doi: 10.3934/dcdsb.2022115
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Global attractor for the periodic generalized Korteweg-De Vries equation through smoothing

Department of Mathematics, University of Illinois Urbana-Champaign, Urbana, IL 61801, USA

Received  February 2022 Revised  May 2022 Early access June 2022

We establish a smoothing result for the generalized KdV (gKdV) on the torus with polynomial non-linearity, damping, and forcing that matches the smoothing level for the gKdV at $ H^1 $. As a consequence, we establish the existence of a global attractor for this equation as well as its compactness in $ H^s(\mathbb{T}) $, $ s\in (1, 2). $

Citation: Ryan McConnell. Global attractor for the periodic generalized Korteweg-De Vries equation through smoothing. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022115
References:
[1]

J. Bao and Y. Wu, Global well-posedness for the periodic generalized Korteweg–de Vries equations, Indiana Univ. Math. J., 66 (2017), 1797-1825.  doi: 10.1512/iumj.2017.66.6135.

[2]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.

[3]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb R$ and $\mathbb T$, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.

[4]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal., 211 (2004), 173-218.  doi: 10.1016/S0022-1236(03)00218-0.

[5]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Resonant decompositions and the $ {I} $-method for the cubic nonlinear Schrödinger equation on $\mathbb{R}^2$, Discrete Contin. Dyn. Syst., 21 (2008), 665-686.  doi: 10.3934/dcds.2008.21.665.

[6]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669.  doi: 10.1137/S0036141001384387.

[7]

S. Correia and J. D. Silva, Nonlinear smoothing for dispersive PDE: A unified approach, J. Differential Equations, 269 (2020), 4253-4285.  doi: 10.1016/j.jde.2020.03.038.

[8]

T. DlotkoM. B. Kania and M. Yang, Generalized Korteweg-de Vries equation in $H^1(\mathbb R)$, Nonlinear Anal., 71 (2009), 3934-3947.  doi: 10.1016/j.na.2009.02.062.

[9]

M. B. ErdoğanT. B. Gurel and N. Tzirakis, Smoothing for the fractional Schrödinger equation on the torus and the real line, Indiana Univ. Math. J., 68 (2019), 369-392.  doi: 10.1512/iumj.2019.68.7618.

[10]

M. B. Erdoğan and N. Tzirakis, Dispersive Partial Differential Equations: Wellposedness and Applications, vol. 86, Cambridge University Press, 2016. doi: 10.1017/CBO9781316563267.

[11]

M. B. Erdoğan and N. Tzirakis, Long time dynamics for forced and weakly damped KdV on the torus, Commun. Pure Appl. Anal., 12 (2013), 2669-2684.  doi: 10.3934/cpaa.2013.12.2669.

[12]

M. B. Erdoğan and N. Tzirakis, Global smoothing for the periodic KdV evolution, Int. Math. Res. Not. IMRN, 2013 (2013), 4589-4614.  doi: 10.1093/imrn/rns189.

[13]

L. G. FarahF. Linares and A. Pastor, The supercritical generalized KdV equation: Global well-posedness in the energy space and below, Math. Res. Lett., 18 (2011), 357-377.  doi: 10.4310/MRL.2011.v18.n2.a13.

[14]

O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete Contin. Dynam. Systems, 6 (2000), 625-644.  doi: 10.3934/dcds.2000.6.625.

[15]

O. Goubet, Analyticity of the global attractor for damped forced periodic Korteweg–de Vries equation, J. Differential Equations, 264 (2018), 3052-3066.  doi: 10.1016/j.jde.2017.11.010.

[16]

P. Goyal, Global attractor for weakly damped, forced mKdV equation below energy space, Nagoya Math. J., 241 (2021), 171-203.  doi: 10.1017/nmj.2019.17.

[17]

P. Goyal, Global attractor for weakly damped, forced mKdV equation in low regularity spaces, São Paulo Journal of Mathematical Sciences, 1–23.

[18]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations, Springer, 1975, 25–70.

[19]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.

[20]

C. E. KenigG. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.  doi: 10.1090/S0894-0347-96-00200-7.

[21]

R. Killip and M. Vişan, KdV is well-posed in $H^{-1}$, Ann. of Math. (2), 190 (2019), 249-305.  doi: 10.4007/annals.2019.190.1.4.

[22]

S. Oh, Resonant phase-shift and global smoothing of the periodic Korteweg-de Vries equation in low regularity settings, Adv. Differential Equations, 18 (2013), 633-662. 

[23]

S. Oh and A. G. Stefanov, Smoothing and growth bound of periodic generalized Korteweg–De Vries equation, J. Hyperbolic Differ. Equ., 18 (2021), 899-930.  doi: 10.1142/S0219891621500260.

[24]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[25]

K. Tsugawa, Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index, Commun. Pure Appl. Anal., 3 (2004), 301-318.  doi: 10.3934/cpaa.2004.3.301.

[26]

M. Wang, Global attractor for weakly damped gKdV equations in higher Sobolev spaces, Discrete Contin. Dyn. Syst., 35 (2015), 3799-3825.  doi: 10.3934/dcds.2015.35.3799.

[27]

M. Wang and J. Huang, The global attractor for the weakly damped KdV equation on $\mathbb R$ has a finite fractal dimension, Math. Methods Appl. Sci., 43 (2020), 4567-4584.  doi: 10.1002/mma.6215.

[28]

M. WangD. LiC. Zhang and Y. Tang, Long time behavior of solutions of gKdV equations, J. Math. Anal. Appl., 390 (2012), 136-150.  doi: 10.1016/j.jmaa.2012.01.031.

[29]

X. Yang, Global attractor for the weakly damped forced KdV equation in Sobolev spaces of low regularity, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 273-285.  doi: 10.1007/s00030-010-0095-9.

show all references

References:
[1]

J. Bao and Y. Wu, Global well-posedness for the periodic generalized Korteweg–de Vries equations, Indiana Univ. Math. J., 66 (2017), 1797-1825.  doi: 10.1512/iumj.2017.66.6135.

[2]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.

[3]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb R$ and $\mathbb T$, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.

[4]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal., 211 (2004), 173-218.  doi: 10.1016/S0022-1236(03)00218-0.

[5]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Resonant decompositions and the $ {I} $-method for the cubic nonlinear Schrödinger equation on $\mathbb{R}^2$, Discrete Contin. Dyn. Syst., 21 (2008), 665-686.  doi: 10.3934/dcds.2008.21.665.

[6]

J. CollianderM. KeelG. StaffilaniH. Takaoka and T. Tao, Global well-posedness for Schrödinger equations with derivative, SIAM J. Math. Anal., 33 (2001), 649-669.  doi: 10.1137/S0036141001384387.

[7]

S. Correia and J. D. Silva, Nonlinear smoothing for dispersive PDE: A unified approach, J. Differential Equations, 269 (2020), 4253-4285.  doi: 10.1016/j.jde.2020.03.038.

[8]

T. DlotkoM. B. Kania and M. Yang, Generalized Korteweg-de Vries equation in $H^1(\mathbb R)$, Nonlinear Anal., 71 (2009), 3934-3947.  doi: 10.1016/j.na.2009.02.062.

[9]

M. B. ErdoğanT. B. Gurel and N. Tzirakis, Smoothing for the fractional Schrödinger equation on the torus and the real line, Indiana Univ. Math. J., 68 (2019), 369-392.  doi: 10.1512/iumj.2019.68.7618.

[10]

M. B. Erdoğan and N. Tzirakis, Dispersive Partial Differential Equations: Wellposedness and Applications, vol. 86, Cambridge University Press, 2016. doi: 10.1017/CBO9781316563267.

[11]

M. B. Erdoğan and N. Tzirakis, Long time dynamics for forced and weakly damped KdV on the torus, Commun. Pure Appl. Anal., 12 (2013), 2669-2684.  doi: 10.3934/cpaa.2013.12.2669.

[12]

M. B. Erdoğan and N. Tzirakis, Global smoothing for the periodic KdV evolution, Int. Math. Res. Not. IMRN, 2013 (2013), 4589-4614.  doi: 10.1093/imrn/rns189.

[13]

L. G. FarahF. Linares and A. Pastor, The supercritical generalized KdV equation: Global well-posedness in the energy space and below, Math. Res. Lett., 18 (2011), 357-377.  doi: 10.4310/MRL.2011.v18.n2.a13.

[14]

O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations, Discrete Contin. Dynam. Systems, 6 (2000), 625-644.  doi: 10.3934/dcds.2000.6.625.

[15]

O. Goubet, Analyticity of the global attractor for damped forced periodic Korteweg–de Vries equation, J. Differential Equations, 264 (2018), 3052-3066.  doi: 10.1016/j.jde.2017.11.010.

[16]

P. Goyal, Global attractor for weakly damped, forced mKdV equation below energy space, Nagoya Math. J., 241 (2021), 171-203.  doi: 10.1017/nmj.2019.17.

[17]

P. Goyal, Global attractor for weakly damped, forced mKdV equation in low regularity spaces, São Paulo Journal of Mathematical Sciences, 1–23.

[18]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations, Springer, 1975, 25–70.

[19]

C. E. KenigG. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.

[20]

C. E. KenigG. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.  doi: 10.1090/S0894-0347-96-00200-7.

[21]

R. Killip and M. Vişan, KdV is well-posed in $H^{-1}$, Ann. of Math. (2), 190 (2019), 249-305.  doi: 10.4007/annals.2019.190.1.4.

[22]

S. Oh, Resonant phase-shift and global smoothing of the periodic Korteweg-de Vries equation in low regularity settings, Adv. Differential Equations, 18 (2013), 633-662. 

[23]

S. Oh and A. G. Stefanov, Smoothing and growth bound of periodic generalized Korteweg–De Vries equation, J. Hyperbolic Differ. Equ., 18 (2021), 899-930.  doi: 10.1142/S0219891621500260.

[24]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, 2nd edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[25]

K. Tsugawa, Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index, Commun. Pure Appl. Anal., 3 (2004), 301-318.  doi: 10.3934/cpaa.2004.3.301.

[26]

M. Wang, Global attractor for weakly damped gKdV equations in higher Sobolev spaces, Discrete Contin. Dyn. Syst., 35 (2015), 3799-3825.  doi: 10.3934/dcds.2015.35.3799.

[27]

M. Wang and J. Huang, The global attractor for the weakly damped KdV equation on $\mathbb R$ has a finite fractal dimension, Math. Methods Appl. Sci., 43 (2020), 4567-4584.  doi: 10.1002/mma.6215.

[28]

M. WangD. LiC. Zhang and Y. Tang, Long time behavior of solutions of gKdV equations, J. Math. Anal. Appl., 390 (2012), 136-150.  doi: 10.1016/j.jmaa.2012.01.031.

[29]

X. Yang, Global attractor for the weakly damped forced KdV equation in Sobolev spaces of low regularity, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 273-285.  doi: 10.1007/s00030-010-0095-9.

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