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On the L∞ stability of Prandtl expansions in the Gevrey class

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Abstract

In this paper, we prove the L∞ ∩ L2 stability of Prandtl expansions of the shear flow type as \((U(y/\sqrt \nu ),0)\) for the initial perturbation in the Gevrey class, where U(y) is a monotone and concave function and ν is the viscosity coefficient. To this end, we develop the direct resolvent estimate method for the linearized Orr-Sommerfeld operator instead of the Rayleigh-Airy iteration method. Our method could be applied to the other relevant problems of hydrodynamic stability.

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References

  1. Abidi H, Danchin R. Optimal bounds for the inviscid limit of Navier-Stokes equations. Asymptot Anal, 2004, 38: 35–46

    MathSciNet  MATH  Google Scholar 

  2. Alexandre R, Wang Y-G, Xu C-J, et al. Well-posedness of the Prandtl equation in Sobolev spaces. J Amer Math Soc, 2015, 28: 745–784

    Article  MathSciNet  MATH  Google Scholar 

  3. Beale J T, Majda A. Rates of convergence for viscous splitting of the Navier-Stokes equations. Math Comp, 1981, 37: 243–259

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen D, Wang Y, Zhang Z. Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow. Ann Inst H Poincaré Anal Non Linéaire, 2018, 35: 1119–1142

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen Q, Li T, Wei D, et al. Transition threshold for the 2-D Couette flow in a finite channel. Arch Ration Mech Anal, 2020, 238: 125–183

    Article  MathSciNet  MATH  Google Scholar 

  6. Clopeau T, Mikelić A, Robert R. On the vanishing viscosity limit for 2D incompressible Navier-Stokes equations with the friction type boundary conditions. Nonlinearity, 1998, 11: 1625–1636

    Article  MathSciNet  MATH  Google Scholar 

  7. Constantin P, Wu J. Inviscid limit for vortex patches. Nonlinearity, 1995, 8: 735–742

    Article  MathSciNet  MATH  Google Scholar 

  8. Dietert H, Gérard-Varet D. Well-posedness of the Prandtl equations without any structural assumption. Ann PDE, 2019, 5: 8

    Article  MathSciNet  MATH  Google Scholar 

  9. Fei M, Tao T, Zhang Z. On the zero-viscosity limit of the Navier-Stokes equations in \(\mathbb{R}_ + ^3\) without analyticity. J Math Pures Appl (9), 2018, 112: 170–229

    Article  MathSciNet  MATH  Google Scholar 

  10. Gérard-Varet D, Dormy E. On the ill-posedness of the Prandtl equation. J Amer Math Soc, 2010, 23: 591–609

    Article  MathSciNet  MATH  Google Scholar 

  11. Gérard-Varet D, Maekawa Y. Sobolev stability of Prandtl expansions for the steady Navier-Stokes equations. Arch Ration Mech Anal, 2019, 223: 1319–1382

    Article  MathSciNet  MATH  Google Scholar 

  12. Gérard-Varet D, Maekawa Y, Masmoudi N. Gevrey stability of Prandtl expansions for 2-dimensional Navier-Stokes flows. Duke Math J, 2018, 167: 2531–2631

    Article  MathSciNet  MATH  Google Scholar 

  13. Gérard-Varet D, Maekawa Y, Masmoudi N. Optimal Prandtl expansion around concave boundary layer. arXiv:2005.05022, 2020

  14. Gérard-Varet D, Masmoudi N. Well-posedness for the Prandtl system without analyticity or monotonicity. Ann Sci Éc Norm Supér (4), 2015, 48: 1273–1325

    Article  MathSciNet  MATH  Google Scholar 

  15. Grenier E. On the nonlinear instability of Euler and Prandtl equations. Comm Pure Appl Math, 2000, 53: 1067–1091

    Article  MathSciNet  MATH  Google Scholar 

  16. Grenier E, Guo Y, Nguyen T. Spectral instability of characteristic boundary layer flows. Duke Math J, 2016, 165: 3085–3146

    Article  MathSciNet  MATH  Google Scholar 

  17. Grenier E, Nguyen T. L∞ instability of Prandtl layers. Ann PDE, 2019, 5: 18

    Article  MathSciNet  MATH  Google Scholar 

  18. Guo Y, Iyer S. Validity of steady Prandtl layer expansions. arXiv:1805.05891, 2018

  19. Guo Y, Nguyen T. Prandtl boundary layer expansions of steady Navier-Stokes flows over a moving plate. Ann PDE, 2017, 3: 10

    Article  MathSciNet  MATH  Google Scholar 

  20. Iftimie D, Planas G. Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions. Nonlinearity, 2006, 19: 899–918

    Article  MathSciNet  MATH  Google Scholar 

  21. Iftimie D, Sueur F. Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions. Arch Ration Mech Anal, 2011, 199: 145–175

    Article  MathSciNet  MATH  Google Scholar 

  22. Kato T. Nonstationary flows of viscous and ideal fluids in ℝ3. J Funct Anal, 1972, 9: 296–305

    Article  MathSciNet  MATH  Google Scholar 

  23. Kato T. Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary. In: Seminar on Nonlinear Partial Differential Equations. Mathematical Sciences Research Institute Publications, vol. 2. New York: Springer, 1984, 85–98

    Chapter  Google Scholar 

  24. Kelliher J P. On Kato’s conditions for vanishing viscosity. Indiana Univ Math J, 2007, 56: 1711–1721

    Article  MathSciNet  MATH  Google Scholar 

  25. Kukavica I, Vicol V, Wang F. The inviscid limit for the Navier-Stokes equations with data analytic only near the boundary. Arch Ration Mech Anal, 2020, 237: 779–827

    Article  MathSciNet  MATH  Google Scholar 

  26. Li W, Yang T. Well-posedness in Gevrey function spaces for the Prandtl equations with non-degenerate critical points. J Eur Math Soc JEMS, 2020, 22: 717–775

    Article  MathSciNet  MATH  Google Scholar 

  27. Lopes Filho M C, Mazzucato A L, Nussenzveig Lopes H J, et al. Vanishing viscosity limits and boundary layers for circularly symmetric 2D flows. Bull Braz Math Soc NS, 2008, 39: 471–513

    Article  MathSciNet  MATH  Google Scholar 

  28. Maekawa Y. On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane. Comm Pure Appl Math, 2014, 67: 1045–1128

    Article  MathSciNet  MATH  Google Scholar 

  29. Marchioro C. On the inviscid limit for a fluid with a concentrated vorticity. Comm Math Phys, 1998, 196: 53–65

    Article  MathSciNet  MATH  Google Scholar 

  30. Masmoudi N. Remarks about the inviscid limit of the Navier-Stokes system. Comm Math Phys, 2007, 270: 777–788

    Article  MathSciNet  MATH  Google Scholar 

  31. Masmoudi N, Rousset F. Uniform regularity for the Navier-Stokes equation with Navier boundary condition. Arch Ration Mech Anal, 2012, 203: 529–575

    Article  MathSciNet  MATH  Google Scholar 

  32. Masmoudi N, Wong T K. Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods. Comm Pure Appl Math, 2015, 68: 1683–1741

    Article  MathSciNet  MATH  Google Scholar 

  33. Mazzucato A, Taylor M. Vanishing viscosity limits for a class of circular pipe flows. Comm Partial Differential Equations, 2011, 36: 328–361

    Article  MathSciNet  MATH  Google Scholar 

  34. Sammartino M, Caflisch R E. Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations. Comm Math Phys, 1998, 192: 433–461

    Article  MathSciNet  MATH  Google Scholar 

  35. Sammartino M, Caflisch R E. Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. II. Construction of the Navier-Stokes solution. Comm Math Phys, 1998, 192: 463–491

    Article  MathSciNet  MATH  Google Scholar 

  36. Swann H. The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in ℝ3. Trans Amer Math Soc, 1971, 157: 373–397

    MathSciNet  MATH  Google Scholar 

  37. Temam R, Wang X. On the behavior of the solutions of the Navier-Stokes equations at vanishing viscosity. Ann Sc Norm Super Pisa Cl Sci (5), 1997, 25: 807–828

    MathSciNet  MATH  Google Scholar 

  38. Wang C, Wang Y, Zhang Z. Zero-viscosity limit of the Navier-Stokes equations in the analytic setting. Arch Ration Mech Anal, 2017, 224: 555–595

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang F. The three-dimensional inviscid limit problem with data analytic near the boundary. SIAM J Math Anal, 2020, 52: 3520–3545

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang L, Xin Z, Zang A. Vanishing viscous limits for 3D Navier-Stokes equations with a Navier slip boundary condition. J Math Fluid Mech, 2012, 14: 791–825

    Article  MathSciNet  MATH  Google Scholar 

  41. Wang X. A Kato type theorem on zero viscosity limit of Navier-Stokes flows. Indiana Univ Math J, 2001, 50: 223–241

    Article  MathSciNet  MATH  Google Scholar 

  42. Xiao Y, Xin Z. On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition. Comm Pure Appl Math, 2007, 60: 1027–1055

    Article  MathSciNet  MATH  Google Scholar 

  43. Xin Z, Zhang L. On the global existence of solutions to the Prandtl’s system. Adv Math, 2004, 181: 88–133

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Qi Chen & Zhifei Zhang

  2. School of Mathematics, South China University of Technology, Guangzhou, 510641, China

    Di Wu

Authors
  1. Qi Chen
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  2. Di Wu
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  3. Zhifei Zhang
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Correspondence to Zhifei Zhang.

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Chen, Q., Wu, D. & Zhang, Z. On the L∞ stability of Prandtl expansions in the Gevrey class. Sci. China Math. 65, 2521–2562 (2022). https://doi.org/10.1007/s11425-021-1896-5

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  • DOI: https://doi.org/10.1007/s11425-021-1896-5

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