Sharp well-posedness for the Chen-Lee equation

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

Sharp well-posedness for the Chen-Lee equation

Ricardo A. Pastrán ^1, and Oscar G. Riaño ^2,

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

Abstract
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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.

Keywords: Cauchy problem, Benjamin-Ono equation., Nonlinear PDE, local and global well-posedness.

Mathematics Subject Classification: Primary: 35A01, 35K55, 35Q53; Secondary: 76B15, 76B0.

Citation: Ricardo A. Pastrán, Oscar G. Riaño. Sharp well-posedness for the Chen-Lee equation. Communications on Pure and 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, Differential Integral Equations, 16 (2003), 1249-1280.
[2]	H. A. Biagioni, J. L. Bona, R. J. Iório Jr. and M. Scialom, On the Korteweg-de Vries-Kuramoto-Sivashinsky equation, Adv. Differential Equations, 1 (1996), 1-20.
[3]	J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262. doi: 10.1007/BF01895688.
[4]	D. B. Dix, Temporal asymptotic behavior of solutions of the Benjamin-Ono-Burgers equation, J. Differential Equations, 90 (1991), 238-287. doi: 10.1016/0022-0396(91)90148-3.
[5]	D. B. Dix, Nonuniqueness and uniqueness in the initial value problem for Burgers' equation, SIAM J. Math. Anal., 27 (1996), 708-724. doi: 10.1137/0527038.
[6]	O. Duque, Sobre una versión bidimensional de la ecuación Benjamin-Ono generalizada, Ph.D thesis, Universidad Nacional de Colombia, sede Bogotá, 2014.
[7]	S. A. Esfahani, High Dimensional Nonlinear Dispersive Models, Ph.D thesis, IMPA, 2008.
[8]	B. -F. Feng and T. Kawahara, Temporal evolutions and stationary waves for dissipative Benjamin-Ono equation, Phys. D, 139 (2000), 301-318. doi: 10.1016/S0167-2789(99)00227-4.
[9]	R. J. Iório Jr., On the Cauchy problem for the Benjamin-Ono equation, Comm. Partial Differential Equations, 11 (1986), 1031-1081. doi: 10.1080/03605308608820456.
[10]	C. E. Kenig, G. 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.
[11]	Y. C. Lee and H. H. Chen, Nonlinear dynamical models of plasma turbulence, Phys. Scr., T2/1 (1982), 41-47.
[12]	R. A. Pastrán, On a Perturbation of the Benjamin-Ono equation, Nonlinear Anal., 93 (2013), 273-296. 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, Rev. Colombiana Mat., arXiv:1510.00447.
[14]	D. Pilod, Sharp well-posedness results for the Kuramoto-Velarde equation, Commun. Pure Appl. Anal., 7 (2008), 867-881. doi: 10.3934/cpaa.2008.7.867.
[15]	S. Qian, H. H. Chen and Y. C. Lee, A turbulence model with stochastic soliton motion, J. Math. Phys., 31 (1990), 506-516. doi: 10.1063/1.528884.
[16]	S. Qian, Y. C. Lee and H. H. Chen, A study of nonlinear dynamical models of plasma turbulence, Phys. Fluids B 1, 1 (1989), 87-98.
[17]	S. Vento, Well-posedness and ill-posedness results for dissipative Benjamin-Ono equations, Osaka J. Math., 48 (2011), 933-958.

show all references

References:

[1]	B. Alvarez Samaniego, The Cauchy problem for a nonlocal perturbation of the KdV equation, Differential Integral Equations, 16 (2003), 1249-1280.
[2]	H. A. Biagioni, J. L. Bona, R. J. Iório Jr. and M. Scialom, On the Korteweg-de Vries-Kuramoto-Sivashinsky equation, Adv. Differential Equations, 1 (1996), 1-20.
[3]	J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262. doi: 10.1007/BF01895688.
[4]	D. B. Dix, Temporal asymptotic behavior of solutions of the Benjamin-Ono-Burgers equation, J. Differential Equations, 90 (1991), 238-287. doi: 10.1016/0022-0396(91)90148-3.
[5]	D. B. Dix, Nonuniqueness and uniqueness in the initial value problem for Burgers' equation, SIAM J. Math. Anal., 27 (1996), 708-724. doi: 10.1137/0527038.
[6]	O. Duque, Sobre una versión bidimensional de la ecuación Benjamin-Ono generalizada, Ph.D thesis, Universidad Nacional de Colombia, sede Bogotá, 2014.
[7]	S. A. Esfahani, High Dimensional Nonlinear Dispersive Models, Ph.D thesis, IMPA, 2008.
[8]	B. -F. Feng and T. Kawahara, Temporal evolutions and stationary waves for dissipative Benjamin-Ono equation, Phys. D, 139 (2000), 301-318. doi: 10.1016/S0167-2789(99)00227-4.
[9]	R. J. Iório Jr., On the Cauchy problem for the Benjamin-Ono equation, Comm. Partial Differential Equations, 11 (1986), 1031-1081. doi: 10.1080/03605308608820456.
[10]	C. E. Kenig, G. 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.
[11]	Y. C. Lee and H. H. Chen, Nonlinear dynamical models of plasma turbulence, Phys. Scr., T2/1 (1982), 41-47.
[12]	R. A. Pastrán, On a Perturbation of the Benjamin-Ono equation, Nonlinear Anal., 93 (2013), 273-296. 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, Rev. Colombiana Mat., arXiv:1510.00447.
[14]	D. Pilod, Sharp well-posedness results for the Kuramoto-Velarde equation, Commun. Pure Appl. Anal., 7 (2008), 867-881. doi: 10.3934/cpaa.2008.7.867.
[15]	S. Qian, H. H. Chen and Y. C. Lee, A turbulence model with stochastic soliton motion, J. Math. Phys., 31 (1990), 506-516. doi: 10.1063/1.528884.
[16]	S. Qian, Y. C. Lee and H. H. Chen, A study of nonlinear dynamical models of plasma turbulence, Phys. Fluids B 1, 1 (1989), 87-98.
[17]	S. Vento, Well-posedness and ill-posedness results for dissipative Benjamin-Ono equations, Osaka J. Math., 48 (2011), 933-958.

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