Partial Differential Equations

Nonlinear Dispersive Equations:

- Well-posedness
- Stability and asymptotic stability
- Blow-up

Most of my papers are available on the arXiv

*Dispersive estimates for full dispersion KP equations,*preprint (2020), 29 pages, joint with Jean-Claude Saut, Sigmund Selberg and Achenef Tesfahun.*On the unique continuation of solutions to non-local non-linear dispersive equations,*to appear in Comm. Part. Diff. Eq. (2020), 15 pages, joint with Carlos Kenig, Gustavo Ponce and Luis Vega.*Full family of flattening solitary waves for the mass critical generalized KdV equation,*preprint (2019), 62 pages, joint with Yvan Martel.

*On the local well-posedness for a full dispersion Boussinesq system with surface tension,*Proc. Amer. Math. Soc.,**147**(2019), 2545-2559 , joint with Henrik Kalisch.*On unconditional well-posedness for the periodic modified Korteweg-De Vries equation,*J. Math. Soc. Japan (2018),**71**(2019), no. 1, 147–201, joint with Luc Molinet and Stéphane Vento.*On well-posedness for some dispersive perturbations of Burgers' equation,*Ann. Inst. H. Poincaré An. Non Lin.,**35**(2018), no. 7, 1719–1756, joint with Luc Molinet and Stéphane Vento.*Unconditional uniqueness for the modified Korteweg- de Vries equation on the line,*Rev. Mat. Iber.,**34**(2018), no. 4, 1563–1608, joint with Luc Molinet and Stéphane Vento.*The Cauchy problem for the fractional Kadomtsev-Petviashvili equations,*SIAM J. Math. Analysis,**50**(2018), 3172–3209, joint with Felipe Linares, and Jean-Claude Saut.*On Whitham and related equations,*Studies Appl. Math.,**140**(2018), 133-177, joint with Christian Klein, Felipe Linares, and Jean-Claude Saut.*Construction of a minimal mass blow up solution of the modified Benjamin-Ono equation,*Mathematische Annalen,**369**(2017), 153-245, joint with Yvan Martel.*Local well-posedness for the KdV hierarchy at high regularity,*Adv. Diff. Eq.,**21**(2016) 801-836, joint with Carlos Kenig.*Asymptotic stability of high-dimensional Zakharov-Kuznetsov solitons,*Arch. Rat. Mech. Anal.,**220**(2016), 639-710, joint with Raphaël Côte, Claudio Muñoz, and Gideon Simpson.*Remarks on the orbital stability of ground state solutions of fKdV e relative equations,*Adv. Diff. Eq.,**20**(2015), 835-858, joint with Felipe Linares, and Jean-Claude Saut.*Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications,*Ann. Inst. H. Poincaré An. Non Lin.,**32**(2015), 347-371, joint with Luc Molinet.*Well-posedness for the fifth-order KdV equation in the energy space,*Trans AMS,**367**(2015), 2555-2612 , joint with Carlos Kenig.*Dispersive perturbations of Burgers and hyperbolic equations I: local theory,*SIAM J. Math. Analysis,**46**(2014), 1505-1537, joint with Felipe Linares, and Jean-Claude Saut.*Global well-posedness and limit behavior for a higher-order Benjamin-Ono equation,*Comm. PDE,**37**(2012), 2050-2080, joint with Luc Molinet.*Well-posednesss of strongly dispersive two-dimensional surface waves Boussinesq systems,*SIAM J. Math. Analysis,**44**(2012), 4195-4221, joint with Felipe Linares and Jean-Claude Saut.*The Cauchy problem for the Benjamin-Ono equation in L^2 revisited,*Analysis&PDE,**5**(2012) 365–395, joint with Luc Molinet.*Well-posedness for a higher-order Benjamin-Ono equation,*J. Diff. Eq.,**250**(2011) 450-475. joint with Felipe Linares and Gustavo Ponce.-
*Local well-posedness for the nonlocal nonlinear Schrödinger equation below the energy space*Adv. Diff. Eq.,**15**(2010) 925-952, joint with Roger Moura. *Stability of solitary-wave solutions to the Hirota-Satsuma equation,*Disc. Cont. Dyn. Syst. A,**27**(2010) 1397-1413, joint with Jerry Bona.*Global well-posedness and non-linear stability of periodic traveling waves for a Schrödinger-Benjamin-Ono system,*Comm. Pure Appl. Anal.,**8**(2009), 815-844, joint with Jaime Angulo and Carlos Matheus.*Well-posedness for Hirota-Stasuma's equation,*Diff. Int. Eq.,**21**(2008) 1177-1192, joint with Rafael Iório.*On the Cauchy problem for higher-order nonlinear dispersive equations,*J. Diff. Eq.,**245**(2008) 2055-2077.*Sharp well-posedness results for the Kuramoto-Velarde equation,*Comm. Pure Appl. Anal.,**7**(2008) 867-881.

Last Update: June 3rd, 2020