March  2020, 40(3): 1411-1433. doi: 10.3934/dcds.2020082

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

* Corresponding author: Lucas C. F. Ferreira

Received  January 2019 Revised  July 2019 Published  December 2019

Fund Project: The first author was supported by CNPq, Brazil. The second author was supported by FAPESP and CNPq, Brazil. The third author was supported by FAPESP, 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.

Citation: Vladimir Angulo-Castillo, Lucas C. F. Ferreira, Leonardo Kosloff. Long-time solvability for the 2D dispersive SQG equation with improved regularity. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1411-1433. doi: 10.3934/dcds.2020082
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. BabinA. 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. BabinA. 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. CannoneC. 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. ChaeP. 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. ChaeP. 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. ConstantinA. 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. ConstantinV. 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. KiselevF. 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. KohS. 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. KozonoT. 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. WuX. 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. BabinA. 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. BabinA. 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. CannoneC. 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. ChaeP. 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. ChaeP. 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. ConstantinA. 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. ConstantinV. 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. KiselevF. 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. KohS. 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. KozonoT. 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. WuX. 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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