American Institute of Mathematical Sciences

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

Long-time solvability for the 2D dispersive SQG equation with improved regularity

Vladimir Angulo-Castillo 1, , Lucas C. F. Ferreira 2,, and  Leonardo Kosloff 2,

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.

Keywords: Dispersive SQG equation, Strichartz estimates, long-time solvability, blow-up, Besov spaces.
Mathematics Subject Classification: Primary: 76U05, 46E35, 35Q35; Secondary: 76B03, 35A01.
Citation: Vladimir Angulo-Castillo, Lucas C. F. Ferreira, Leonardo Kosloff. Long-time solvability for the 2D dispersive SQG equation with improved regularity. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. Chae, Local existence and blow-up criterion for the Euler equations in the Besov spaces, Asymptot. Anal., 38 (2004), 339-358.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

A. Kiselev, Nonlocal maximum principles for active scalars, Adv. Math., 227 (2011), 1806-1826.  doi: 10.1016/j.aim.2011.03.019.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

G. Lapeyre, Surface quasi-geostrophy, Fluids, 2 (2017), 1-28.  doi: 10.3390/fluids2010007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

S. Resnick, Dynamical Problems in Nonlinear Advective Partial Differential Equations, Ph.D. thesis, University of Chicago, 1995.  Google Scholar

[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.  Google Scholar

[31]

M. Vishik, Hydrodynamics in Besov spaces, Arch. Ration. Mech. Anal., 145 (1998), 197-214.  doi: 10.1007/s002050050128.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. Chae, Local existence and blow-up criterion for the Euler equations in the Besov spaces, Asymptot. Anal., 38 (2004), 339-358.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

A. Kiselev, Nonlocal maximum principles for active scalars, Adv. Math., 227 (2011), 1806-1826.  doi: 10.1016/j.aim.2011.03.019.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

G. Lapeyre, Surface quasi-geostrophy, Fluids, 2 (2017), 1-28.  doi: 10.3390/fluids2010007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

S. Resnick, Dynamical Problems in Nonlinear Advective Partial Differential Equations, Ph.D. thesis, University of Chicago, 1995.  Google Scholar

[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.  Google Scholar

[31]

M. Vishik, Hydrodynamics in Besov spaces, Arch. Ration. Mech. Anal., 145 (1998), 197-214.  doi: 10.1007/s002050050128.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Min Li, Zhaoyang Yin. Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6471-6485. doi: 10.3934/dcds.2017280
[2]

C. I. Christov, M. D. Todorov. Investigation of the long-time evolution of localized solutions of a dispersive wave system. Conference Publications, 2013, 2013 (special) : 139-148. doi: 10.3934/proc.2013.2013.139
[3]

María J. Cáceres, Ricarda Schneider. Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models. Kinetic & Related Models, 2017, 10 (3) : 587-612. doi: 10.3934/krm.2017024
[4]

Frédéric Abergel, Jean-Michel Rakotoson. Gradient blow-up in Zygmund spaces for the very weak solution of a linear elliptic equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1809-1818. doi: 10.3934/dcds.2013.33.1809
[5]

Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1
[6]

Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113
[7]

Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831
[8]

Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903
[9]

Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54
[10]

Hongwei Chen. Blow-up estimates of positive solutions of a reaction-diffusion system. Conference Publications, 2003, 2003 (Special) : 182-188. doi: 10.3934/proc.2003.2003.182
[11]

Yi-hang Hao, Xian-Gao Liu. The existence and blow-up criterion of liquid crystals system in critical Besov space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 225-236. doi: 10.3934/cpaa.2014.13.225
[12]

Baoquan Yuan, Xiao Li. Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2167-2179. doi: 10.3934/dcdss.2016090
[13]

José M. Arrieta, Raúl Ferreira, Arturo de Pablo, Julio D. Rossi. Stability of the blow-up time and the blow-up set under perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 43-61. doi: 10.3934/dcds.2010.26.43
[14]

Vural Bayrak, Emil Novruzov, Ibrahim Ozkol. Local-in-space blow-up criteria for two-component nonlinear dispersive wave system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6023-6037. doi: 10.3934/dcds.2019263
[15]

Monica Marras, Stella Vernier Piro. Bounds for blow-up time in nonlinear parabolic systems. Conference Publications, 2011, 2011 (Special) : 1025-1031. doi: 10.3934/proc.2011.2011.1025
[16]

Jean-Paul Chehab, Pierre Garnier, Youcef Mammeri. Long-time behavior of solutions of a BBM equation with generalized damping. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1897-1915. doi: 10.3934/dcdsb.2015.20.1897
[17]

Yihong Du, Yoshio Yamada. On the long-time limit of positive solutions to the degenerate logistic equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 123-132. doi: 10.3934/dcds.2009.25.123
[18]

Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163
[19]

Pelin G. Geredeli, Azer Khanmamedov. Long-time dynamics of the parabolic $p$-Laplacian equation. Communications on Pure & Applied Analysis, 2013, 12 (2) : 735-754. doi: 10.3934/cpaa.2013.12.735
[20]

Vladimir Varlamov. Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 675-702. doi: 10.3934/dcds.2001.7.675

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (62)
  • HTML views (47)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences