MR 785713, DOI https://doi.org/10.1016/0362-546X%2885%2990001-X
- Yasushi Shizuta and Shuichi Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985), no. 2, 249–275. MR 798756, DOI https://doi.org/10.14492/hokmj/1381757663
- Wen-An Yong, Entropy and global existence for hyperbolic balance laws, Arch. Ration. Mech. Anal. 172 (2004), no. 2, 247–266. MR 2058165, DOI https://doi.org/10.1007/s00205-003-0304-3
- Shih-Hsien Yu, Nonlinear wave propagations over a Boltzmann shock profile, J. Amer. Math. Soc. 23 (2010), no. 4, 1041–1118. MR 2669708, DOI https://doi.org/10.1090/S0894-0347-2010-00671-6
- Shih-Hsien Yu, Initial and shock layers for Boltzmann equation, Arch. Ration. Mech. Anal. 211 (2014), no. 1, 1–60. MR 3182477, DOI https://doi.org/10.1007/s00205-013-0684-y
- Yanni Zeng, $L^1$ asymptotic behavior of compressible, isentropic, viscous $1$-D flow, Comm. Pure Appl. Math. 47 (1994), no. 8, 1053–1082. MR 1288632, DOI https://doi.org/10.1002/cpa.3160470804
- Stefano Bianchini, Bernard Hanouzet, and Roberto Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math. 60 (2007), no. 11, 1559–1622. MR 2349349, DOI https://doi.org/10.1002/cpa.20195
- Shijin Deng, Weike Wang, and Shih-Hsien Yu, Pointwise convergence to Knudsen layers of the Boltzmann equation, Comm. Math. Phys. 281 (2008), no. 2, 287–347. MR 2410897, DOI https://doi.org/10.1007/s00220-008-0496-3
- Jonathan Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal. 95 (1986), no. 4, 325–344. MR 853782, DOI https://doi.org/10.1007/BF00276840
- Jonathan Goodman and Judith R. Miller, Long-time behavior of scalar viscous shock fronts in two dimensions, J. Dynam. Differential Equations 11 (1999), no. 2, 255–277. MR 1695245, DOI https://doi.org/10.1023/A%3A1021977329306
- B. Hanouzet and R. Natalini, Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Arch. Ration. Mech. Anal. 169 (2003), no. 2, 89–117. MR 2005637, DOI https://doi.org/10.1007/s00205-003-0257-6
- David Hoff and Kevin Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J. 44 (1995), no. 2, 603–676. MR 1355414, DOI https://doi.org/10.1512/iumj.1995.44.2003
- David Hoff and Kevin Zumbrun, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. Angew. Math. Phys. 48 (1997), no. 4, 597–614. MR 1471469, DOI https://doi.org/10.1007/s000330050049
- David Hoff and Kevin Zumbrun, Asymptotic behavior of multidimensional scalar viscous shock fronts, Indiana Univ. Math. J. 49 (2000), no. 2, 427–474. MR 1793680, DOI https://doi.org/10.1512/iumj.2000.49.1942
- David Hoff and Kevin Zumbrun, Pointwise Green’s function bounds for multidimensional scalar viscous shock fronts, J. Differential Equations 183 (2002), no. 2, 368–408. MR 1919784, DOI https://doi.org/10.1006/jdeq.2001.4125
- Feimin Huang, Akitaka Matsumura, and Xiaoding Shi, Viscous shock wave and boundary layer solution to an inflow problem for compressible viscous gas, Comm. Math. Phys. 239 (2003), no. 1-2, 261–285. MR 1997442, DOI https://doi.org/10.1007/s00220-003-0874-9
- Yoshiyuki Kagei and Takayuki Kobayashi, Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space, Arch. Ration. Mech. Anal. 177 (2005), no. 2, 231–330. MR 2188049, DOI https://doi.org/10.1007/s00205-005-0365-6
- S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, thesis, Kyoto University, 1983.
- Shuichi Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh Sect. A 106 (1987), no. 1-2, 169–194. MR 899951, DOI https://doi.org/10.1017/S0308210500018308
- Shuichi Kawashima and Akitaka Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys. 101 (1985), no. 1, 97–127. MR 814544
- David Linnan Li, The Green’s function of the Navier-Stokes equations for gas dynamics in $\mathbb {R}^3$, Comm. Math. Phys. 257 (2005), no. 3, 579–619. MR 2164944, DOI https://doi.org/10.1007/s00220-005-1351-4
- Tai-Ping Liu, Nonlinear stability of shock waves for viscous conservation laws, Mem. Amer. Math. Soc. 56 (1985), no. 328, v+108. MR 791863, DOI https://doi.org/10.1090/memo/0328
- Tai-Ping Liu, Pointwise convergence to shock waves for viscous conservation laws, Comm. Pure Appl. Math. 50 (1997), no. 11, 1113–1182. MR 1470318, DOI https://doi.org/10.1002/%28SICI%291097-0312%28199711%2950%3A11%24%5Clangle%241113%3A%3AAID-CPA3%24%5Crangle%243.3.CO%3B2-8
- Tai-Ping Liu and Weike Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Comm. Math. Phys. 196 (1998), no. 1, 145–173. MR 1643525, DOI https://doi.org/10.1007/s002200050418
- Tai-Ping Liu and Shih-Hsien Yu, The Green’s function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math. 57 (2004), no. 12, 1543–1608. MR 2082240, DOI https://doi.org/10.1002/cpa.20011
- Tai-Ping Liu and Shih-Hsien Yu, Green’s function of Boltzmann equation, 3-D waves, Bull. Inst. Math. Acad. Sin. (N.S.) 1 (2006), no. 1, 1–78. MR 2230121
- Tai-Ping Liu and Shih-Hsien Yu, Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation, Comm. Pure Appl. Math. 60 (2007), no. 3, 295–356. MR 2284213, DOI https://doi.org/10.1002/cpa.20172
- T.-P. Liu and S.-H. Yu, Multi-Dimensional Wave Propagation Over Burgers Shock Profile , preprint.
- Tai-Ping Liu and Yanni Zeng, Shock waves in conservation laws with physical viscosity, Mem. Amer. Math. Soc. 234 (2015), no. 1105, vi+168. MR 3244333, DOI https://doi.org/10.1090/memo/1105
- Akitaka Matsumura and Takaaki Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 9, 337–342. MR 555060
- Akitaka Matsumura and Takaaki Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ. 20 (1980), no. 1, 67–104. MR 564670
- Akitaka Matsumura, Takaaki Nishida, and Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys. 89 (1983), no. 4, 445–464. MR 713680
- Akitaka Matsumura and Kenji Nishihara, Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas, Comm. Math. Phys. 222 (2001), no. 3, 449–474. MR 1888084, DOI https://doi.org/10.1007/s002200100517
- O. Guès, G. Métivier, M. Williams, and K. Zumbrun, Boundary layer and long time stability for multidimensional viscous shocks, Discrete Contin. Dyn. Syst. 11 (2004), no. 1, 131–160. MR 2073950, DOI https://doi.org/10.3934/dcds.2004.11.131
- Olivier Guès, Guy Métivier, Mark Williams, and Kevin Zumbrun, Multidimensional viscous shocks. I. Degenerate symmetrizers and long time stability, J. Amer. Math. Soc. 18 (2005), no. 1, 61–120. MR 2114817, DOI https://doi.org/10.1090/S0894-0347-04-00470-9
- Gustavo Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal. 9 (1985), no. 5, 399–418. MR 785713, DOI https://doi.org/10.1016/0362-546X%2885%2990001-X
- Yasushi Shizuta and Shuichi Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985), no. 2, 249–275. MR 798756, DOI https://doi.org/10.14492/hokmj/1381757663
- Wen-An Yong, Entropy and global existence for hyperbolic balance laws, Arch. Ration. Mech. Anal. 172 (2004), no. 2, 247–266. MR 2058165, DOI https://doi.org/10.1007/s00205-003-0304-3
- Shih-Hsien Yu, Nonlinear wave propagations over a Boltzmann shock profile, J. Amer. Math. Soc. 23 (2010), no. 4, 1041–1118. MR 2669708, DOI https://doi.org/10.1090/S0894-0347-2010-00671-6
- Shih-Hsien Yu, Initial and shock layers for Boltzmann equation, Arch. Ration. Mech. Anal. 211 (2014), no. 1, 1–60. MR 3182477, DOI https://doi.org/10.1007/s00205-013-0684-y
- Yanni Zeng, $L^1$ asymptotic behavior of compressible, isentropic, viscous $1$-D flow, Comm. Pure Appl. Math. 47 (1994), no. 8, 1053–1082. MR 1288632, DOI https://doi.org/10.1002/cpa.3160470804
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Additional Information
Shijin Deng
Affiliation:
Department of Mathematics, Shanghai Jiao Tong University
MR Author ID:
817488
Email:
matdengs@sjtu.edu.cn
Shih-Hsien Yu
Affiliation:
Department of Mathematics, National University of Singapore
MR Author ID:
616205
Email:
matysh@nus.edu.sg
Received by editor(s):
November 20, 2016
Published electronically:
February 2, 2017
Article copyright:
© Copyright 2017
Brown University