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Existence of periodically invariant tori on resonant surfaces for twist mappings
Long-time solvability for the 2D dispersive SQG equation with improved regularity
1. | Departamento de Matemáticas, Universidad Nacional de Colombia - Sede Orinoquia, Kilometro 9 vía a Caño Limón, Arauca, Colombia |
2. | IMECC-Departamento de Matemática, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda, CEP 13083-859, Campinas, SP, Brazil |
In this paper, we study the long-time existence and uniqueness (solvability) for the initial value problem of the 2D inviscid dispersive SQG equation. First we obtain the local solvability with existence-time independent of the amplitude parameter $ A $. Then, assuming more regularity and using a blow-up criterion of BKM type and a space-time estimate of Strichartz type, we prove long-time solvability of solutions in Besov spaces for large $ A $ and arbitrary initial data. In comparison with previous results, we are able to consider improved cases of the regularity and larger initial data classes.
References:
[1] |
V. Angulo-Castillo and L. C. F. Ferreira,
On the 3D Euler equations with Coriolis force in borderline Besov spaces, Commun. Math. Sci, 16 (2018), 145-164.
doi: 10.4310/CMS.2018.v16.n1.a7. |
[2] |
A. Babin, A. Mahalov and B. Nicolaenko,
Global regularity of 3D rotating Navier-Stokes equations for resonant domains, Indiana Univ. Math. J., 48 (1999), 1133-1176.
|
[3] |
A. Babin, A. Mahalov and B. Nicolaenko,
3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity, Indiana Univ. Math. J., 50 (2001), 1-35.
doi: 10.1512/iumj.2001.50.2155. |
[4] |
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, 223, Springer-Verlag, Berlin-New York, 1976.
doi: 10.1007/978-3-642-66451-9. |
[5] |
J. Bourgain and D. Li,
Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces, Invent. Math., 201 (2015), 97-157.
doi: 10.1007/s00222-014-0548-6. |
[6] |
L. Caffarelli and V. Vasseur,
Drift diffusion equations with fractional diffusion and quasi-geostrophic equations, Ann. of Math. (2), 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[7] |
M. Cannone, C. Miao and L. Xue,
Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing, Proc. Lond. Math. Soc. (3), 106 (2013), 650-674.
doi: 10.1112/plms/pds046. |
[8] |
J. A. Carrillo and L. C. F. Ferreira,
The asymptotic behaviour of subcritical dissipative quasi-geostrophic equations, Nonlinearity, 21 (2008), 1001-1018.
doi: 10.1088/0951-7715/21/5/006. |
[9] |
D. Chae,
Local existence and blow-up criterion for the Euler equations in the Besov spaces, Asymptot. Anal., 38 (2004), 339-358.
|
[10] |
D. Chae, P. Constantin and J. Wu,
Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations, Arch. Ration. Mech. Anal., 202 (2011), 35-62.
doi: 10.1007/s00205-011-0411-5. |
[11] |
D. Chae, P. Constantin and J. Wu,
Dissipative models generalizing the 2D Navier-Stokes and surface quasi-geostrophic equations, Indiana Univ. Math. J., 61 (2012), 1997-2018.
doi: 10.1512/iumj.2012.61.4756. |
[12] |
D. Chae and J. Lee,
Global well-posedness in the super-critical dissipative quasi-geostrophic equations, Comm. Math. Phys., 233 (2003), 297-311.
doi: 10.1007/s00220-002-0750-z. |
[13] |
J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Mathematical Geophysics. An Introduction to Rotating Fluids and the Navier-Stokes Equations, Oxford Lecture Series in Mathematics and its Applications, 32, The Clarendon Press, Oxford University Press, Oxford, 2006. |
[14] |
P. Constantin, A. J. Majda and E. Tabak,
Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533.
doi: 10.1088/0951-7715/7/6/001. |
[15] |
P. Constantin, V. Vicol and A. Tarfulea,
Long time dynamics of forced critical SQG, Comm. Math. Phys., 335 (2015), 93-141.
doi: 10.1007/s00220-014-2129-3. |
[16] |
P. Constantin and J. Wu,
Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948.
doi: 10.1137/S0036141098337333. |
[17] |
A. Dutrifoy,
Examples of dispersive effects in non-viscous rotating fluids, J. Math. Pures Appl. (9), 84 (2005), 331-356.
doi: 10.1016/j.matpur.2004.09.007. |
[18] |
T. M. Elgindi and K. Widmayer,
Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-gepstrophic and inviscid Boussinesq systems, SIAM J. Math. Anal., 47 (2015), 4672-4684.
doi: 10.1137/14099036X. |
[19] |
T. Iwabuchi and R. Takada,
Global solutions for the Navier-Stokes equations in the rotational framework, Math. Ann., 357 (2013), 727-741.
doi: 10.1007/s00208-013-0923-4. |
[20] |
T. Kato,
Nonstationary flows of viscous and ideal fluids in R3, J. Functional Analysis, 9 (1972), 296-305.
doi: 10.1016/0022-1236(72)90003-1. |
[21] |
A. Kiselev,
Nonlocal maximum principles for active scalars, Adv. Math., 227 (2011), 1806-1826.
doi: 10.1016/j.aim.2011.03.019. |
[22] |
A. Kiselev and F. Nazarov,
Global regularity for the critical dispersive dissipative surface quasi-geostrophic equation, Nonlinearity, 23 (2010), 549-554.
doi: 10.1088/0951-7715/23/3/006. |
[23] |
A. Kiselev, F. Nazarov and A. Volberg,
Global well-posedness for the critical 2D dissipative quasigeostrophic equation, Invent. Math., 167 (2007), 445-453.
doi: 10.1007/s00222-006-0020-3. |
[24] |
Y. Koh, S. Lee and R. Takada,
Strichartz estimates for the Euler equations in the rotational framework, J. Differential Equations, 256 (2014), 707-744.
doi: 10.1016/j.jde.2013.09.017. |
[25] |
H. Kozono, T. Ogawa and Y. Tanuichi,
Navier-Stokes equations in the Besov space near $L^{\infty}$ and $BMO$, Kyushu J. Math., 57 (2003), 303-324.
doi: 10.2206/kyushujm.57.303. |
[26] |
G. Lapeyre,
Surface quasi-geostrophy, Fluids, 2 (2017), 1-28.
doi: 10.3390/fluids2010007. |
[27] |
H. Pak and Y. Park,
Existence of solution for the Euler equations in a critical Besov space $B_{\infty, 1}^{1}(R^{n})$, Comm. Partial Differential Equations, 29 (2004), 1149-1166.
doi: 10.1081/PDE-200033764. |
[28] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[29] |
S. Resnick, Dynamical Problems in Nonlinear Advective Partial Differential Equations, Ph.D. thesis, University of Chicago, 1995. |
[30] |
R. Takada,
Local existence and blow-up criterion for the Euler equations in Besov spaces of weak type, J. Evol. Equ., 8 (2008), 693-725.
doi: 10.1007/s00028-008-0403-6. |
[31] |
M. Vishik,
Hydrodynamics in Besov spaces, Arch. Ration. Mech. Anal., 145 (1998), 197-214.
doi: 10.1007/s002050050128. |
[32] |
R. Wan, Long time stability for the dispersive SQG equation and Boussinesq equations in Sobolev space $H^{s}$, Commun. Contemporary Math., (2018).
doi: 10.1142/S0219199718500633. |
[33] |
R. Wan and J. Chen, Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations, Z. Angew. Math. Phys., 67 (2016), 22pp.
doi: 10.1007/s00033-016-0697-0. |
[34] |
J. Wu, X. Xu and Z. Ye,
Global smooth solutions to the $n$-dimensional damped models of incompressible fluid mechanics with small initial datum, J. Nonlinear Sci., 25 (2015), 157-192.
doi: 10.1007/s00332-014-9224-7. |
[35] |
M. Zelati and V. Vicol,
On the global regularity for the supercritical SQG equation, Indiana Univ. Math. J., 65 (2016), 535-552.
doi: 10.1512/iumj.2016.65.5807. |
[36] |
Y. Zhou,
Local well-posedness for the incompressible Euler equations in the critical Besov spaces, Ann. Inst. Fourier (Grenoble), 54 (2004), 773-786.
doi: 10.5802/aif.2033. |
show all references
References:
[1] |
V. Angulo-Castillo and L. C. F. Ferreira,
On the 3D Euler equations with Coriolis force in borderline Besov spaces, Commun. Math. Sci, 16 (2018), 145-164.
doi: 10.4310/CMS.2018.v16.n1.a7. |
[2] |
A. Babin, A. Mahalov and B. Nicolaenko,
Global regularity of 3D rotating Navier-Stokes equations for resonant domains, Indiana Univ. Math. J., 48 (1999), 1133-1176.
|
[3] |
A. Babin, A. Mahalov and B. Nicolaenko,
3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity, Indiana Univ. Math. J., 50 (2001), 1-35.
doi: 10.1512/iumj.2001.50.2155. |
[4] |
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, 223, Springer-Verlag, Berlin-New York, 1976.
doi: 10.1007/978-3-642-66451-9. |
[5] |
J. Bourgain and D. Li,
Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces, Invent. Math., 201 (2015), 97-157.
doi: 10.1007/s00222-014-0548-6. |
[6] |
L. Caffarelli and V. Vasseur,
Drift diffusion equations with fractional diffusion and quasi-geostrophic equations, Ann. of Math. (2), 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[7] |
M. Cannone, C. Miao and L. Xue,
Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing, Proc. Lond. Math. Soc. (3), 106 (2013), 650-674.
doi: 10.1112/plms/pds046. |
[8] |
J. A. Carrillo and L. C. F. Ferreira,
The asymptotic behaviour of subcritical dissipative quasi-geostrophic equations, Nonlinearity, 21 (2008), 1001-1018.
doi: 10.1088/0951-7715/21/5/006. |
[9] |
D. Chae,
Local existence and blow-up criterion for the Euler equations in the Besov spaces, Asymptot. Anal., 38 (2004), 339-358.
|
[10] |
D. Chae, P. Constantin and J. Wu,
Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations, Arch. Ration. Mech. Anal., 202 (2011), 35-62.
doi: 10.1007/s00205-011-0411-5. |
[11] |
D. Chae, P. Constantin and J. Wu,
Dissipative models generalizing the 2D Navier-Stokes and surface quasi-geostrophic equations, Indiana Univ. Math. J., 61 (2012), 1997-2018.
doi: 10.1512/iumj.2012.61.4756. |
[12] |
D. Chae and J. Lee,
Global well-posedness in the super-critical dissipative quasi-geostrophic equations, Comm. Math. Phys., 233 (2003), 297-311.
doi: 10.1007/s00220-002-0750-z. |
[13] |
J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier, Mathematical Geophysics. An Introduction to Rotating Fluids and the Navier-Stokes Equations, Oxford Lecture Series in Mathematics and its Applications, 32, The Clarendon Press, Oxford University Press, Oxford, 2006. |
[14] |
P. Constantin, A. J. Majda and E. Tabak,
Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533.
doi: 10.1088/0951-7715/7/6/001. |
[15] |
P. Constantin, V. Vicol and A. Tarfulea,
Long time dynamics of forced critical SQG, Comm. Math. Phys., 335 (2015), 93-141.
doi: 10.1007/s00220-014-2129-3. |
[16] |
P. Constantin and J. Wu,
Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948.
doi: 10.1137/S0036141098337333. |
[17] |
A. Dutrifoy,
Examples of dispersive effects in non-viscous rotating fluids, J. Math. Pures Appl. (9), 84 (2005), 331-356.
doi: 10.1016/j.matpur.2004.09.007. |
[18] |
T. M. Elgindi and K. Widmayer,
Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-gepstrophic and inviscid Boussinesq systems, SIAM J. Math. Anal., 47 (2015), 4672-4684.
doi: 10.1137/14099036X. |
[19] |
T. Iwabuchi and R. Takada,
Global solutions for the Navier-Stokes equations in the rotational framework, Math. Ann., 357 (2013), 727-741.
doi: 10.1007/s00208-013-0923-4. |
[20] |
T. Kato,
Nonstationary flows of viscous and ideal fluids in R3, J. Functional Analysis, 9 (1972), 296-305.
doi: 10.1016/0022-1236(72)90003-1. |
[21] |
A. Kiselev,
Nonlocal maximum principles for active scalars, Adv. Math., 227 (2011), 1806-1826.
doi: 10.1016/j.aim.2011.03.019. |
[22] |
A. Kiselev and F. Nazarov,
Global regularity for the critical dispersive dissipative surface quasi-geostrophic equation, Nonlinearity, 23 (2010), 549-554.
doi: 10.1088/0951-7715/23/3/006. |
[23] |
A. Kiselev, F. Nazarov and A. Volberg,
Global well-posedness for the critical 2D dissipative quasigeostrophic equation, Invent. Math., 167 (2007), 445-453.
doi: 10.1007/s00222-006-0020-3. |
[24] |
Y. Koh, S. Lee and R. Takada,
Strichartz estimates for the Euler equations in the rotational framework, J. Differential Equations, 256 (2014), 707-744.
doi: 10.1016/j.jde.2013.09.017. |
[25] |
H. Kozono, T. Ogawa and Y. Tanuichi,
Navier-Stokes equations in the Besov space near $L^{\infty}$ and $BMO$, Kyushu J. Math., 57 (2003), 303-324.
doi: 10.2206/kyushujm.57.303. |
[26] |
G. Lapeyre,
Surface quasi-geostrophy, Fluids, 2 (2017), 1-28.
doi: 10.3390/fluids2010007. |
[27] |
H. Pak and Y. Park,
Existence of solution for the Euler equations in a critical Besov space $B_{\infty, 1}^{1}(R^{n})$, Comm. Partial Differential Equations, 29 (2004), 1149-1166.
doi: 10.1081/PDE-200033764. |
[28] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[29] |
S. Resnick, Dynamical Problems in Nonlinear Advective Partial Differential Equations, Ph.D. thesis, University of Chicago, 1995. |
[30] |
R. Takada,
Local existence and blow-up criterion for the Euler equations in Besov spaces of weak type, J. Evol. Equ., 8 (2008), 693-725.
doi: 10.1007/s00028-008-0403-6. |
[31] |
M. Vishik,
Hydrodynamics in Besov spaces, Arch. Ration. Mech. Anal., 145 (1998), 197-214.
doi: 10.1007/s002050050128. |
[32] |
R. Wan, Long time stability for the dispersive SQG equation and Boussinesq equations in Sobolev space $H^{s}$, Commun. Contemporary Math., (2018).
doi: 10.1142/S0219199718500633. |
[33] |
R. Wan and J. Chen, Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations, Z. Angew. Math. Phys., 67 (2016), 22pp.
doi: 10.1007/s00033-016-0697-0. |
[34] |
J. Wu, X. Xu and Z. Ye,
Global smooth solutions to the $n$-dimensional damped models of incompressible fluid mechanics with small initial datum, J. Nonlinear Sci., 25 (2015), 157-192.
doi: 10.1007/s00332-014-9224-7. |
[35] |
M. Zelati and V. Vicol,
On the global regularity for the supercritical SQG equation, Indiana Univ. Math. J., 65 (2016), 535-552.
doi: 10.1512/iumj.2016.65.5807. |
[36] |
Y. Zhou,
Local well-posedness for the incompressible Euler equations in the critical Besov spaces, Ann. Inst. Fourier (Grenoble), 54 (2004), 773-786.
doi: 10.5802/aif.2033. |
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