Dynamical behavior for the solutions of the Navier-Stokes equation

Kuijie Li; Tohru Ozawa; Baoxiang Wang

doi:10.3934/cpaa.2018073

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

Dynamical behavior for the solutions of the Navier-Stokes equation

Kuijie Li ^1, , Tohru Ozawa ^2, and Baoxiang Wang ^1,,

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

$3<p< ∞$

, 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

$\dot B^{-1+6/p}_{p/2, ∞}({{\mathbb{R}}^{3}})$

Keywords: Navier-Stokes equation, concentration phenomena, blowup profile, Type-Ⅰ blowup solution, $L^p$-minimal singularity-generating data.

Mathematics Subject Classification: Primary: 35Q30; Secondary: 76D03.

Citation: Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073

References:

[1]	B. Abe, The Navier-Stokes equations in a space of bounded functions, Commun. Math. Phys., 338 (2015), 849-865.
[2]	D. Albritton, Blow-up criteria for the Navier-Stokes equations in non-endpoint critical Besov spaces, preprint, arXiv: 1612.04439.
[3]	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.
[4]	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.
[5]	H. Bahouri, J. -Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343, Springer, Heidelberg, 2011.
[6]	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.
[7]	T. Barker and G. Seregin, On global solutions to the Navier-Stokes system with large $L^{3, ∞}$ initial data, preprint, arXiv: 1603.03211.
[8]	J. Bergh and J. Löfström, Interpolation Spaces, Springer-Verlag, 1976.
[9]	J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171.
[10]	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.
[11]	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.
[12]	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.
[13]	M. Cannone, Ondelettes, Paraproduits et Navier-Stokes, (French) [Wavelets, Paraproducts and Navier-Stokes], Diderot Editeur, Paris, 1995.
[14]	M. Cannone and Y. Meyer, Littlewood-Paley decomposition and Navier-Stokes equations, Methods Appl. Anal., 2 (1995), 307-319.
[15]	M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamericana, 13 (1997), 515-541.
[16]	J.-Y. Chemin, Théorémes d'unicité pour le systéme de Navier-Stokes tridimensionnel, J. Anal. Math., 77 (1999), 27-50.
[17]	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.
[18]	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.
[19]	H. Dong and D. Du, The Navier-Stokes equation in the critical Lebesgue space, Commun. Math. Phys., 292 (2009), 811-827.
[20]	L. Escauriaza, G. Seregin and V. Sverak, $L_{3,∞}$ solutions of Navier-Stokes equations and backward uniquness, Uspekhi Mat. Nauk., 58 (2003), 3-44.
[21]	C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369.
[22]	I. Gallagher, Profile decomposition for solutions of the Navier-Stokes equations, Bull. Soc. Math. France, 129 (2001), 285-316.
[23]	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.
[24]	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.
[25]	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.
[26]	P. Germain, The second iterate for the Navier-Stokes equation, J. Funct. Anal., 255 (2008), 2248-2264.
[27]	P. Gérard, Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233.
[28]	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.
[29]	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).
[30]	Y. Giga and T. Miyakawa, Solutions in $L^r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267-281.
[31]	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.
[32]	C. Huang and B. Wang, Analyticity for the (generalized) Navier-Stokes equations with rough initial data, preprint, arXiv: 1310.2141.
[33]	T. Iwabuchi, Navier-Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices, J. Differential Equations, 248 (2010), 1972-2002.
[34]	H. Jia and V. Sverak, Minimal $L_3$ -initial data for potential Navier-Stokes singularities, SIAM J. Math. Anal., 45 (2013), 1448-1459.
[35]	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.
[36]	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.
[37]	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.
[38]	G. S. Koch, Profile decompositions for critical Lebesgue and Besov space embeddings, Indiana Univ. Math.J., 59 (2010), 1801-1830.
[39]	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.
[40]	H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.
[41]	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.
[42]	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.
[43]	J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.
[44]	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.
[45]	P. G. Lemarié-Rieusset, The Navier-Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL, 2016.
[46]	F. H. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51 (1998), 241-257.
[47]	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.
[48]	F. Planchon, Asymptotic behavior of global solutions to the Navier-Stokes equations in ${{\mathbb{R}}^{3}}$, Rev. Mat. Iberoamericana, 14 (1998), 71-93.
[49]	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.
[50]	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.
[51]	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.
[52]	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.
[53]	W. Rusin and V. Sverak, Minimal initial data for potential Navier-Stokes singularities, J. Funct. Anal., 260 (2011), 879-891.
[54]	G. Seregin, A certain necessary condition of potential blow up for Navier-Stokes equations, Comm. Math. Phys., 312 (2012), 833-845.
[55]	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.
[56]	G. Seregin, Necessary conditions of potential blow up for the Navier-Stokes equations, Zap. Nauchn. Sem. POMI, 385 (2010), 187-199.
[57]	G. Seregin, Lecture Notes on Regularity Theory for the Navier-Stokes Equations World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.
[58]	H. Triebel, Theory of Function Spaces, Birkhäuser–Verlag, 1983.
[59]	B. Wang, Exponential Besov spaces and their applications to certain evolution equations with dissipations, Commun. Pure Appl. Anal., 3 (2004), 883-919.
[60]	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.
[61]	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.
[62]	B. Wang, Ill-posedness for the Navier-Stokes equation in critical Besov spaces $\dot B^{-1}_{∞, q}$, Adv. in Math., 268 (2015), 350-372.
[63]	F. B. Weissler, The Navier-Stokes initial value problem in $L^p$, Arch. Rational Mech. Anal., 74 (1980), 219-230.
[64]	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:

[1]	B. Abe, The Navier-Stokes equations in a space of bounded functions, Commun. Math. Phys., 338 (2015), 849-865.
[2]	D. Albritton, Blow-up criteria for the Navier-Stokes equations in non-endpoint critical Besov spaces, preprint, arXiv: 1612.04439.
[3]	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.
[4]	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.
[5]	H. Bahouri, J. -Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343, Springer, Heidelberg, 2011.
[6]	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.
[7]	T. Barker and G. Seregin, On global solutions to the Navier-Stokes system with large $L^{3, ∞}$ initial data, preprint, arXiv: 1603.03211.
[8]	J. Bergh and J. Löfström, Interpolation Spaces, Springer-Verlag, 1976.
[9]	J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171.
[10]	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.
[11]	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.
[12]	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.
[13]	M. Cannone, Ondelettes, Paraproduits et Navier-Stokes, (French) [Wavelets, Paraproducts and Navier-Stokes], Diderot Editeur, Paris, 1995.
[14]	M. Cannone and Y. Meyer, Littlewood-Paley decomposition and Navier-Stokes equations, Methods Appl. Anal., 2 (1995), 307-319.
[15]	M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamericana, 13 (1997), 515-541.
[16]	J.-Y. Chemin, Théorémes d'unicité pour le systéme de Navier-Stokes tridimensionnel, J. Anal. Math., 77 (1999), 27-50.
[17]	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.
[18]	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.
[19]	H. Dong and D. Du, The Navier-Stokes equation in the critical Lebesgue space, Commun. Math. Phys., 292 (2009), 811-827.
[20]	L. Escauriaza, G. Seregin and V. Sverak, $L_{3,∞}$ solutions of Navier-Stokes equations and backward uniquness, Uspekhi Mat. Nauk., 58 (2003), 3-44.
[21]	C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369.
[22]	I. Gallagher, Profile decomposition for solutions of the Navier-Stokes equations, Bull. Soc. Math. France, 129 (2001), 285-316.
[23]	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.
[24]	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.
[25]	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.
[26]	P. Germain, The second iterate for the Navier-Stokes equation, J. Funct. Anal., 255 (2008), 2248-2264.
[27]	P. Gérard, Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233.
[28]	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.
[29]	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).
[30]	Y. Giga and T. Miyakawa, Solutions in $L^r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267-281.
[31]	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.
[32]	C. Huang and B. Wang, Analyticity for the (generalized) Navier-Stokes equations with rough initial data, preprint, arXiv: 1310.2141.
[33]	T. Iwabuchi, Navier-Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices, J. Differential Equations, 248 (2010), 1972-2002.
[34]	H. Jia and V. Sverak, Minimal $L_3$ -initial data for potential Navier-Stokes singularities, SIAM J. Math. Anal., 45 (2013), 1448-1459.
[35]	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.
[36]	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.
[37]	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.
[38]	G. S. Koch, Profile decompositions for critical Lebesgue and Besov space embeddings, Indiana Univ. Math.J., 59 (2010), 1801-1830.
[39]	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.
[40]	H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.
[41]	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.
[42]	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.
[43]	J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248.
[44]	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.
[45]	P. G. Lemarié-Rieusset, The Navier-Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL, 2016.
[46]	F. H. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51 (1998), 241-257.
[47]	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.
[48]	F. Planchon, Asymptotic behavior of global solutions to the Navier-Stokes equations in ${{\mathbb{R}}^{3}}$, Rev. Mat. Iberoamericana, 14 (1998), 71-93.
[49]	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.
[50]	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.
[51]	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.
[52]	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.
[53]	W. Rusin and V. Sverak, Minimal initial data for potential Navier-Stokes singularities, J. Funct. Anal., 260 (2011), 879-891.
[54]	G. Seregin, A certain necessary condition of potential blow up for Navier-Stokes equations, Comm. Math. Phys., 312 (2012), 833-845.
[55]	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.
[56]	G. Seregin, Necessary conditions of potential blow up for the Navier-Stokes equations, Zap. Nauchn. Sem. POMI, 385 (2010), 187-199.
[57]	G. Seregin, Lecture Notes on Regularity Theory for the Navier-Stokes Equations World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.
[58]	H. Triebel, Theory of Function Spaces, Birkhäuser–Verlag, 1983.
[59]	B. Wang, Exponential Besov spaces and their applications to certain evolution equations with dissipations, Commun. Pure Appl. Anal., 3 (2004), 883-919.
[60]	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.
[61]	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.
[62]	B. Wang, Ill-posedness for the Navier-Stokes equation in critical Besov spaces $\dot B^{-1}_{∞, q}$, Adv. in Math., 268 (2015), 350-372.
[63]	F. B. Weissler, The Navier-Stokes initial value problem in $L^p$, Arch. Rational Mech. Anal., 74 (1980), 219-230.
[64]	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.

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American Institute of Mathematical Sciences

Dynamical behavior for the solutions of the Navier-Stokes equation

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Dynamical behavior for the solutions of the Navier-Stokes equation

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