Abstract
In this paper, we consider the zero-viscosity limit of the 2-D Navier–Stokes equations in a simply-connected bounded domain with non-slip boundary condition. Based on the energy method in Wang et al. (Arch Ration Mech Anal 224(2):555–595, 2017), we justify the zero-viscosity limit under the analytic setting.
Similar content being viewed by others
References
Abidi, H., Danchin, R.: Optimal bounds for the inviscid limit of Navier–Stokes equations. Asymptot. Anal. 38, 35–46 (2004)
Alexandre, R., Wang, Y., Xu, C.-J., Yang, T.: Well-posedness of the Prandtl equation in Sobolev spaces. J. Am. Math. Soc. 28, 745–784 (2015)
Beale, J.T., Majda, A.: Rates of convergence for viscous splitting of the Navier–Stokes. Math. Comput. 37, 243–259 (1981)
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 35(4), 1119–1142 (2018)
Constantin, P., Kukavica, I., Vicol, V.: On the inviscid limit of the Navier–Stokes equations. Proc. Am. Math. Soc. 143(7), 3075–3090 (2015)
Constantine, G.M., Savits, T.H.: A multivariate Fa di Bruno formula with applications. Trans. Am. Math. Soc. 348(2), 503–520 (1996)
Fei, M., Tao, T., Zhang, Z.: On the zero-viscosity limit of the Navier–Stokes equations in \(\mathbf{R}^3_{+}\) without analyticity. J. Math. Pures Appl. 112, 170–229 (2018)
Gerard-Varet, D., Dormy, E.: On the ill-posedness of the Prandtl equation. J. Am. Math. Soc. 23, 591–609 (2010)
Gerard-Varet, D., Maekawa, Y., Masmoudi, N.: Gevrey stability of Prandtl expansions for 2D Navier–Stokes flows. arXiv:1607.06434
Gerard-Varet, D., Masmoudi, N.: Well-posedness for the Prandtl system without analyticity or monotonicity. Ann. Sci. šŠc. Norm. Super. 48(6), 1273–1325 (2015)
Gerard-Varet, D., Nguyen, T.: Remarks on the ill-posedness of the Prandtl equation. Asymptot. Anal. 77, 71–88 (2012)
Guo, Y., Nguyen, T.: A note on Prandtl boundary layers. Commun. Pure Appl. Math. 64, 1416–1438 (2011)
Iftimie, D., Planas, G.: Inviscid limits for the Navier–Stokes equations with Navier friction boundary conditions. Nonlinearity 19, 899–918 (2006)
Iftimie, D., Sueur, F.: Viscous boundary layer for the Navier–Stokes equations with the Navier slip conditions. Arch. Rational Mech. Anal. 199, 145–175 (2011)
Kato, T.: Nonstationary flows of viscous and ideal fluids in \(R^3\). J. Functional Anal. 9, 296–305 (1972)
Kato, T.: Remarks on zero viscosity limit for nonstationary Navier–Stokes flows with boundary. In: Seminar on nonlinear partial differential equations , pp. 85–98. Mathematical Sciences Research Institute Publications, 2. Springer, New York (1984)
Kelliher, J.P.: On Kato’s conditions for vanishing viscosity. Indiana Univ. Math. J. 56, 1711–1721 (2007)
Kelliher, J.P.: On the vanishing viscosity limit in a disk. Math. Ann. 343, 701–726 (2009)
Kukavica, I., Masmoudi, N., Vicol, V., Wong, T.: On the local well-posedness of the Prandtl and hydrostatic Euler equations with multiple monotonicity regions. SIAM J. Math. Anal. 46, 3865–3890 (2014)
Li, W., Yang, T.: Well-posedness in Gevrey space for the Prandtl equations with non-degenerate critical points. arXiv:1609.08430
Liu, C., Wang, Y., Yang, T.: On the ill-posedness of the Prandtl equations in three-dimensional space. Arch. Ration. Mech. Anal. 220(1), 83–108 (2016)
Liu, C., Wang, Y., Yang, T.: A well-posedness theory for the Prandtl equations in three space variables. Adv. Math. 308, 1074–1126 (2017)
Lombardo, M.C., Cannone, M., Sammartino, M.: Well-posedness of the boundary layer equations. SIAM J. Math. Anal. 35, 987–1004 (2003)
Maekawa, Y.: On the inviscid limit problem of the vorticity equations for viscous incompressible flows in the half-plane. Commun. Pure Appl. Math. 67, 1045–1128 (2014)
Masmoudi, N., Rousset, F.: Uniform regularity for the Navier–Stokes equation with Navier boundary condition. Arch. Ration. Mech. Anal. 203, 529–575 (2012)
Masmoudi, N., Rousset, F.: Uniform regularity and vanishing viscosity limit for the free surface Navier–Stokes equations. Arch. Ration. Mech. Anal. 223(1), 301–417 (2017)
Masmoudi, N., Wong, T.K.: Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods. Commun. Pure Appl. Math. 68, 1683–1741 (2015)
Oleinik, O.A., Samokhin, V.N.: Mathematical Models in Boundary Layer Theory. Applied Mathematics and Mathematical Computation, vol. 15. Chapman & Hall/CRC, Boca Raton, FL (1999)
Prandtl, L.: Uber flüssigkeits-bewegung bei sehr kleiner reibung. Actes du 3me Congrés international dse Mathématiciens, Heidelberg. Teubner, leipzig, 484–491 (1904)
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. Commun. Math. Phys. 192, 433–461 (1998)
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. Commun. Math. Phys. 192, 463–491 (1998)
Swann, H.S.G.: The convergence with vanishing viscosity of nonstationary Navier–Stokes flow to ideal flow in \( \mathbf{R}^3\). Trans. Am. Math. Soc. 157, 373–397 (1971)
Temam, R., Wang, X.: On the behavior of the solutions of the Navier–Stokes equations at vanishing viscosity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 25, 807–828 (1997)
Wang, C., Wang, Y., Zhang, Z.F.: Zero-viscosity limit of the Navier–Stokes equations in the analytic setting. Arch. Ration. Mech. Anal. 224(2), 555–595 (2017)
Wang, L., Xin, Z., Zang, A.: Vanishing viscous limits for 3D Navier–Stokes equations with a Navier slip boundary condition. J. Math. Fluid Mech. 14, 791–825 (2012)
Wang, X.: A Kato type theorem on zero viscosity limit of Navier–Stokes flows. Indiana Univ. Math. J. 50, 223–241 (2001)
Wang, Y., Xin, Z., Yong, Y.: Uniform regularity and vanishing viscosity limit for the compressible Navier–Stokes with general Navier–Slip boundary conditions in three-dimensional domains. SIAM J. Math. Anal. 47, 4123–4191 (2015)
Xiao, Y., Xin, Z.: On the vanishing viscosity limit for the 3D Navier–Stokes equations with a slip boundary condition. Commun. Pure Appl. Math. 60, 1027–1055 (2007)
Xin, Z., Zhang, L.: On the global existence of solutions to the Prandtl system. Adv. Math. 181, 88–133 (2004)
Zhang, P., Zhang, Z.: Long time well-posedness of Prandtl system with small and analytic initial data. J. Funct. Anal. 270(7), 2591–2615 (2016)
Acknowledgements
C. Wang is supported by NSF of China under Grant 11701016. Y. Wang is supported by China Postdoctoral Science Foundation 8206200009.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Communicated by Y. Maekawa.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: Well-Posedness of the Euler Equations and the Prandtl Equation in the Analytic Space
Appendix: Well-Posedness of the Euler Equations and the Prandtl Equation in the Analytic Space
In this appendix, we prove the well-posedness of the Euler equations and the Prandtl equation in the analytic space. The proof of well-poseness of the linearized Euler equations and Prandtl equation is similar, thus they are omitted.
1.1 Well-Posedness of the Euler Equations
Let us first introduce analytic norms
where \(\rho _E(t)=\delta (2-\frac{\lambda _E}{\delta } t)\ge \delta \) with \(\lambda _E\) defined later. What’ s more, operator \(\partial ^\gamma =\partial _\theta ^{\gamma _1}Z^{\gamma _2},~\nabla _{\theta ,r}^\alpha =(\partial _\theta ^{\alpha _1},\partial _r^{\alpha _2})\) . The main difference between definition \(\Vert \cdot \Vert _{X_b^k}\) and \(\Vert \cdot \Vert _{X_b^{k,\frac{1}{2}}}\) is that we add finite derivatives \(\nabla _{\theta ,r}^\alpha .\)
Proposition 9.1
Let initial data \((u_0, v_0)\) (and \((u^0, v^0)\) in polar coordinates) satisfies compatible condition (1.8), (1.9) and bound (1.10). Then there exists \(T_E>0\) so that the Euler equations (2.1) has a unique solution \(U_e^0=(u_e^0,v_e^0)\) in \([0,T_E]\), which satisfies
for any \(t\in [0,T_E]\), where \(f_0^e\) is given in (2.16) and \(U^e_0\) is given in (2.1) with relation
Proof
Here we just give a priori estimates of the solution. Set \(\omega _{i,e}^0\buildrel \hbox {def}\over =\partial _y \mathfrak {R}f_e^0-\partial _x\mathfrak {I}f_e^0\) be the vorticity of velocity \(f_e^0\) under Eulerian coordinates and \(\omega _{b,e}^0\buildrel \hbox {def}\over =-\frac{\partial _r}{r}(r u^0_e)+\frac{\partial _\theta }{r}v_e^0\) be the vorticity of velocity \((u_e^0,v_e^0)\) under polar coordinates. It is easy to see \(\omega _{i,e}^0\) and \(\omega _{b,e}^0\) satisfy
and
respectively.
Taking \(\langle \cdot , \cdot \rangle _{X^{20}_i}\) on both sides of (9.1) with \(\omega _{i,e}^0\), it is easy to get
Similarly, taking \(\langle \cdot , \cdot \rangle _{Y^{20}_b}\) on both sides of (9.2) with \(\omega _{b,e}^0\), it is easy to obtain
To close estimates, it remains to estimate \(\Vert \nabla _{r,\theta }U_e^0\Vert _{Y^{19}_b}^2\) and \(\Vert (f_e^0)_{\rho _E}\Vert _{L^2}^2.\) By calculation, we get equation for \(v_e^0:\)
We first estimate norm \(\Vert \nabla _{r,\theta }U_e^0\Vert _{X^{19}_b}^2\) due to there is no boundary term by integrating by parts. We do similar argument as Lemma 6.1 to arrive at the following estimates:
Then we take \(\rho _E\) small enough and use Young’s inequality to obtain
By the relation:
and along with estimate (9.4), we have
Taking \(\rho _E\) small enough, we have
Next, we estimate norm \(\Vert \nabla _{r,\theta }U_e^0\Vert _{Y^{19}_b}^2.\) Here we heavily use (9.5) and the equation of \(v_e^0\) [ see (9.3)] to obtain
On the other hand, we use
and an easy argument which is same as Lemma 5.13 that
In all, we get that
Here, we take \(\lambda _E=4CM\) and \(T_E>0\) so that
Then continuous argument ensures that
This gives the first estimate. The second estimate can be deduced by using the equation. \(\square \)
1.2 Well-Posedness of the Prandtl Equation
Let us introduce some weighted analytic norms
where \(\rho _P(t)=\delta (2-\frac{\lambda _p}{\delta } t)\ge \delta \) and \(\phi (t,R)=\rho _P(t)R^2\) with \(\lambda _P\) defined later.
Proposition 9.2
Let \((u_e^0,v_e^0)\) be given by Proposition 9.1. There exists \(T_P>0\) so that the Prandtl equation (2.6) has a unique solution \((u^p,v^p)\) in \([0,T_P]\), which satisfies
Proof
Again, we just present a priori estimates of the solution. We introduce a new function
It is easy to verify that \({\overline{u}}^p\) satisfies
where \(F^p\) is given by
Taking \(\langle \cdot , \cdot \rangle _{X^{17}_w}\) on both sides of (9.9) with \({\overline{u}}^p\), it is easy to get
For the dissipation term, we have
For the other term, \((u_{e}^0,v_{e}^0)\) is estimated in Proposition 9.1 and is bounded by constant in norm \(\Vert \cdot \Vert _{Y_b^k}\) and \(\Vert \cdot \Vert _{Y_b^{k,\frac{1}{2}}}\). Similar argument in Lemma 4.8, we have
In all, we deduce that
With this, a continuous argument ensures that there exists \(T_p>0\) so that
for any \(t\in [0,T_p]\). This implies the first estimate. For the second estimate, one can first prove that
Then the desired estimate can be deduced by using the equation of \(u^p\). \(\square \)
1.3 Proof of Lemmas 5.1, 5.3 and 5.4
First of all, Proposition 9.1 gives the existence of the solution \(U_e^0\) of (2.1) and the solution \(f_0^e\) of (2.16) with the bound
With \(U_e^0\) in hand, Proposition 9.2 gives the existence of the solution \((u_p^{0},v^{1}_p)\) of (2.6), (2.7) with the bound
Next, we can solve the linearized Euler equation (2.2), (2.3) of \(U_e^1\) in \(Y^{15}_b\) and \(f_e^1\) in \(X^{15}_i\). Finally, we solve the linearized Prandtl equation (2.10) of \((u_p^1, v_p^2)\) in \(X^{12}_w\). Then Lemma 5.1 follows easily.
While, recalling the definition of \((R_1,R_2,R_3)\) in (3.3)–(3.5) and \((f^a,\widetilde{{\widetilde{R}}})\) in (3.20) and (3.23), we know \((R_1,R_2,R_3)\) and \((f^a,\widetilde{{\widetilde{R}}})\) are composed of approximate solutions which are estimated in Lemma 5.1. As a result, Lemmas 5.3 and 5.4 can be deduced by using Lemmas 4.11 and 5.1. Here we omit the details.
Rights and permissions
About this article
Cite this article
Wang, C., Wang, Y. Zero-Viscosity Limit of the Navier–Stokes Equations in a Simply-Connected Bounded Domain Under the Analytic Setting. J. Math. Fluid Mech. 22, 8 (2020). https://doi.org/10.1007/s00021-019-0471-0
Accepted:
Published:
DOI: https://doi.org/10.1007/s00021-019-0471-0