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Blow up for initial boundary value problem of critical semilinear wave equation in two space dimensions
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 |
$ \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}$ |
$L^{∞}(\mathbb{R}^d)$ |
$H^{d/2 -1}(\mathbb{R}^d)$ |
${\rm supp} \ \widehat{u_0} \subset \{ξ∈ \mathbb{R}^n: ξ_1≥ L \}$ |
$ \|u_0\|_{∞} \ll L$ |
$L >0$ |
$u∈ C(\mathbb{R}_+, L^∞)$ |
$L^p$ |
$3<p< ∞$ |
$L^p({{\mathbb{R}}^{3}})$ |
$\dot B^{-1+6/p}_{p/2, ∞}({{\mathbb{R}}^{3}})$ |
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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