• Previous Article
    3D-2D asymptotic observation for minimization problems associated with degenerate energy-coefficients
  • PROC Home
  • This Issue
  • Next Article
    Pohozaev-Ôtani type inequalities for weak solutions of quasilinear elliptic equations with homogeneous coefficients
2011, 2011(Special): 634-642. doi: 10.3934/proc.2011.2011.634

Anisotropic Gevrey regularity for mKdV on the circle

1. 

Department of Mathematics, East Central University, Ada, OK 74820, United States

2. 

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States

3. 

Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP 13565-905, Brazil

Received  July 2010 Revised  April 2011 Published  October 2011

It is shown that the solution to the Cauchy problem for the modifi ed Korteweg-de Vries equation with initial data in an analytic Gevrey space $G^\sigma$, $\sigma \>= 1$, as a function of the spacial variable belongs to the same Gevrey space. However, considered as function of time the solution does not belong to $G^\sigma$. In fact, it belong to $G^(3\sigma)$ and not to any Gevrey space $G^r$, 1 $\<= r$ < 3$\sigma$.
Citation: Heather Hannah, A. Alexandrou Himonas, Gerson Petronilho. Anisotropic Gevrey regularity for mKdV on the circle. Conference Publications, 2011, 2011 (Special) : 634-642. doi: 10.3934/proc.2011.2011.634
[1]

Van Duong Dinh. On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1127-1143. doi: 10.3934/dcds.2018047

[2]

Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi. Local well-posedness in low regularity of the MKDV equation with periodic boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1635-1654. doi: 10.3934/dcds.2010.28.1635

[3]

Kotaro Tsugawa. Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index. Communications on Pure & Applied Analysis, 2004, 3 (2) : 301-318. doi: 10.3934/cpaa.2004.3.301

[4]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511

[5]

Nadia Lekrine, Chao-Jiang Xu. Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation. Kinetic & Related Models, 2009, 2 (4) : 647-666. doi: 10.3934/krm.2009.2.647

[6]

Felipe Linares, M. Panthee. On the Cauchy problem for a coupled system of KdV equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 417-431. doi: 10.3934/cpaa.2004.3.417

[7]

Hao Tang, Zhengrong Liu. On the Cauchy problem for the Boltzmann equation in Chemin-Lerner type spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2229-2256. doi: 10.3934/dcds.2016.36.2229

[8]

Cedric Galusinski, Serguei Zelik. Uniform Gevrey regularity for the attractor of a damped wave equation. Conference Publications, 2003, 2003 (Special) : 305-312. doi: 10.3934/proc.2003.2003.305

[9]

Siqi Xu, Dongfeng Yan. Smooth quasi-periodic solutions for the perturbed mKdV equation. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1857-1869. doi: 10.3934/cpaa.2016019

[10]

Shouming Zhou. The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4967-4986. doi: 10.3934/dcds.2014.34.4967

[11]

S. Raynor, G. Staffilani. Low regularity stability of solitons for the KDV equation. Communications on Pure & Applied Analysis, 2003, 2 (3) : 277-296. doi: 10.3934/cpaa.2003.2.277

[12]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597

[13]

V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 731-753. doi: 10.3934/dcds.2004.10.731

[14]

Rudong Zheng, Zhaoyang Yin. The Cauchy problem for a generalized Novikov equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3503-3519. doi: 10.3934/dcds.2017149

[15]

M. Nakamura, Tohru Ozawa. The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 215-231. doi: 10.3934/dcds.1999.5.215

[16]

Alberto Fiorenza, Anna Mercaldo, Jean Michel Rakotoson. Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 893-906. doi: 10.3934/dcds.2002.8.893

[17]

Duchao Liu, Beibei Wang, Peihao Zhao. On the trace regularity results of Musielak-Orlicz-Sobolev spaces in a bounded domain. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1643-1659. doi: 10.3934/cpaa.2016018

[18]

Minghua Yang, Jinyi Sun. Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1617-1639. doi: 10.3934/cpaa.2017078

[19]

Yuta Kugo, Motohiro Sobajima, Toshiyuki Suzuki, Tomomi Yokota, Kentarou Yoshii. Solvability of a class of complex Ginzburg-Landau equations in periodic Sobolev spaces. Conference Publications, 2015, 2015 (special) : 754-763. doi: 10.3934/proc.2015.0754

[20]

Guillermo Reyes, Juan-Luis Vázquez. The Cauchy problem for the inhomogeneous porous medium equation. Networks & Heterogeneous Media, 2006, 1 (2) : 337-351. doi: 10.3934/nhm.2006.1.337

 Impact Factor: 

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (0)

[Back to Top]