June  2020, 19(6): 3233-3244. doi: 10.3934/cpaa.2020141

Blowup of solutions to the thermal boundary layer problem in two-dimensional incompressible heat conducting flow

1. 

School of Mathematical Sciences, MOE-LSC and SHL-MAC, , Shanghai Jiao Tong University, Shanghai 200240, China

2. 

School of Mathematical Sciences, , Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author

Received  June 2019 Revised  November 2019 Published  March 2020

Fund Project: This research was partially supported by National Natural Science Foundation of China (NNSFC) under Grant No. 11631008

In this paper, we study the formation of singularities in a finite time for the solution of the boundary layer equations in the two-dimensional incompressible heat conducting flow. We obtain that the first order spacial derivative of the solution blows up in a finite time for the thermal boundary layer problem, for a kind of data which are analytic in the tangential variable but do not satisfy the Oleinik monotonicity condition, by using a Lyapunov functional approach. It is observed that the buoyancy coming from the temperature difference in the flow may destabilize the thermal boundary layer.

Citation: Yaguang Wang, Shiyong Zhu. Blowup of solutions to the thermal boundary layer problem in two-dimensional incompressible heat conducting flow. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3233-3244. doi: 10.3934/cpaa.2020141
References:
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R. AlexandreY. G. WangC. J. Xu and T. Yang, Well posedness of the Prandtl equation in Sobolev spaces, J. Amer. Math. Soc., 28 (2015), 745-784.  doi: 10.1090/S0894-0347-2014-00813-4.  Google Scholar

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D. Gérard-Varet and N. Masmoudi, Well-posedness for the Prandtl system without analyticity or monotoncity, Ann. Sci. Éc. Norm. Supér., 48 (2015), 1273–1325. doi: 10.24033/asens.2270.  Google Scholar

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Y. Guo and T. Nguyen, A note on the Prandtl boundary layers, Commun. Pure Appl. Math., 64 (2011), 1416-1438.  doi: 10.1002/cpa.20377.  Google Scholar

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I. KukavicaV. Vicol and F. Wang, The van Dommelen and Shen singularity in the Prandtl equations, Commun. Pure Appl. Math., 307 (2017), 288-311.  doi: 10.1016/j.aim.2016.11.013.  Google Scholar

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M. C. LombardoM. Cannone and M. Sammartino, Well-posedness of the boundary layer equations, SIAM J. Math. Anal., 35 (2003), 987-1004.  doi: 10.1137/S0036141002412057.  Google Scholar

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W. Li and T. Yang, Well-posedness in Gevrey function space for the Prandtl equations with non-degenerate critical points, J. Eur. Math. Soc., 22 (2020), 717-775.  doi: 10.4171/jems/931.  Google Scholar

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C. J. Liu and T. Yang, Ill-posedness of the Prandtl equations in Sobolev spaces around a shear flow with general decay, J. Math. Pure Appl., 108 (2017), 150-162.  doi: 10.1016/j.matpur.2016.10.014.  Google Scholar

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N. Masmoudi and T. K. Wong, Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods, Commun. Pure Appl. Math., 68 (2015), 1683-1741.  doi: 10.1002/cpa.21595.  Google Scholar

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O. A. Oleinik, The Prandtl system of equations in boundary layer theory, Soviet Math. Dokl., 4 (1963), 583-586.   Google Scholar

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O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, Chapman & Hall/CRC, Boca Raton, FL, 1999.  Google Scholar

[14]

L. Prandtl, Über Flüssigkeitsbewegungen bei sehr kleiner Reibung, Verh. III Intern. Math. Kongr., Heidelberg, (1904), 485–491. Google Scholar

[15]

M. Sammartino and R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half-space, Ⅰ. Existence for Euler and Prandtl equations, Commun. Math. Phys., 192 (1998), 433-461.  doi: 10.1007/s002200050304.  Google Scholar

[16]

M. Sammartino and R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half-space, Ⅱ. Construction of the Navier-Stokes solution, Commun. Math. Phys., 192 (1998), 463-491.  doi: 10.1007/s002200050305.  Google Scholar

[17]

H. Schlichting and K. Gersten., Boundary Layer Theory, 9$^th$ edition, Springer, Berlin, Heidelberg, 2017. doi: 10.1007/978-3-662-52919-5.  Google Scholar

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Z. Xin and L. Zhang, On the global existence of solutions to the Prandtl's system, Adv. Math., 181 (2004), 88-133.  doi: 10.1016/S0001-8708(03)00046-X.  Google Scholar

[19]

Y. G. Wang and S. Y. Zhu, Mathematical analysis of boundary layers in two-dimensional incompressible viscous heat conducting flows (in Chinese), Sci. Sin. Math., 49 (2019), 267-280.   Google Scholar

[20]

Y. G. Wang and S. Y. Zhu, Well-posedness of the boundary layer equation in incompressible heat conducting flow with analytic datum, to appear in Math. Meth. Appl. Sci.. doi: 10.1002/mma.6226.  Google Scholar

[21]

P. Zhang and Z. Zhang, Long time well-posedness of Prandtl system with small and analytic initial data, J. Funct. Anal., 270 (2016), 2591-2615.  doi: 10.1016/j.jfa.2016.01.004.  Google Scholar

show all references

References:
[1]

R. AlexandreY. G. WangC. J. Xu and T. Yang, Well posedness of the Prandtl equation in Sobolev spaces, J. Amer. Math. Soc., 28 (2015), 745-784.  doi: 10.1090/S0894-0347-2014-00813-4.  Google Scholar

[2]

D. ChenY. Wang and Z. Zhang, Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow, Ann. Inst. Henri Poincare - Anal. Non Lineaire, 35 (2018), 1119-1142.  doi: 10.1016/j.anihpc.2017.11.001.  Google Scholar

[3]

W. E and B. Engquist, Blowup of solutions of the unsteady Prandtl's equation, Commun. Pure Appl. Math., 50 (1997), 1287-1293.  doi: 10.1002/(SICI)1097-0312(199712)50:12<1287::AID-CPA4>3.0.CO;2-4.  Google Scholar

[4]

D. Gérard-Varet and E. Dormy, On the ill-posedness of the Prandtl equation, J. Amer. Math. Soc., 23 (2010), 591-609.  doi: 10.1090/S0894-0347-09-00652-3.  Google Scholar

[5]

D. Gérard-Varet and N. Masmoudi, Well-posedness for the Prandtl system without analyticity or monotoncity, Ann. Sci. Éc. Norm. Supér., 48 (2015), 1273–1325. doi: 10.24033/asens.2270.  Google Scholar

[6]

Y. Guo and T. Nguyen, A note on the Prandtl boundary layers, Commun. Pure Appl. Math., 64 (2011), 1416-1438.  doi: 10.1002/cpa.20377.  Google Scholar

[7]

I. KukavicaV. Vicol and F. Wang, The van Dommelen and Shen singularity in the Prandtl equations, Commun. Pure Appl. Math., 307 (2017), 288-311.  doi: 10.1016/j.aim.2016.11.013.  Google Scholar

[8]

M. C. LombardoM. Cannone and M. Sammartino, Well-posedness of the boundary layer equations, SIAM J. Math. Anal., 35 (2003), 987-1004.  doi: 10.1137/S0036141002412057.  Google Scholar

[9]

W. Li and T. Yang, Well-posedness in Gevrey function space for the Prandtl equations with non-degenerate critical points, J. Eur. Math. Soc., 22 (2020), 717-775.  doi: 10.4171/jems/931.  Google Scholar

[10]

C. J. Liu and T. Yang, Ill-posedness of the Prandtl equations in Sobolev spaces around a shear flow with general decay, J. Math. Pure Appl., 108 (2017), 150-162.  doi: 10.1016/j.matpur.2016.10.014.  Google Scholar

[11]

N. Masmoudi and T. K. Wong, Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods, Commun. Pure Appl. Math., 68 (2015), 1683-1741.  doi: 10.1002/cpa.21595.  Google Scholar

[12]

O. A. Oleinik, The Prandtl system of equations in boundary layer theory, Soviet Math. Dokl., 4 (1963), 583-586.   Google Scholar

[13]

O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, Chapman & Hall/CRC, Boca Raton, FL, 1999.  Google Scholar

[14]

L. Prandtl, Über Flüssigkeitsbewegungen bei sehr kleiner Reibung, Verh. III Intern. Math. Kongr., Heidelberg, (1904), 485–491. Google Scholar

[15]

M. Sammartino and R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half-space, Ⅰ. Existence for Euler and Prandtl equations, Commun. Math. Phys., 192 (1998), 433-461.  doi: 10.1007/s002200050304.  Google Scholar

[16]

M. Sammartino and R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half-space, Ⅱ. Construction of the Navier-Stokes solution, Commun. Math. Phys., 192 (1998), 463-491.  doi: 10.1007/s002200050305.  Google Scholar

[17]

H. Schlichting and K. Gersten., Boundary Layer Theory, 9$^th$ edition, Springer, Berlin, Heidelberg, 2017. doi: 10.1007/978-3-662-52919-5.  Google Scholar

[18]

Z. Xin and L. Zhang, On the global existence of solutions to the Prandtl's system, Adv. Math., 181 (2004), 88-133.  doi: 10.1016/S0001-8708(03)00046-X.  Google Scholar

[19]

Y. G. Wang and S. Y. Zhu, Mathematical analysis of boundary layers in two-dimensional incompressible viscous heat conducting flows (in Chinese), Sci. Sin. Math., 49 (2019), 267-280.   Google Scholar

[20]

Y. G. Wang and S. Y. Zhu, Well-posedness of the boundary layer equation in incompressible heat conducting flow with analytic datum, to appear in Math. Meth. Appl. Sci.. doi: 10.1002/mma.6226.  Google Scholar

[21]

P. Zhang and Z. Zhang, Long time well-posedness of Prandtl system with small and analytic initial data, J. Funct. Anal., 270 (2016), 2591-2615.  doi: 10.1016/j.jfa.2016.01.004.  Google Scholar

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