August  2022, 42(8): 3629-3659. doi: 10.3934/dcds.2022027

Mach limits in analytic spaces on exterior domains

Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA

*Corresponding author

Received  July 2021 Revised  January 2022 Published  August 2022 Early access  March 2022

We address the Mach limit problem for the Euler equations in an exterior domain with an analytic boundary. We first prove the existence of tangential analytic vector fields for the exterior domain with constant analyticity radii and introduce an analytic norm in which we distinguish derivatives taken from different directions. Then we prove the uniform boundedness of the solutions in the analytic space on a time interval independent of the Mach number, and Mach limit holds in the analytic norm. The results extend more generally to Gevrey initial data with convergence in a Gevrey norm.

Citation: Juhi Jang, Igor Kukavica, Linfeng Li. Mach limits in analytic spaces on exterior domains. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 3629-3659. doi: 10.3934/dcds.2022027
References:
[1]

T. Alazard, Incompressible limit of the nonisentropic Euler equations with the solid wall boundary conditions, Adv. Differential Equations, 10 (2005), 19-44. 

[2]

T. Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal., 180 (2006), 1-73.  doi: 10.1007/s00205-005-0393-2.

[3]

T. Alazard, A minicourse on the Low Mach number limit, Discret. Contin. Dyn. Syst. Ser. S, 1 (2008), 365-404.  doi: 10.3934/dcdss.2008.1.365.

[4]

K. Asano, On the incompressible limit of the compressible Euler equation, Japan J. Appl. Math., 4 (1987), 455-488.  doi: 10.1007/BF03167815.

[5]

C. Bardos, Analyticité de la solution de l'équation d'Euler dans un ouvert de $R^{n}$, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), A255–A258.

[6]

C. Bardos and S. Benachour, Domaine d'analycité des solutions de l'équation d'Euler dans un ouvert de $R^{n}$, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4 (1977), 647-687. 

[7]

A. Biswas, Local existence and Gevrey regularity of 3-D Navier-Stokes equations with $l_p$ initial data, J. Differential Equations, 215 (2005), 429-447.  doi: 10.1016/j.jde.2004.12.012.

[8]

A. Biswas and C. Foias, On the maximal space analyticity radius for the 3D Navier-Stokes equations and energy cascades, Ann. Mat. Pura Appl., 193 (2014), 739-777.  doi: 10.1007/s10231-012-0300-z.

[9]

J. L. BonaZ. Grujić and H. Kalisch, A KdV-type Boussinesq system: From the energy level to analytic spaces, Discrete Contin. Dyn. Syst., 26 (2010), 1121-1139.  doi: 10.3934/dcds.2010.26.1121.

[10]

J. P. Bourguignon and H. Brezis, Remarks on the Euler equation, J. Functional Analysis, 15 (1974), 341-363.  doi: 10.1016/0022-1236(74)90027-5.

[11]

Z. BradshawZ. Grujić and I. Kukavica, Local analyticity radii of solutions to the 3D Navier-Stokes equations with locally analytic forcing, J. Differential Equations, 259 (2015), 3955-3975.  doi: 10.1016/j.jde.2015.05.009.

[12]

G. CamliyurtI. Kukavica and V. Vicol, Analyticity up to the boundary for the Stokes and the Navier-Stokes systems, Trans. Amer. Math. Soc., 373 (2020), 3375-3422.  doi: 10.1090/tran/7990.

[13]

R. Danchin, Zero Mach number limit for compressible flows with periodic boundary conditions, Amer. J. Math., 124 (2002), 1153-1219.  doi: 10.1353/ajm.2002.0036.

[14]

R. Danchin, Zero Mach number limit in critical spaces for compressible Navier-Stokes equations, Ann. Sci. École Norm. Sup., 35 (2002), 27–75. doi: 10.1016/S0012-9593(01)01085-0.

[15]

R. Danchin and P. B. Mucha, From compressible to incompressible inhomogeneous flows in the case of large data, Tunis. J. Math., 1 (2019), 127-149.  doi: 10.2140/tunis.2019.1.127.

[16]

B. Desjardins and E. Grenier, Low Mach number limit of viscous compressible flows in the whole space, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 2271-2279.  doi: 10.1098/rspa.1999.0403.

[17]

M. M. Disconzi and D. G. Ebin, Motion of slightly compressible fluids in a bounded domain, II, Commun. Contemp. Math., 19 (2017), 1650054, 57 pp. doi: 10.1142/S0219199716500541.

[18]

M. M. Disconzi and C. Luo, On the incompressible limit for the compressible free-boundary Euler equations with surface tension in the case of a liquid, Arch. Ration. Mech. Anal., 237 (2020), 829-897.  doi: 10.1007/s00205-020-01516-4.

[19]

D. G. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraining force, Ann. Math., 105 (1977), 141-200.  doi: 10.2307/1971029.

[20]

E. Feireisl, Flows of viscous compressible fluids under strong stratification: Incompressible limits for long-range potential forces, Math. Models Methods Appl. Sci., 21 (2011), 7-27.  doi: 10.1142/S0218202511004964.

[21]

E. FeireislC. Klingenberg and S. Markfelder, On the low Mach number limit for the compressible Euler system, SIAM J. Math. Anal., 51 (2019), 1496-1513.  doi: 10.1137/17M1131799.

[22]

E. Feireisl and A. Novotný, Inviscid incompressible limits of the full Navier-Stokes-Fourier system, Comm. Math. Phys., 321 (2013), 605-628.  doi: 10.1007/s00220-013-1691-4.

[23]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369.  doi: 10.1016/0022-1236(89)90015-3.

[24]

Y. Giga, Time and spatial analyticity of solutions of the Navier-Stokes equations, Comm. Partial Differential Equations, 8 (1983), 929-948.  doi: 10.1080/03605308308820290.

[25]

Z. Grujić and I. Kukavica, Space analyticity for the Navier-Stokes and related equations with initial data in $L^p$, J. Funct. Anal., 152 (1998), 447-466.  doi: 10.1006/jfan.1997.3167.

[26]

D. Hoff, The zero-Mach limit of compressible flows, Comm. Math. Phys., 192 (1998), 543-554.  doi: 10.1007/s002200050308.

[27]

T. Iguchi, The incompressible limit and the initial layer of the compressible Euler equation in $\mathbf R^n_+$, Math. Methods Appl. Sci., 20 (1997), 945-958.  doi: 10.1002/(SICI)1099-1476(19970725)20:11<945::AID-MMA894>3.0.CO;2-T.

[28]

H. Isozaki, Wave operators and the incompressible limit of the compressible Euler equation, Commun. Math. Phys., 110 (1987), 519-524.  doi: 10.1007/BF01212426.

[29]

H. Isozaki, Singular limits for the compressible Euler equation in an exterior domain, J. Reine Angew. Math., 381 (1987), 1-36.  doi: 10.1515/crll.1987.381.1.

[30]

H. Isozaki, Singular limits for the compressible Euler equation in an exterior domain. II. Bodies in a uniform flow, Osaka J. Math., 26 (1989), 399-410. 

[31]

J. JangI. Kukavica and L. Li, Mach limits in analytic spaces, J. Differential Equations, 299 (2021), 284-332.  doi: 10.1016/j.jde.2021.07.014.

[32]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.  doi: 10.1002/cpa.3160340405.

[33]

S. Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math., 35 (1982), 629-651.  doi: 10.1002/cpa.3160350503.

[34]

G. Komatsu, Analyticity up to the boundary of solutions of nonliear parabolic equations, Comm. Pure Appl. Math., 32 (1979), 669-720.  doi: 10.1002/cpa.3160320504.

[35]

S. G. Krantz and H. R. Parks, A Primer of Real Analytic Functions, Basler Lehrbücher [Basel Textbooks], 1992, vol. 4, Birkhäuser Verlag, Basel. doi: 10.1007/978-3-0348-7644-5.

[36]

I. Kukavica and V. C. Vicol, The domain of analyticity of solutions to the three-dimensional Euler equations in a half space, Discrete Contin. Dyn. Syst., 29 (2011), 285-303.  doi: 10.3934/dcds.2011.29.285.

[37]

I. Kukavica and V. Vicol, On the analyticity and Gevrey-class regularity up to the boundary for the Euler equations, Nonlinearity, 24 (2011), 765-796.  doi: 10.1088/0951-7715/24/3/004.

[38]

C. D. Levermore and M. Oliver, Analyticity of solutions for a generalized Euler equation, J. Differential Equations, 133 (1997), 321-339.  doi: 10.1006/jdeq.1996.3200.

[39]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications: Volume III, 2012, Springer Science & Business Media.

[40]

P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl., 77 (1998), 585-627.  doi: 10.1016/S0021-7824(98)80139-6.

[41]

N. Masmoudi, Incompressible, inviscid limit of the compressible Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 199-224.  doi: 10.1016/s0294-1449(00)00123-2.

[42]

G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158 (2001), 61-90.  doi: 10.1007/PL00004241.

[43]

M. Oliver and E. S. Titi, On the domain of analyticity of solutions of second order analytic nonlinear differential equations, J. Differential Equations, 174 (2001), 55-74.  doi: 10.1006/jdeq.2000.3927.

[44]

S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Comm. Math. Phys., 104 (1986), 49-75.  doi: 10.1007/BF01210792.

[45]

S. Schochet, The mathematical theory of low Mach number flows, M2AN Math. Model. Numer. Anal., 39 (2005), 441-458.  doi: 10.1051/m2an:2005017.

[46]

S. Ukai, The incompressible limit and the initial layer of the compressible Euler equation, J. Math. Kyoto Univ., 26 (1986), 323-331.  doi: 10.1215/kjm/1250520925.

show all references

References:
[1]

T. Alazard, Incompressible limit of the nonisentropic Euler equations with the solid wall boundary conditions, Adv. Differential Equations, 10 (2005), 19-44. 

[2]

T. Alazard, Low Mach number limit of the full Navier-Stokes equations, Arch. Ration. Mech. Anal., 180 (2006), 1-73.  doi: 10.1007/s00205-005-0393-2.

[3]

T. Alazard, A minicourse on the Low Mach number limit, Discret. Contin. Dyn. Syst. Ser. S, 1 (2008), 365-404.  doi: 10.3934/dcdss.2008.1.365.

[4]

K. Asano, On the incompressible limit of the compressible Euler equation, Japan J. Appl. Math., 4 (1987), 455-488.  doi: 10.1007/BF03167815.

[5]

C. Bardos, Analyticité de la solution de l'équation d'Euler dans un ouvert de $R^{n}$, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), A255–A258.

[6]

C. Bardos and S. Benachour, Domaine d'analycité des solutions de l'équation d'Euler dans un ouvert de $R^{n}$, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 4 (1977), 647-687. 

[7]

A. Biswas, Local existence and Gevrey regularity of 3-D Navier-Stokes equations with $l_p$ initial data, J. Differential Equations, 215 (2005), 429-447.  doi: 10.1016/j.jde.2004.12.012.

[8]

A. Biswas and C. Foias, On the maximal space analyticity radius for the 3D Navier-Stokes equations and energy cascades, Ann. Mat. Pura Appl., 193 (2014), 739-777.  doi: 10.1007/s10231-012-0300-z.

[9]

J. L. BonaZ. Grujić and H. Kalisch, A KdV-type Boussinesq system: From the energy level to analytic spaces, Discrete Contin. Dyn. Syst., 26 (2010), 1121-1139.  doi: 10.3934/dcds.2010.26.1121.

[10]

J. P. Bourguignon and H. Brezis, Remarks on the Euler equation, J. Functional Analysis, 15 (1974), 341-363.  doi: 10.1016/0022-1236(74)90027-5.

[11]

Z. BradshawZ. Grujić and I. Kukavica, Local analyticity radii of solutions to the 3D Navier-Stokes equations with locally analytic forcing, J. Differential Equations, 259 (2015), 3955-3975.  doi: 10.1016/j.jde.2015.05.009.

[12]

G. CamliyurtI. Kukavica and V. Vicol, Analyticity up to the boundary for the Stokes and the Navier-Stokes systems, Trans. Amer. Math. Soc., 373 (2020), 3375-3422.  doi: 10.1090/tran/7990.

[13]

R. Danchin, Zero Mach number limit for compressible flows with periodic boundary conditions, Amer. J. Math., 124 (2002), 1153-1219.  doi: 10.1353/ajm.2002.0036.

[14]

R. Danchin, Zero Mach number limit in critical spaces for compressible Navier-Stokes equations, Ann. Sci. École Norm. Sup., 35 (2002), 27–75. doi: 10.1016/S0012-9593(01)01085-0.

[15]

R. Danchin and P. B. Mucha, From compressible to incompressible inhomogeneous flows in the case of large data, Tunis. J. Math., 1 (2019), 127-149.  doi: 10.2140/tunis.2019.1.127.

[16]

B. Desjardins and E. Grenier, Low Mach number limit of viscous compressible flows in the whole space, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 2271-2279.  doi: 10.1098/rspa.1999.0403.

[17]

M. M. Disconzi and D. G. Ebin, Motion of slightly compressible fluids in a bounded domain, II, Commun. Contemp. Math., 19 (2017), 1650054, 57 pp. doi: 10.1142/S0219199716500541.

[18]

M. M. Disconzi and C. Luo, On the incompressible limit for the compressible free-boundary Euler equations with surface tension in the case of a liquid, Arch. Ration. Mech. Anal., 237 (2020), 829-897.  doi: 10.1007/s00205-020-01516-4.

[19]

D. G. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraining force, Ann. Math., 105 (1977), 141-200.  doi: 10.2307/1971029.

[20]

E. Feireisl, Flows of viscous compressible fluids under strong stratification: Incompressible limits for long-range potential forces, Math. Models Methods Appl. Sci., 21 (2011), 7-27.  doi: 10.1142/S0218202511004964.

[21]

E. FeireislC. Klingenberg and S. Markfelder, On the low Mach number limit for the compressible Euler system, SIAM J. Math. Anal., 51 (2019), 1496-1513.  doi: 10.1137/17M1131799.

[22]

E. Feireisl and A. Novotný, Inviscid incompressible limits of the full Navier-Stokes-Fourier system, Comm. Math. Phys., 321 (2013), 605-628.  doi: 10.1007/s00220-013-1691-4.

[23]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369.  doi: 10.1016/0022-1236(89)90015-3.

[24]

Y. Giga, Time and spatial analyticity of solutions of the Navier-Stokes equations, Comm. Partial Differential Equations, 8 (1983), 929-948.  doi: 10.1080/03605308308820290.

[25]

Z. Grujić and I. Kukavica, Space analyticity for the Navier-Stokes and related equations with initial data in $L^p$, J. Funct. Anal., 152 (1998), 447-466.  doi: 10.1006/jfan.1997.3167.

[26]

D. Hoff, The zero-Mach limit of compressible flows, Comm. Math. Phys., 192 (1998), 543-554.  doi: 10.1007/s002200050308.

[27]

T. Iguchi, The incompressible limit and the initial layer of the compressible Euler equation in $\mathbf R^n_+$, Math. Methods Appl. Sci., 20 (1997), 945-958.  doi: 10.1002/(SICI)1099-1476(19970725)20:11<945::AID-MMA894>3.0.CO;2-T.

[28]

H. Isozaki, Wave operators and the incompressible limit of the compressible Euler equation, Commun. Math. Phys., 110 (1987), 519-524.  doi: 10.1007/BF01212426.

[29]

H. Isozaki, Singular limits for the compressible Euler equation in an exterior domain, J. Reine Angew. Math., 381 (1987), 1-36.  doi: 10.1515/crll.1987.381.1.

[30]

H. Isozaki, Singular limits for the compressible Euler equation in an exterior domain. II. Bodies in a uniform flow, Osaka J. Math., 26 (1989), 399-410. 

[31]

J. JangI. Kukavica and L. Li, Mach limits in analytic spaces, J. Differential Equations, 299 (2021), 284-332.  doi: 10.1016/j.jde.2021.07.014.

[32]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.  doi: 10.1002/cpa.3160340405.

[33]

S. Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math., 35 (1982), 629-651.  doi: 10.1002/cpa.3160350503.

[34]

G. Komatsu, Analyticity up to the boundary of solutions of nonliear parabolic equations, Comm. Pure Appl. Math., 32 (1979), 669-720.  doi: 10.1002/cpa.3160320504.

[35]

S. G. Krantz and H. R. Parks, A Primer of Real Analytic Functions, Basler Lehrbücher [Basel Textbooks], 1992, vol. 4, Birkhäuser Verlag, Basel. doi: 10.1007/978-3-0348-7644-5.

[36]

I. Kukavica and V. C. Vicol, The domain of analyticity of solutions to the three-dimensional Euler equations in a half space, Discrete Contin. Dyn. Syst., 29 (2011), 285-303.  doi: 10.3934/dcds.2011.29.285.

[37]

I. Kukavica and V. Vicol, On the analyticity and Gevrey-class regularity up to the boundary for the Euler equations, Nonlinearity, 24 (2011), 765-796.  doi: 10.1088/0951-7715/24/3/004.

[38]

C. D. Levermore and M. Oliver, Analyticity of solutions for a generalized Euler equation, J. Differential Equations, 133 (1997), 321-339.  doi: 10.1006/jdeq.1996.3200.

[39]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications: Volume III, 2012, Springer Science & Business Media.

[40]

P.-L. Lions and N. Masmoudi, Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl., 77 (1998), 585-627.  doi: 10.1016/S0021-7824(98)80139-6.

[41]

N. Masmoudi, Incompressible, inviscid limit of the compressible Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 199-224.  doi: 10.1016/s0294-1449(00)00123-2.

[42]

G. Métivier and S. Schochet, The incompressible limit of the non-isentropic Euler equations, Arch. Ration. Mech. Anal., 158 (2001), 61-90.  doi: 10.1007/PL00004241.

[43]

M. Oliver and E. S. Titi, On the domain of analyticity of solutions of second order analytic nonlinear differential equations, J. Differential Equations, 174 (2001), 55-74.  doi: 10.1006/jdeq.2000.3927.

[44]

S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Comm. Math. Phys., 104 (1986), 49-75.  doi: 10.1007/BF01210792.

[45]

S. Schochet, The mathematical theory of low Mach number flows, M2AN Math. Model. Numer. Anal., 39 (2005), 441-458.  doi: 10.1051/m2an:2005017.

[46]

S. Ukai, The incompressible limit and the initial layer of the compressible Euler equation, J. Math. Kyoto Univ., 26 (1986), 323-331.  doi: 10.1215/kjm/1250520925.

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