    July 2018, 17(4): 1511-1560. doi: 10.3934/cpaa.2018073

## Dynamical behavior for the solutions of the Navier-Stokes equation

 1 LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China 2 Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan

* Corresponding author: Baoxiang Wang

Received  August 2016 Revised  April 2017 Published  April 2018

We study several quantitative properties of solutions to the incompressible Navier-Stokes equation in three and higher dimensions:
 \begin{align} u_t -Δ u+u· \nabla u +\nabla p = 0, \ \ {\rm div} u = 0, \ \ u(0, x) = u_0(x). \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left( 1 \right)\label{NSa} \end{align}
More precisely, for the blow up mild solutions with initial data in
 $L^{∞}(\mathbb{R}^d)$
and
 $H^{d/2 -1}(\mathbb{R}^d)$
, we obtain a concentration phenomenon and blowup profile decomposition respectively. On the other hand, if the Fourier support has the form
 ${\rm supp} \ \widehat{u_0} \subset \{ξ∈ \mathbb{R}^n: ξ_1≥ L \}$
and
 $\|u_0\|_{∞} \ll L$
for some
 $L >0$
, then (1) has a unique global solution
 $u∈ C(\mathbb{R}_+, L^∞)$
. In 3D, we show the compactness of the set consisting of minimal-
 $L^p$
singularity-generating initial data with
 $3 , furthermore, if the mild solution with data in $L^p({{\mathbb{R}}^{3}})$blows up in a Type-Ⅰ manner, we prove the existence of a blowup solution which is uniformly bounded in critical Besov spaces $\dot B^{-1+6/p}_{p/2, ∞}({{\mathbb{R}}^{3}})$. Citation: Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073 ##### References:   B. Abe, The Navier-Stokes equations in a space of bounded functions, Commun. Math. Phys., 338 (2015), 849-865.  D. Albritton, Blow-up criteria for the Navier-Stokes equations in non-endpoint critical Besov spaces, preprint, arXiv: 1612.04439.  P. Auscher, S. Dubois and P. Tchamitchian, On the stability of global solutions to Navier-Stokes equations in the space, J. Math. Pures Appl., 83 (2004), 673-697.  H. Bae, A. Biswas and E. Tadmor, Analyticity and decay estimates of the Navier-Stokes equations in critical Besov spaces, Arch. Rational Mech. Anal., 205 (2012), 963-991.  H. Bahouri, J. -Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343, Springer, Heidelberg, 2011.  T. Barker, Uniqueness results for weak Leray-Hopf solutions of the Navier-Stokes system with initial values in critical spaces, J. Math. Fluid Mech., 20 (2018), 133-160.  T. Barker and G. Seregin, On global solutions to the Navier-Stokes system with large$L^{3, ∞}$initial data, preprint, arXiv: 1603.03211.  J. Bergh and J. Löfström, Interpolation Spaces, Springer-Verlag, 1976.  J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171.  J. Bourgain and N. Pavlovic, Ill-posedness of the Navier-Stokes equations in a critical space in 3D, J. Funct. Anal., 255 (2008), 2233-2247.  L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831.  C. P. Calderón, Existence of weak solutions for the Navier-Stokes equations with initial data in$L^p$, Trans. Amer. Math. Soc., 318 (1990), 179-200.  M. Cannone, Ondelettes, Paraproduits et Navier-Stokes, (French) [Wavelets, Paraproducts and Navier-Stokes], Diderot Editeur, Paris, 1995.  M. Cannone and Y. Meyer, Littlewood-Paley decomposition and Navier-Stokes equations, Methods Appl. Anal., 2 (1995), 307-319.  M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamericana, 13 (1997), 515-541.  J.-Y. Chemin, Théorémes d'unicité pour le systéme de Navier-Stokes tridimensionnel, J. Anal. Math., 77 (1999), 27-50.  J.-Y. Chemin, I. Gallagher and M. Paicu, Global regularity for some classes of large solutions to the Navier-Stokes equations, Ann. of Math., 173 (2011), 983-1012.  J. C. Cortissoz, J. A. Montero and C. E. Pinilla, On lower bounds for possible blow-up solutions to the periodic Navier-Stokes equation, J. Math. Phys. , 55 (2014), 033101.  H. Dong and D. Du, The Navier-Stokes equation in the critical Lebesgue space, Commun. Math. Phys., 292 (2009), 811-827.  L. Escauriaza, G. Seregin and V. Sverak,$L_{3,∞}$solutions of Navier-Stokes equations and backward uniquness, Uspekhi Mat. Nauk., 58 (2003), 3-44.  C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369.  I. Gallagher, Profile decomposition for solutions of the Navier-Stokes equations, Bull. Soc. Math. France, 129 (2001), 285-316.  I. Gallagher, D. Iftimie and F. Planchon, Asympototics and stability for global solutions to the Navier-Stokes equations, Ann. Inst. Fourier(Grenoble), 53 (2003), 1387-1424.  I. Gallagher, G. S. Koch and F. Planchnon, A profile decomposition approach to the$L^∞_t(L^3_x)$Navier-Stokes regularity criterion, Math. Ann., 355 (2013), 1527-1559.  I. Gallagher, G. S. Koch and F. Planchon, Blow-up of critical Besov norms at a potential Navier-Stokes singularity, Comm. Math. Phys., 343 (2016), 39-82.  P. Germain, The second iterate for the Navier-Stokes equation, J. Funct. Anal., 255 (2008), 2248-2264.  P. Gérard, Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233.  Y. Giga, Solutions for semilinear parabolic equations in$L_p$and regularity of weak solutions of the Navier-Stokes system, J. Differ. Equations, 62 (1986), 182-212.  Y. Giga, K. Inui and S. Matsui, On the Cauchy problem for the Navier-Stokes equations with nondecaying initial data, in Advances in Fluid Dynamics, vol. 4 of Quad. Mat., pp. 27–68. Dept. Math., Seconda Univ. Napoli, Caserta (1999).  Y. Giga and T. Miyakawa, Solutions in$L^r$of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267-281.  Y. Giga and T. Miyakawa, Navier-Stokes flow in$\mathbb{R}^3$with measures as initial vorticity and Morrey spaces, Comm. Partial Differential Equations, 14 (1989), 577-618.  C. Huang and B. Wang, Analyticity for the (generalized) Navier-Stokes equations with rough initial data, preprint, arXiv: 1310.2141.  T. Iwabuchi, Navier-Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices, J. Differential Equations, 248 (2010), 1972-2002.  H. Jia and V. Sverak, Minimal$L_3$-initial data for potential Navier-Stokes singularities, SIAM J. Math. Anal., 45 (2013), 1448-1459.  T. Kato, Strong$L^p$solutions of the Navier-Stokes equations in$ \mathbb{{R}}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.  C. E. Kenig and G. S. Koch, An alternative approach to regularity for the Navier-Stokes equations in critical spaces, Ann. l'Inst. H. Poincare (C) Non Linear Anal., 28 (2011), 159-187.  C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy critical focusing nonlinear wave equations, Acta Math., 201 (2008), 147-212.  G. S. Koch, Profile decompositions for critical Lebesgue and Besov space embeddings, Indiana Univ. Math.J., 59 (2010), 1801-1830.  G. S. Koch, N. Nadirashvili, G. A. Seregin and V. Sverak, Liouville theorems for the Navier-Stokes equations and applications, Acta Math., 203 (2009), 83-105.  H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.  H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z., 242 (2002), 251-278.  O. A. Ladyzhenskaya and G. A. Seregin, On partial Regularity of Suitable Weak Solutions to the Three-Dimensional Navier-Stokes equations, J. Math. Fluid Mech., 1 (1999), 365-387.  J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431. Chapman & Hall/CRC, Boca Raton, FL, 2002.  P. G. Lemarié-Rieusset, The Navier-Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL, 2016.  F. H. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51 (1998), 241-257.  F. Planchon, Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier-Stokes equations in$\mathbb{R}^3$, Ann, Inst. H. Poincare, AN, 13 (1996), 319-336.  F. Planchon, Asymptotic behavior of global solutions to the Navier-Stokes equations in${{\mathbb{R}}^{3}}$, Rev. Mat. Iberoamericana, 14 (1998), 71-93.  G. Ponce, R. Racke, T. C. Sideris and E. S. Titi, Global stability of large solutions to the 3D Navier-Stokes equations, Comm. Math. Phys., 159 (1994), 329-341.  E. Poulon, About the possibility of minimal blow up for Navier-Stokes solutions with data in$\dot{H}^s(\mathbb{R}^3)$, preprint, arXiv: 1505.06197.  E. Poulon, Etude Qualitative d'Eventuelles Singularités dans les Equation de Navier-Stokes Tridimensionnelles pour un Fluide Visqueux, Ph. D thesis, Université Pierre et Marie Curie, 2015.  J. C. Robinson, W. Sadowski and R. P. Silva, Lower bounds on blow up solutions of the three dimensional Navier-Stokes equations in homogeneous Sobolev spaces, Journal of Mathematical Physics, 260 (2011), 879-891.  W. Rusin and V. Sverak, Minimal initial data for potential Navier-Stokes singularities, J. Funct. Anal., 260 (2011), 879-891.  G. Seregin, A certain necessary condition of potential blow up for Navier-Stokes equations, Comm. Math. Phys., 312 (2012), 833-845.  G. Seregin and V. Sverak, On global weak solutions to the Cauchy problem ˇ for the Navier-Stokes equations with large$L_3$-initial data, Nonlinear Analysis, Theory, Methods and Applications, 154 (2017), 269-296.  G. Seregin, Necessary conditions of potential blow up for the Navier-Stokes equations, Zap. Nauchn. Sem. POMI, 385 (2010), 187-199.  G. Seregin, Lecture Notes on Regularity Theory for the Navier-Stokes Equations World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.  H. Triebel, Theory of Function Spaces, Birkhäuser–Verlag, 1983.  B. Wang, Exponential Besov spaces and their applications to certain evolution equations with dissipations, Commun. Pure Appl. Anal., 3 (2004), 883-919.  B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations. I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011.  B. Wang, L. Zhao and B. Guo, Isometric decomposition operators, function spaces$E^λ_{p, q}$and their applications to nonlinear evolution equations, J. Funct. Anal., 233 (2006), 1-39.  B. Wang, Ill-posedness for the Navier-Stokes equation in critical Besov spaces$\dot B^{-1}_{∞, q}$, Adv. in Math., 268 (2015), 350-372.  F. B. Weissler, The Navier-Stokes initial value problem in$L^p$, Arch. Rational Mech. Anal., 74 (1980), 219-230.  T. Yoneda, Ill-posedness of the 3D Navier-Stokes equations in a generalized Besov space near$BMO^{-1}$, J. Funct. Anal., 258 (2010), 3376-3387. show all references ##### References:   B. Abe, The Navier-Stokes equations in a space of bounded functions, Commun. Math. Phys., 338 (2015), 849-865.  D. Albritton, Blow-up criteria for the Navier-Stokes equations in non-endpoint critical Besov spaces, preprint, arXiv: 1612.04439.  P. Auscher, S. Dubois and P. Tchamitchian, On the stability of global solutions to Navier-Stokes equations in the space, J. Math. Pures Appl., 83 (2004), 673-697.  H. Bae, A. Biswas and E. Tadmor, Analyticity and decay estimates of the Navier-Stokes equations in critical Besov spaces, Arch. Rational Mech. Anal., 205 (2012), 963-991.  H. Bahouri, J. -Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343, Springer, Heidelberg, 2011.  T. Barker, Uniqueness results for weak Leray-Hopf solutions of the Navier-Stokes system with initial values in critical spaces, J. Math. Fluid Mech., 20 (2018), 133-160.  T. Barker and G. Seregin, On global solutions to the Navier-Stokes system with large$L^{3, ∞}$initial data, preprint, arXiv: 1603.03211.  J. Bergh and J. Löfström, Interpolation Spaces, Springer-Verlag, 1976.  J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171.  J. Bourgain and N. Pavlovic, Ill-posedness of the Navier-Stokes equations in a critical space in 3D, J. Funct. Anal., 255 (2008), 2233-2247.  L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831.  C. P. Calderón, Existence of weak solutions for the Navier-Stokes equations with initial data in$L^p$, Trans. Amer. Math. Soc., 318 (1990), 179-200.  M. Cannone, Ondelettes, Paraproduits et Navier-Stokes, (French) [Wavelets, Paraproducts and Navier-Stokes], Diderot Editeur, Paris, 1995.  M. Cannone and Y. Meyer, Littlewood-Paley decomposition and Navier-Stokes equations, Methods Appl. Anal., 2 (1995), 307-319.  M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamericana, 13 (1997), 515-541.  J.-Y. Chemin, Théorémes d'unicité pour le systéme de Navier-Stokes tridimensionnel, J. Anal. Math., 77 (1999), 27-50.  J.-Y. Chemin, I. Gallagher and M. Paicu, Global regularity for some classes of large solutions to the Navier-Stokes equations, Ann. of Math., 173 (2011), 983-1012.  J. C. Cortissoz, J. A. Montero and C. E. Pinilla, On lower bounds for possible blow-up solutions to the periodic Navier-Stokes equation, J. Math. Phys. , 55 (2014), 033101.  H. Dong and D. Du, The Navier-Stokes equation in the critical Lebesgue space, Commun. Math. Phys., 292 (2009), 811-827.  L. Escauriaza, G. Seregin and V. Sverak,$L_{3,∞}$solutions of Navier-Stokes equations and backward uniquness, Uspekhi Mat. Nauk., 58 (2003), 3-44.  C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369.  I. Gallagher, Profile decomposition for solutions of the Navier-Stokes equations, Bull. Soc. Math. France, 129 (2001), 285-316.  I. Gallagher, D. Iftimie and F. Planchon, Asympototics and stability for global solutions to the Navier-Stokes equations, Ann. Inst. Fourier(Grenoble), 53 (2003), 1387-1424.  I. Gallagher, G. S. Koch and F. Planchnon, A profile decomposition approach to the$L^∞_t(L^3_x)$Navier-Stokes regularity criterion, Math. Ann., 355 (2013), 1527-1559.  I. Gallagher, G. S. Koch and F. Planchon, Blow-up of critical Besov norms at a potential Navier-Stokes singularity, Comm. Math. Phys., 343 (2016), 39-82.  P. Germain, The second iterate for the Navier-Stokes equation, J. Funct. Anal., 255 (2008), 2248-2264.  P. Gérard, Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233.  Y. Giga, Solutions for semilinear parabolic equations in$L_p$and regularity of weak solutions of the Navier-Stokes system, J. Differ. Equations, 62 (1986), 182-212.  Y. Giga, K. Inui and S. Matsui, On the Cauchy problem for the Navier-Stokes equations with nondecaying initial data, in Advances in Fluid Dynamics, vol. 4 of Quad. Mat., pp. 27–68. Dept. Math., Seconda Univ. Napoli, Caserta (1999).  Y. Giga and T. Miyakawa, Solutions in$L^r$of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267-281.  Y. Giga and T. Miyakawa, Navier-Stokes flow in$\mathbb{R}^3$with measures as initial vorticity and Morrey spaces, Comm. Partial Differential Equations, 14 (1989), 577-618.  C. Huang and B. Wang, Analyticity for the (generalized) Navier-Stokes equations with rough initial data, preprint, arXiv: 1310.2141.  T. Iwabuchi, Navier-Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices, J. Differential Equations, 248 (2010), 1972-2002.  H. Jia and V. Sverak, Minimal$L_3$-initial data for potential Navier-Stokes singularities, SIAM J. Math. Anal., 45 (2013), 1448-1459.  T. Kato, Strong$L^p$solutions of the Navier-Stokes equations in$ \mathbb{{R}}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.  C. E. Kenig and G. S. Koch, An alternative approach to regularity for the Navier-Stokes equations in critical spaces, Ann. l'Inst. H. Poincare (C) Non Linear Anal., 28 (2011), 159-187.  C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy critical focusing nonlinear wave equations, Acta Math., 201 (2008), 147-212.  G. S. Koch, Profile decompositions for critical Lebesgue and Besov space embeddings, Indiana Univ. Math.J., 59 (2010), 1801-1830.  G. S. Koch, N. Nadirashvili, G. A. Seregin and V. Sverak, Liouville theorems for the Navier-Stokes equations and applications, Acta Math., 203 (2009), 83-105.  H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.  H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z., 242 (2002), 251-278.  O. A. Ladyzhenskaya and G. A. Seregin, On partial Regularity of Suitable Weak Solutions to the Three-Dimensional Navier-Stokes equations, J. Math. Fluid Mech., 1 (1999), 365-387.  J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.  P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431. Chapman & Hall/CRC, Boca Raton, FL, 2002.  P. G. Lemarié-Rieusset, The Navier-Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL, 2016.  F. H. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51 (1998), 241-257.  F. Planchon, Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier-Stokes equations in$\mathbb{R}^3$, Ann, Inst. H. Poincare, AN, 13 (1996), 319-336.  F. Planchon, Asymptotic behavior of global solutions to the Navier-Stokes equations in${{\mathbb{R}}^{3}}$, Rev. Mat. Iberoamericana, 14 (1998), 71-93.  G. Ponce, R. Racke, T. C. Sideris and E. S. Titi, Global stability of large solutions to the 3D Navier-Stokes equations, Comm. Math. Phys., 159 (1994), 329-341.  E. Poulon, About the possibility of minimal blow up for Navier-Stokes solutions with data in$\dot{H}^s(\mathbb{R}^3)$, preprint, arXiv: 1505.06197.  E. Poulon, Etude Qualitative d'Eventuelles Singularités dans les Equation de Navier-Stokes Tridimensionnelles pour un Fluide Visqueux, Ph. D thesis, Université Pierre et Marie Curie, 2015.  J. C. Robinson, W. Sadowski and R. P. Silva, Lower bounds on blow up solutions of the three dimensional Navier-Stokes equations in homogeneous Sobolev spaces, Journal of Mathematical Physics, 260 (2011), 879-891.  W. Rusin and V. Sverak, Minimal initial data for potential Navier-Stokes singularities, J. Funct. Anal., 260 (2011), 879-891.  G. Seregin, A certain necessary condition of potential blow up for Navier-Stokes equations, Comm. Math. Phys., 312 (2012), 833-845.  G. Seregin and V. Sverak, On global weak solutions to the Cauchy problem ˇ for the Navier-Stokes equations with large$L_3$-initial data, Nonlinear Analysis, Theory, Methods and Applications, 154 (2017), 269-296.  G. Seregin, Necessary conditions of potential blow up for the Navier-Stokes equations, Zap. Nauchn. Sem. POMI, 385 (2010), 187-199.  G. Seregin, Lecture Notes on Regularity Theory for the Navier-Stokes Equations World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.  H. Triebel, Theory of Function Spaces, Birkhäuser–Verlag, 1983.  B. Wang, Exponential Besov spaces and their applications to certain evolution equations with dissipations, Commun. Pure Appl. Anal., 3 (2004), 883-919.  B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations. I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011.  B. Wang, L. Zhao and B. 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