D. Albritton, Blow-up criteria for the Navier-Stokes equations in non-endpoint critical Besov spaces

R. Alexandre, Y. Wang, C. Xu, and T. Yang, Well-posedness of The Prandtl Equation in Sobolev Spaces, J. Amer. Math. Soc, vol.40, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00682867

C. Bjorland, L. Brandolese, D. &. Iftimie, and . Schonbek, L p -solutions of the steady-state Navier-Stokes equations with rough external forces. Communications in Partial Differential Equations, vol.36, pp.216-246, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00378603

C. P. Calderón, Existence of weak solutions for the Navier-Stokes equations with initial data in L p, Trans. Amer. Math. Soc, vol.318, issue.1, pp.179-200, 1990.

M. Cannone, M. C. Lombardo, and M. Sammartino, Well-posedness of the Prandtl equation with non compatible data, Nonlinearity, vol.26, pp.3077-3100, 2013.

M. Cannone and &. Karch, About the regularized Navier-Stokes equations, J. Math. Fluid Mech, vol.7, issue.1, pp.1-28, 2005.

M. Cannone-&-f.-planchon, On the non-stationary Navier-Stokes equations with an external force, Adv. Differential Equations, vol.4, issue.5, pp.697-730, 1999.

J. Chemin, Théorèmes d'unicité pour le système de Navier-Stokes tridimensionnel. (French) [Uniqueness theorems for the three-dimensional NavierStokes system

, J. Anal. Math, vol.77, pp.27-50, 1999.

J. Chemin, Remarques sur l'existence globale pour le système de NavierStokes incompressible, SIAM J. Math. Anal, vol.23, issue.1, pp.20-28, 1992.

J. Chemin and &. N. Lerner, Flot de champs de vecteurs non lipschitziens et équa-tions de Navier-Stokes, J. Differential Equations, vol.121, issue.2, pp.314-328, 1995.

W. E. and B. Enquist, Blow up of solutions of the unsteady Prandtl's equation, Comm. Pure Appl. Math, vol.50, pp.1287-1293, 1997.

L. Escauriaza, G. Seregin, and &. ?verák, ? -solutions of Navier-Stokes equations and backward uniqueness. Uspekhi Mat, Nauk, vol.3, issue.2, pp.3-44, 2003.

I. Gallagher, D. Iftimie, and &. Planchon, Asymptotics and stability for global solutions to the Navier-Stokes equations, Ann. Inst. Fourier (Grenoble), vol.53, issue.5, pp.1387-1424, 2003.

I. Gallagher, G. S. Koch, and &. Planchon, Blow-up of critical Besov norms at a potential Navier-Stokes singularity, Comm. Math. Phys, vol.343, issue.1, pp.39-82, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01231551

I. Gallagher, G. S. Koch, and &. Planchon, A profile decomposition approach to the L ? t (L 3 x ) Navier-Stokes regularity criterion, Math. Ann, vol.355, issue.4, pp.1527-1559, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00936368

D. Gérard-varet and E. Dormy, On the ill-posedness of the Prandtl equation, J. Amer. Math. Soc, vol.23, pp.591-609, 2010.

D. Gérard-varet and N. Masmoudi, Well-posedness for the Prandtl system without analyticity or monotonicity

T. Kato, Strong L p solutions of the Navier-Stokes equation in R m , with applications to weak solutions, Mathematische Zeitschrift, vol.187, pp.471-480, 1984.

C. E. Kenig-&-g and . Koch, An alternative approach to regularity for the NavierStokes equations in critical spaces, Ann. Inst. H. Poincaré Anal. Non, vol.28, issue.2, pp.159-187, 2011.

H. Koch and &. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math, vol.157, issue.1, pp.22-35, 2001.

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math, vol.63, issue.1, pp.193-248, 1934.

N. Masmoudi and T. K. Wong, Local-in-time existence and uniqueness of solution to the Prandtl equation by energy method

O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layers Theory, 1999.

F. Planchon, Asymptotic behavior of global solutions to the Navier-Stokes equations in R 3, Rev. Mat. Iberoamericana, vol.14, issue.1, pp.71-93, 1998.

G. Tian-&-z.-xin, One-point singular solutions to the Navier-Stokes equations, Topol. Meth. Nonlinear Anal, vol.11, pp.135-145, 1998.

Z. Xin and L. Zhang, On the global existence of solutions to the Prandtl's system, Adv. Math, vol.181, pp.88-133, 2004.

C. Xu, Hypoellipticity of nonlinear second order partial differential equations, J. Partial Differential Equations Ser. A, vol.1, pp.85-95, 1988.

C. Xu, Régularité des solutions pour les équations aux dérivées partielles quasi linéaires non elliptiques du second ordre, C. R. Acad. Sci. Paris Sér. I Math, vol.300, issue.9, pp.267-270, 1985.

C. Xu, Hypoellipticité d'équations aux dérivées partielles non linéaires. Proceedings of the conference on partial differential equations, vol.1, 1985.

H. Bahouri, J. -y-chemin, and &. R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00732127

O. A. Barraza, Self-similar solutions in weak L p -spaces of the Navier-Stokes equations, Rev. Mat. Iberoamericana, vol.12, issue.2, pp.411-439, 1996.

O. A. Barraza, Regularity and stability for the solutions of the Navier-Stokes equations in Lorentz spaces, Ser. A: Theory Methods, pp.747-764

J. Bergh and &. Löfström, Interpolation spaces: an introduction, 1976.

J. Bourgain and &. Pavlovi´cpavlovi´c, Ill-posedness of the Navier-Stokes equations in a critical space in 3D, J. Funct. Anal, vol.255, issue.9, pp.2233-2247, 2008.

C. P. Calderón, Existence of weak solutions for the Navier-Stokes equations with initial data in L p, Trans. Amer. Math. Soc, vol.318, issue.1, pp.179-200, 1990.

M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamericana, vol.13, issue.3, pp.515-541, 1997.

M. Cannone and &. Karch, About the regularized Navier-Stokes equations, J. Math. Fluid Mech, vol.7, issue.1, pp.1-28, 2005.

M. Cannone, Y. Meyer, and &. Planchon, Solutions auto-similaires des équations de Navier-Stokes, Séminaire sur les Équations aux Dérivées Partielles, vol.12, 1993.

M. Cannone-&-f.-planchon, On the non-stationary Navier-Stokes equations with an external force, Adv. Differential Equations, vol.4, issue.5, pp.697-730, 1999.

, Comm. Partial Differential Equations, vol.36, issue.2, pp.216-246, 2011.

J. Chemin, Théorèmes d'unicité pour le système de Navier-Stokes tridimensionnel. (French) [Uniqueness theorems for the three-dimensional NavierStokes system

, J. Anal. Math, vol.77, pp.27-50, 1999.

J. Chemin, Remarques sur l'existence globale pour le système de NavierStokes incompressible, SIAM J. Math. Anal, vol.23, issue.1, pp.20-28, 1992.

J. Chemin and &. N. Lerner, Flot de champs de vecteurs non lipschitziens et équa-tions de Navier-Stokes, J. Differential Equations, vol.121, issue.2, pp.314-328, 1995.

H. Fujita and &. Kato, On the Navier-Stokes initial value problem, I. Arch. Rational Mech. Anal, vol.16, pp.269-315, 1964.

I. Gallagher, D. Iftimie, and &. Planchon, Asymptotics and stability for global solutions to the Navier-Stokes equations, Ann. Inst. Fourier (Grenoble), vol.53, issue.5, pp.1387-1424, 2003.

I. Gallagher-&-f.-planchon, On global infinite energy solutions to the NavierStokes equations in two dimensions, Arch. Ration. Mech. Anal, vol.161, issue.4, pp.307-337, 2002.

P. Germain, The second iterate for the Navier-Stokes equation, J. Funct. Anal, vol.255, issue.9, pp.2248-2264, 2008.

T. Kato, Strong L p solutions of the Navier-Stokes equation in R m , with applications to weak solutions, Mathematische Zeitschrift, vol.187, pp.471-480, 1984.

H. Koch and &. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math, vol.157, issue.1, pp.22-35, 2001.

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math, vol.63, issue.1, pp.193-248, 1934.

J. Ne?as, M. R-?-u?i?ka, and &. V. ?verák, On Leray's self-similar solutions of the Navier-Stokes equations

F. Planchon, Asymptotic behavior of global solutions to the Navier-Stokes equations in R 3, Rev. Mat. Iberoamericana, vol.14, issue.1, pp.71-93, 1998.

T. Yoneda, Ill-posedness of the 3D-Navier-Stokes equations in a generalized Besov space near BMO ?1, J. Funct. Anal, vol.258, issue.10, pp.3376-3387, 2010.

D. Albritton, Blow-up criteria for the Navier-Stokes equations in non-endpoint critical Besov spaces

J. Bourgain and &. Pavlovi´cpavlovi´c, Ill-posedness of the Navier-Stokes equations in a critical space in 3D, J. Funct. Anal, vol.255, issue.9, pp.2233-2247, 2008.

M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamericana, vol.13, issue.3, pp.515-541, 1997.

M. Cannone, Y. Meyer, and &. Planchon, Solutions auto-similaires des équations de Navier-Stokes, Séminaire sur les Équations aux Dérivées Partielles, vol.12, 1993.

M. Cannone-&-f.-planchon, On the non-stationary Navier-Stokes equations with an external force, Adv. Differential Equations, vol.4, issue.5, pp.697-730, 1999.

J. Chemin, Théorèmes d'unicité pour le système de Navier-Stokes tridimensionnel. (French) [Uniqueness theorems for the three-dimensional NavierStokes system

, J. Anal. Math, vol.77, pp.27-50, 1999.

J. Chemin and &. N. Lerner, Flot de champs de vecteurs non lipschitziens et équa-tions de Navier-Stokes, J. Differential Equations, vol.121, issue.2, pp.314-328, 1995.

L. Escauriaza, G. Seregin, and &. ?verák, ? -solutions of Navier-Stokes equations and backward uniqueness. Uspekhi Mat, Nauk, vol.3, issue.2, pp.3-44, 2003.

L. Escauriaza, G. Seregin, and &. ?verák, Backward uniqueness for the heat operator in half-space, Algebra i Analiz, vol.15, issue.1, pp.201-214, 2003.

H. Fujita and &. Kato, On the Navier-Stokes initial value problem, I. Arch. Rational Mech. Anal, vol.16, pp.269-315, 1964.

I. Gallagher, Profile decomposition for solutions of the Navier-Stokes equations, Bull. Soc. Math. France, vol.129, issue.2, pp.285-316, 2001.

I. Gallagher, D. Iftimie, and &. Planchon, Asymptotics and stability for global solutions to the Navier-Stokes equations, Ann. Inst. Fourier (Grenoble), vol.53, issue.5, pp.1387-1424, 2003.

P. Gérard, Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var, vol.3, pp.213-233, 1998.

T. Kato, Strong L p solutions of the Navier-Stokes equation in R m , with applications to weak solutions, Mathematische Zeitschrift, vol.187, pp.471-480, 1984.

C. E. Kenig-&-g and . Koch, An alternative approach to regularity for the NavierStokes equations in critical spaces, Ann. Inst. H. Poincaré Anal. Non, vol.28, issue.2, pp.159-187, 2011.

G. S. Koch, Profile decompositions for critical Lebesgue and Besov space embeddings, Indiana Univ. Math. J, vol.59, issue.5, pp.1801-1830, 2010.

H. Koch and &. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math, vol.157, issue.1, pp.22-35, 2001.

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math, vol.63, issue.1, pp.193-248, 1934.

J. Ne?as, M. R-?-u?i?ka, and &. V. ?verák, On Leray's self-similar solutions of the Navier-Stokes equations, Acta Math, vol.176, issue.2, pp.283-294, 1996.

F. Planchon, Asymptotic behavior of global solutions to the Navier-Stokes equations in R 3, Rev. Mat. Iberoamericana, vol.14, issue.1, pp.71-93, 1998.

G. Seregin, A certain necessary condition of potential blow up for Navier-Stokes equations, Comm. Math. Phys, vol.312, issue.3, pp.833-845, 2012.

R. Alexandre, Y. Wang, C. Xu, and T. Yang, Well-posedness of The Prandtl Equation in Sobolev Spaces, J. Amer. Math. Soc, vol.40, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01116737

R. E. Caflisch and M. Sammartino, Existence and singularities for the Prandtl boundary layer equations, Z. Angew. Math. Mech, vol.80, pp.733-744, 2000.

M. Cannone, M. C. Lombardo, and M. Sammartino, Well-posedness of the Prandtl equation with non compatible data, Nonlinearity, vol.26, pp.3077-3100, 2013.

W. E. , Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation, Acta Math. Sin. (Engl. Ser.), vol.16, pp.207-218, 2000.

H. Chen, W. Li, and C. Xu, Gevrey hypoellipticity for linear and non-linear Fokker-Planck equations, J. Differential Equations, vol.246, pp.320-339, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00369846

H. Chen, W. Li, and C. Xu, Analytic smoothness effect of solutions for spatially homogeneous Landau equation, J. Differential Equations, pp.77-94, 2010.
URL : https://hal.archives-ouvertes.fr/hal-01116731

H. Chen, W. Li, and C. Xu, Gevrey hypoellipticity for a class of kinetic equations, Comm. Partial Differential Equations, vol.36, pp.693-728, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00570082

M. Derridj and C. Zuily, Sur la régularité Gevrey des opérateurs de Hörman-der, J.Math.Pures et Appl, vol.52, pp.309-336, 1973.

Y. Ding and N. Jiang, On Analytic Solutions of the Prandtl Equations with Robin Boundary Condition in Half Space, 2014.

W. E. and B. Enquist, Blow up of solutions of the unsteady Prandtl's equation, Comm. Pure Appl. Math, vol.50, pp.1287-1293, 1997.

D. Gérard-varet and E. Dormy, On the ill-posedness of the Prandtl equation, J. Amer. Math. Soc, vol.23, pp.591-609, 2010.

D. Gérard-varet and N. Masmoudi, Well-posedness for the Prandtl system without analyticity or monotonicity

D. Gérard-varet and T. Nguyen, Remarks on the ill-posedness of the Prandtl equation, Asymptot. Anal, vol.77, pp.71-88, 2012.

Y. Guo and T. Nguyen, A note on the Prandtl boundary layers, Comm. Pure Appl. Math, vol.64, pp.1416-1438, 2011.

L. Hong and J. K. Hunter, Singularity formation and instability in the unsteady inviscid and viscous Prandtl equations, Commun. Math. Sci, vol.1, pp.293-316, 2003.

L. Hörmander, The analysis of linear partial differential operators, Grundlehren der Mathematischen Wissenschaften, vol.275, 1985.

I. Kukavica, N. Masmoudi, V. Vicol, and T. Wong, On the Local Well-posedness of the Prandtl and Hydrostatic Euler Equations with Multiple Monotonicity Regions, SIAM J. Math. Anal, vol.46, pp.3865-3890, 2014.

I. Kukavica and V. Vicol, On the analyticity and Gevrey-class regularity up to the boundary for the Euler equations, Nonlinearity, vol.24, pp.765-796, 2011.

N. Lerner, Metrics on the phase space and non-selfadjoint pseudo-differential operators, Theory and Applications, vol.3, 2010.

M. C. Lombardo, M. Cannone, and M. Sammartino, Well-posedness of the boundary layer equations, SIAM J. Math. Anal, vol.35, pp.987-1004, 2003.

N. Masmoudi and T. K. Wong, Local-in-time existence and uniqueness of solution to the Prandtl equation by energy method

G. Métivier, Small Viscosity and Boundary Layer Methods. Theory, Stability Analysis, and Applications. Modeling and Simulation in Science, Engineering and Technology, 2004.

O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layers Theory, 1999.

L. , Über Flüssigkeitsbewegungen bei sehr kleiner Reibung, Teubner 1905, pp.484-494, 1904.

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, Comm. Math. Phys, vol.192, pp.433-461, 1998.

. Ii, Construction of the NavierStokes solution, Comm. Math. Phys, vol.192, pp.463-491, 1998.

Z. Xin and L. Zhang, On the global existence of solutions to the Prandtl's system, Adv. Math, vol.181, pp.88-133, 2004.

C. Xu, Hypoellipticity of nonlinear second order partial differential equations, J. Partial Differential Equations Ser. A, vol.1, pp.85-95, 1988.

C. Xu, Hypoellipticité d'équations aux dérivées partielles non linéaires. Proceedings of the conference on partial differential equations, vol.1, 1985.

P. Zhang and Z. Zhang, Long time well-posdness of Prandtl system with small and analytic initial data, 2014.