    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. Google Scholar  D. Albritton, Blow-up criteria for the Navier-Stokes equations in non-endpoint critical Besov spaces, preprint, arXiv: 1612.04439. Google Scholar  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. Google Scholar  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. Google Scholar  H. Bahouri, J. -Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343, Springer, Heidelberg, 2011. Google Scholar  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. Google Scholar  T. Barker and G. Seregin, On global solutions to the Navier-Stokes system with large$L^{3, ∞}$initial data, preprint, arXiv: 1603.03211. Google Scholar  J. Bergh and J. Löfström, Interpolation Spaces, Springer-Verlag, 1976. Google Scholar  J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  M. Cannone, Ondelettes, Paraproduits et Navier-Stokes, (French) [Wavelets, Paraproducts and Navier-Stokes], Diderot Editeur, Paris, 1995. Google Scholar  M. Cannone and Y. Meyer, Littlewood-Paley decomposition and Navier-Stokes equations, Methods Appl. Anal., 2 (1995), 307-319. Google Scholar  M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamericana, 13 (1997), 515-541. Google Scholar  J.-Y. Chemin, Théorémes d'unicité pour le systéme de Navier-Stokes tridimensionnel, J. Anal. Math., 77 (1999), 27-50. Google Scholar  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. Google Scholar  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. Google Scholar  H. Dong and D. Du, The Navier-Stokes equation in the critical Lebesgue space, Commun. Math. Phys., 292 (2009), 811-827. Google Scholar  L. Escauriaza, G. Seregin and V. Sverak,$L_{3,∞}$solutions of Navier-Stokes equations and backward uniquness, Uspekhi Mat. Nauk., 58 (2003), 3-44. Google Scholar  C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369. Google Scholar  I. Gallagher, Profile decomposition for solutions of the Navier-Stokes equations, Bull. Soc. Math. France, 129 (2001), 285-316. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  P. Germain, The second iterate for the Navier-Stokes equation, J. Funct. Anal., 255 (2008), 2248-2264. Google Scholar  P. Gérard, Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233. Google Scholar  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. Google Scholar  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). Google Scholar  Y. Giga and T. Miyakawa, Solutions in$L^r$of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267-281. Google Scholar  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. Google Scholar  C. Huang and B. Wang, Analyticity for the (generalized) Navier-Stokes equations with rough initial data, preprint, arXiv: 1310.2141. Google Scholar  T. Iwabuchi, Navier-Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices, J. Differential Equations, 248 (2010), 1972-2002. Google Scholar  H. Jia and V. Sverak, Minimal$L_3$-initial data for potential Navier-Stokes singularities, SIAM J. Math. Anal., 45 (2013), 1448-1459. Google Scholar  T. Kato, Strong$L^p$solutions of the Navier-Stokes equations in$ \mathbb{{R}}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480. Google Scholar  C. E. Kenig and G. S. Koch, An alternative approach to regularity for the Navier-Stokes equations in critical spaces, Ann. l'Inst. H. Poincare (C) Non Linear Anal., 28 (2011), 159-187. Google Scholar  C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy critical focusing nonlinear wave equations, Acta Math., 201 (2008), 147-212. Google Scholar  G. S. Koch, Profile decompositions for critical Lebesgue and Besov space embeddings, Indiana Univ. Math.J., 59 (2010), 1801-1830. Google Scholar  G. S. Koch, N. Nadirashvili, G. A. Seregin and V. Sverak, Liouville theorems for the Navier-Stokes equations and applications, Acta Math., 203 (2009), 83-105. Google Scholar  H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. Google Scholar  H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z., 242 (2002), 251-278. Google Scholar  O. A. Ladyzhenskaya and G. A. Seregin, On partial Regularity of Suitable Weak Solutions to the Three-Dimensional Navier-Stokes equations, J. Math. Fluid Mech., 1 (1999), 365-387. Google Scholar  J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. Google Scholar  P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431. Chapman & Hall/CRC, Boca Raton, FL, 2002. Google Scholar  P. G. Lemarié-Rieusset, The Navier-Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL, 2016. Google Scholar  F. H. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51 (1998), 241-257. Google Scholar  F. Planchon, Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier-Stokes equations in$\mathbb{R}^3$, Ann, Inst. H. Poincare, AN, 13 (1996), 319-336. Google Scholar  F. Planchon, Asymptotic behavior of global solutions to the Navier-Stokes equations in${{\mathbb{R}}^{3}}$, Rev. Mat. Iberoamericana, 14 (1998), 71-93. Google Scholar  G. Ponce, R. Racke, T. C. Sideris and E. S. Titi, Global stability of large solutions to the 3D Navier-Stokes equations, Comm. Math. Phys., 159 (1994), 329-341. Google Scholar  E. Poulon, About the possibility of minimal blow up for Navier-Stokes solutions with data in$\dot{H}^s(\mathbb{R}^3)$, preprint, arXiv: 1505.06197. Google Scholar  E. Poulon, Etude Qualitative d'Eventuelles Singularités dans les Equation de Navier-Stokes Tridimensionnelles pour un Fluide Visqueux, Ph. D thesis, Université Pierre et Marie Curie, 2015. Google Scholar  J. C. Robinson, W. Sadowski and R. P. Silva, Lower bounds on blow up solutions of the three dimensional Navier-Stokes equations in homogeneous Sobolev spaces, Journal of Mathematical Physics, 260 (2011), 879-891. Google Scholar  W. Rusin and V. Sverak, Minimal initial data for potential Navier-Stokes singularities, J. Funct. Anal., 260 (2011), 879-891. Google Scholar  G. Seregin, A certain necessary condition of potential blow up for Navier-Stokes equations, Comm. Math. Phys., 312 (2012), 833-845. Google Scholar  G. Seregin and V. Sverak, On global weak solutions to the Cauchy problem ˇ for the Navier-Stokes equations with large$L_3$-initial data, Nonlinear Analysis, Theory, Methods and Applications, 154 (2017), 269-296. Google Scholar  G. Seregin, Necessary conditions of potential blow up for the Navier-Stokes equations, Zap. Nauchn. Sem. POMI, 385 (2010), 187-199. Google Scholar  G. Seregin, Lecture Notes on Regularity Theory for the Navier-Stokes Equations World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. Google Scholar  H. Triebel, Theory of Function Spaces, Birkhäuser–Verlag, 1983. Google Scholar  B. Wang, Exponential Besov spaces and their applications to certain evolution equations with dissipations, Commun. Pure Appl. Anal., 3 (2004), 883-919. Google Scholar  B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations. I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. Google Scholar  B. Wang, L. Zhao and B. Guo, Isometric decomposition operators, function spaces$E^λ_{p, q}$and their applications to nonlinear evolution equations, J. Funct. Anal., 233 (2006), 1-39. Google Scholar  B. Wang, Ill-posedness for the Navier-Stokes equation in critical Besov spaces$\dot B^{-1}_{∞, q}$, Adv. in Math., 268 (2015), 350-372. Google Scholar  F. B. Weissler, The Navier-Stokes initial value problem in$L^p$, Arch. Rational Mech. Anal., 74 (1980), 219-230. Google Scholar  T. Yoneda, Ill-posedness of the 3D Navier-Stokes equations in a generalized Besov space near$BMO^{-1}$, J. Funct. Anal., 258 (2010), 3376-3387. Google Scholar show all references ##### References:   B. Abe, The Navier-Stokes equations in a space of bounded functions, Commun. Math. Phys., 338 (2015), 849-865. Google Scholar  D. Albritton, Blow-up criteria for the Navier-Stokes equations in non-endpoint critical Besov spaces, preprint, arXiv: 1612.04439. Google Scholar  P. Auscher, S. Dubois and P. Tchamitchian, On the stability of global solutions to Navier-Stokes equations in the space, J. Math. Pures Appl., 83 (2004), 673-697. Google Scholar  H. Bae, A. Biswas and E. Tadmor, Analyticity and decay estimates of the Navier-Stokes equations in critical Besov spaces, Arch. Rational Mech. Anal., 205 (2012), 963-991. Google Scholar  H. Bahouri, J. -Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343, Springer, Heidelberg, 2011. Google Scholar  T. Barker, Uniqueness results for weak Leray-Hopf solutions of the Navier-Stokes system with initial values in critical spaces, J. Math. Fluid Mech., 20 (2018), 133-160. Google Scholar  T. Barker and G. Seregin, On global solutions to the Navier-Stokes system with large$L^{3, ∞}$initial data, preprint, arXiv: 1603.03211. Google Scholar  J. Bergh and J. Löfström, Interpolation Spaces, Springer-Verlag, 1976. Google Scholar  J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12 (1999), 145-171. Google Scholar  J. Bourgain and N. Pavlovic, Ill-posedness of the Navier-Stokes equations in a critical space in 3D, J. Funct. Anal., 255 (2008), 2233-2247. Google Scholar  L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. Google Scholar  C. P. Calderón, Existence of weak solutions for the Navier-Stokes equations with initial data in$L^p$, Trans. Amer. Math. Soc., 318 (1990), 179-200. Google Scholar  M. Cannone, Ondelettes, Paraproduits et Navier-Stokes, (French) [Wavelets, Paraproducts and Navier-Stokes], Diderot Editeur, Paris, 1995. Google Scholar  M. Cannone and Y. Meyer, Littlewood-Paley decomposition and Navier-Stokes equations, Methods Appl. Anal., 2 (1995), 307-319. Google Scholar  M. Cannone, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamericana, 13 (1997), 515-541. Google Scholar  J.-Y. Chemin, Théorémes d'unicité pour le systéme de Navier-Stokes tridimensionnel, J. Anal. Math., 77 (1999), 27-50. Google Scholar  J.-Y. Chemin, I. Gallagher and M. Paicu, Global regularity for some classes of large solutions to the Navier-Stokes equations, Ann. of Math., 173 (2011), 983-1012. Google Scholar  J. C. Cortissoz, J. A. Montero and C. E. Pinilla, On lower bounds for possible blow-up solutions to the periodic Navier-Stokes equation, J. Math. Phys. , 55 (2014), 033101. Google Scholar  H. Dong and D. Du, The Navier-Stokes equation in the critical Lebesgue space, Commun. Math. Phys., 292 (2009), 811-827. Google Scholar  L. Escauriaza, G. Seregin and V. Sverak,$L_{3,∞}$solutions of Navier-Stokes equations and backward uniquness, Uspekhi Mat. Nauk., 58 (2003), 3-44. Google Scholar  C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369. Google Scholar  I. Gallagher, Profile decomposition for solutions of the Navier-Stokes equations, Bull. Soc. Math. France, 129 (2001), 285-316. Google Scholar  I. Gallagher, D. Iftimie and F. Planchon, Asympototics and stability for global solutions to the Navier-Stokes equations, Ann. Inst. Fourier(Grenoble), 53 (2003), 1387-1424. Google Scholar  I. Gallagher, G. S. Koch and F. Planchnon, A profile decomposition approach to the$L^∞_t(L^3_x)$Navier-Stokes regularity criterion, Math. Ann., 355 (2013), 1527-1559. Google Scholar  I. Gallagher, G. S. Koch and F. Planchon, Blow-up of critical Besov norms at a potential Navier-Stokes singularity, Comm. Math. Phys., 343 (2016), 39-82. Google Scholar  P. Germain, The second iterate for the Navier-Stokes equation, J. Funct. Anal., 255 (2008), 2248-2264. Google Scholar  P. Gérard, Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233. Google Scholar  Y. Giga, Solutions for semilinear parabolic equations in$L_p$and regularity of weak solutions of the Navier-Stokes system, J. Differ. Equations, 62 (1986), 182-212. Google Scholar  Y. Giga, K. Inui and S. Matsui, On the Cauchy problem for the Navier-Stokes equations with nondecaying initial data, in Advances in Fluid Dynamics, vol. 4 of Quad. Mat., pp. 27–68. Dept. Math., Seconda Univ. Napoli, Caserta (1999). Google Scholar  Y. Giga and T. Miyakawa, Solutions in$L^r$of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267-281. Google Scholar  Y. Giga and T. Miyakawa, Navier-Stokes flow in$\mathbb{R}^3$with measures as initial vorticity and Morrey spaces, Comm. Partial Differential Equations, 14 (1989), 577-618. Google Scholar  C. Huang and B. Wang, Analyticity for the (generalized) Navier-Stokes equations with rough initial data, preprint, arXiv: 1310.2141. Google Scholar  T. Iwabuchi, Navier-Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices, J. Differential Equations, 248 (2010), 1972-2002. Google Scholar  H. Jia and V. Sverak, Minimal$L_3$-initial data for potential Navier-Stokes singularities, SIAM J. Math. Anal., 45 (2013), 1448-1459. Google Scholar  T. Kato, Strong$L^p$solutions of the Navier-Stokes equations in$ \mathbb{{R}}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480. Google Scholar  C. E. Kenig and G. S. Koch, An alternative approach to regularity for the Navier-Stokes equations in critical spaces, Ann. l'Inst. H. Poincare (C) Non Linear Anal., 28 (2011), 159-187. Google Scholar  C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy critical focusing nonlinear wave equations, Acta Math., 201 (2008), 147-212. Google Scholar  G. S. Koch, Profile decompositions for critical Lebesgue and Besov space embeddings, Indiana Univ. Math.J., 59 (2010), 1801-1830. Google Scholar  G. S. Koch, N. Nadirashvili, G. A. Seregin and V. Sverak, Liouville theorems for the Navier-Stokes equations and applications, Acta Math., 203 (2009), 83-105. Google Scholar  H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35. Google Scholar  H. Kozono, T. Ogawa and Y. Taniuchi, The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations, Math. Z., 242 (2002), 251-278. Google Scholar  O. A. Ladyzhenskaya and G. A. Seregin, On partial Regularity of Suitable Weak Solutions to the Three-Dimensional Navier-Stokes equations, J. Math. Fluid Mech., 1 (1999), 365-387. Google Scholar  J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. Google Scholar  P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431. Chapman & Hall/CRC, Boca Raton, FL, 2002. Google Scholar  P. G. Lemarié-Rieusset, The Navier-Stokes Problem in the 21st Century, CRC Press, Boca Raton, FL, 2016. Google Scholar  F. H. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51 (1998), 241-257. Google Scholar  F. Planchon, Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier-Stokes equations in$\mathbb{R}^3$, Ann, Inst. H. Poincare, AN, 13 (1996), 319-336. Google Scholar  F. Planchon, Asymptotic behavior of global solutions to the Navier-Stokes equations in${{\mathbb{R}}^{3}}$, Rev. Mat. Iberoamericana, 14 (1998), 71-93. Google Scholar  G. Ponce, R. Racke, T. C. Sideris and E. S. Titi, Global stability of large solutions to the 3D Navier-Stokes equations, Comm. Math. Phys., 159 (1994), 329-341. Google Scholar  E. Poulon, About the possibility of minimal blow up for Navier-Stokes solutions with data in$\dot{H}^s(\mathbb{R}^3)$, preprint, arXiv: 1505.06197. Google Scholar  E. Poulon, Etude Qualitative d'Eventuelles Singularités dans les Equation de Navier-Stokes Tridimensionnelles pour un Fluide Visqueux, Ph. D thesis, Université Pierre et Marie Curie, 2015. Google Scholar  J. C. Robinson, W. Sadowski and R. P. Silva, Lower bounds on blow up solutions of the three dimensional Navier-Stokes equations in homogeneous Sobolev spaces, Journal of Mathematical Physics, 260 (2011), 879-891. Google Scholar  W. Rusin and V. Sverak, Minimal initial data for potential Navier-Stokes singularities, J. Funct. Anal., 260 (2011), 879-891. Google Scholar  G. Seregin, A certain necessary condition of potential blow up for Navier-Stokes equations, Comm. Math. Phys., 312 (2012), 833-845. Google Scholar  G. Seregin and V. Sverak, On global weak solutions to the Cauchy problem ˇ for the Navier-Stokes equations with large$L_3$-initial data, Nonlinear Analysis, Theory, Methods and Applications, 154 (2017), 269-296. Google Scholar  G. Seregin, Necessary conditions of potential blow up for the Navier-Stokes equations, Zap. Nauchn. Sem. POMI, 385 (2010), 187-199. Google Scholar  G. Seregin, Lecture Notes on Regularity Theory for the Navier-Stokes Equations World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. Google Scholar  H. Triebel, Theory of Function Spaces, Birkhäuser–Verlag, 1983. Google Scholar  B. Wang, Exponential Besov spaces and their applications to certain evolution equations with dissipations, Commun. Pure Appl. Anal., 3 (2004), 883-919. Google Scholar  B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations. I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. Google Scholar  B. Wang, L. Zhao and B. Guo, Isometric decomposition operators, function spaces$E^λ_{p, q}$and their applications to nonlinear evolution equations, J. Funct. Anal., 233 (2006), 1-39. Google Scholar  B. Wang, Ill-posedness for the Navier-Stokes equation in critical Besov spaces$\dot B^{-1}_{∞, q}$, Adv. in Math., 268 (2015), 350-372. Google Scholar  F. B. Weissler, The Navier-Stokes initial value problem in$L^p$, Arch. Rational Mech. Anal., 74 (1980), 219-230. Google Scholar  T. Yoneda, Ill-posedness of the 3D Navier-Stokes equations in a generalized Besov space near$BMO^{-1}$, J. Funct. Anal., 258 (2010), 3376-3387. Google Scholar   Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020348  Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241  Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in$ \mathbb{R}^2 $. 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