November  2016, 15(6): 2179-2202. doi: 10.3934/cpaa.2016033

Sharp well-posedness for the Chen-Lee equation

1. 

Departamento de Matemáticas, Universidad Nacional de Colombia, AK 30 45-03, 111321, Bogotá, Colombia

2. 

Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, RJ, Brazil

Received  December 2015 Revised  May 2016 Published  September 2016

We study the initial value problem associated to a perturbation of the Benjamin-Ono equation or Chen-Lee equation. We prove that results about local and global well-posedness for initial data in $H^s(\mathbb{R})$, with $s>-1/2$, are sharp in the sense that the flow-map data-solution fails to be $C^3$ in $H^s(\mathbb{R})$ when $s<-\frac{1}{2}$. Also, we determine the limiting behavior of the solutions when the dispersive and dissipative parameters goes to zero. In addition, we will discuss the asymptotic behavior (as $|x|\to \infty$) of the solutions by solving the equation in weighted Sobolev spaces.
Citation: Ricardo A. Pastrán, Oscar G. Riaño. Sharp well-posedness for the Chen-Lee equation. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2179-2202. doi: 10.3934/cpaa.2016033
References:
[1]

B. Alvarez Samaniego, The Cauchy problem for a nonlocal perturbation of the KdV equation,, \emph{Differential Integral Equations}, 16 (2003), 1249.

[2]

H. A. Biagioni, J. L. Bona, R. J. Iório Jr. and M. Scialom, On the Korteweg-de Vries-Kuramoto-Sivashinsky equation,, \emph{Adv. Differential Equations}, 1 (1996), 1.

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation,, \emph{Geom. Funct. Anal.}, 3 (1993), 209. doi: 10.1007/BF01895688.

[4]

D. B. Dix, Temporal asymptotic behavior of solutions of the Benjamin-Ono-Burgers equation,, \emph{J. Differential Equations}, 90 (1991), 238. doi: 10.1016/0022-0396(91)90148-3.

[5]

D. B. Dix, Nonuniqueness and uniqueness in the initial value problem for Burgers' equation,, \emph{SIAM J. Math. Anal.}, 27 (1996), 708. doi: 10.1137/0527038.

[6]

O. Duque, Sobre una versión bidimensional de la ecuación Benjamin-Ono generalizada,, Ph.D thesis, (2014).

[7]

S. A. Esfahani, High Dimensional Nonlinear Dispersive Models,, Ph.D thesis, (2008).

[8]

B. -F. Feng and T. Kawahara, Temporal evolutions and stationary waves for dissipative Benjamin-Ono equation,, \emph{Phys. D}, 139 (2000), 301. doi: 10.1016/S0167-2789(99)00227-4.

[9]

R. J. Iório Jr., On the Cauchy problem for the Benjamin-Ono equation,, \emph{Comm. Partial Differential Equations}, 11 (1986), 1031. doi: 10.1080/03605308608820456.

[10]

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

[11]

Y. C. Lee and H. H. Chen, Nonlinear dynamical models of plasma turbulence,, \emph{Phys. Scr.}, T2/1 (1982), 41.

[12]

R. A. Pastrán, On a Perturbation of the Benjamin-Ono equation,, \emph{Nonlinear Anal.}, 93 (2013), 273. doi: 10.1016/j.na.2013.07.014.

[13]

R. A. Pastrán and O. G. Riaño, On the well-posedness for the Chen-Lee equation in periodic Sobolev spaces, preprint,, \emph{Rev. Colombiana Mat.}, ().

[14]

D. Pilod, Sharp well-posedness results for the Kuramoto-Velarde equation,, \emph{Commun. Pure Appl. Anal.}, 7 (2008), 867. doi: 10.3934/cpaa.2008.7.867.

[15]

S. Qian, H. H. Chen and Y. C. Lee, A turbulence model with stochastic soliton motion,, \emph{J. Math. Phys.}, 31 (1990), 506. doi: 10.1063/1.528884.

[16]

S. Qian, Y. C. Lee and H. H. Chen, A study of nonlinear dynamical models of plasma turbulence,, \emph{Phys. Fluids B 1}, 1 (1989), 87.

[17]

S. Vento, Well-posedness and ill-posedness results for dissipative Benjamin-Ono equations,, \emph{Osaka J. Math.}, 48 (2011), 933.

show all references

References:
[1]

B. Alvarez Samaniego, The Cauchy problem for a nonlocal perturbation of the KdV equation,, \emph{Differential Integral Equations}, 16 (2003), 1249.

[2]

H. A. Biagioni, J. L. Bona, R. J. Iório Jr. and M. Scialom, On the Korteweg-de Vries-Kuramoto-Sivashinsky equation,, \emph{Adv. Differential Equations}, 1 (1996), 1.

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation,, \emph{Geom. Funct. Anal.}, 3 (1993), 209. doi: 10.1007/BF01895688.

[4]

D. B. Dix, Temporal asymptotic behavior of solutions of the Benjamin-Ono-Burgers equation,, \emph{J. Differential Equations}, 90 (1991), 238. doi: 10.1016/0022-0396(91)90148-3.

[5]

D. B. Dix, Nonuniqueness and uniqueness in the initial value problem for Burgers' equation,, \emph{SIAM J. Math. Anal.}, 27 (1996), 708. doi: 10.1137/0527038.

[6]

O. Duque, Sobre una versión bidimensional de la ecuación Benjamin-Ono generalizada,, Ph.D thesis, (2014).

[7]

S. A. Esfahani, High Dimensional Nonlinear Dispersive Models,, Ph.D thesis, (2008).

[8]

B. -F. Feng and T. Kawahara, Temporal evolutions and stationary waves for dissipative Benjamin-Ono equation,, \emph{Phys. D}, 139 (2000), 301. doi: 10.1016/S0167-2789(99)00227-4.

[9]

R. J. Iório Jr., On the Cauchy problem for the Benjamin-Ono equation,, \emph{Comm. Partial Differential Equations}, 11 (1986), 1031. doi: 10.1080/03605308608820456.

[10]

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

[11]

Y. C. Lee and H. H. Chen, Nonlinear dynamical models of plasma turbulence,, \emph{Phys. Scr.}, T2/1 (1982), 41.

[12]

R. A. Pastrán, On a Perturbation of the Benjamin-Ono equation,, \emph{Nonlinear Anal.}, 93 (2013), 273. doi: 10.1016/j.na.2013.07.014.

[13]

R. A. Pastrán and O. G. Riaño, On the well-posedness for the Chen-Lee equation in periodic Sobolev spaces, preprint,, \emph{Rev. Colombiana Mat.}, ().

[14]

D. Pilod, Sharp well-posedness results for the Kuramoto-Velarde equation,, \emph{Commun. Pure Appl. Anal.}, 7 (2008), 867. doi: 10.3934/cpaa.2008.7.867.

[15]

S. Qian, H. H. Chen and Y. C. Lee, A turbulence model with stochastic soliton motion,, \emph{J. Math. Phys.}, 31 (1990), 506. doi: 10.1063/1.528884.

[16]

S. Qian, Y. C. Lee and H. H. Chen, A study of nonlinear dynamical models of plasma turbulence,, \emph{Phys. Fluids B 1}, 1 (1989), 87.

[17]

S. Vento, Well-posedness and ill-posedness results for dissipative Benjamin-Ono equations,, \emph{Osaka J. Math.}, 48 (2011), 933.

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