Luc Molinet

Ill-Posedness Issues for the Benjamin--Ono and Related Equations

We establish that the Cauchy problem for the Benjamin--Ono equation and for a rather general class of nonlinear dispersive equations with dispersion slightly weaker than that of the Korteweg--de Vries equation cannot be solved by an iteration scheme based on the Duhamel formula. As a consequence, the flow map fails to be smooth.

Orbital stability of solitary waves for a shallow water equation

We prove the orbital stability of the solitary waves for a shallow water equation by means of variational methods, considering a minimization problem with an appropriate constraint.

On Well-Posedness Results for Camassa-Holm Equation on the Line: A Survey

Abstract We survey recent results on well-posedness, blow-up phenomena, lifespan and global existence for the Camassa-Holm equation. Results on weak solutions are also considered.

Stability of multipeakons

Abstract The Camassa–Holm equation possesses well-known peaked solitary waves that are called peakons. Their orbital stability has been established by Constantin and Strauss in [A. Constantin, W. Strauss, Stability of peakons, Comm. Pure Appl. Math. 53 (2000) 603–610]. We prove here the stability of ordered trains of peakons. We also establish a result on the stability of multipeakons.

Well-posedness results for the generalized Benjamin-Ono equation with arbitrary large initial data

We prove new local well-posedness results for the generalized Benjamin-Ono equation (GBO) ∂tu+ℋ∂x2u+uk∂xu=0, k ≥ 2. By combining a gauge transformation with dispersive estimates, we establish the local well-posedness of GBO in H s (ℝ) for s ≥ 1/2 if k ≥ 5, s > 1/2 if k=2,4, and s≥ 3/4 if k=3. Moreover, we prove that in all these cases, the flow map is locally Lipschitz on H s (ℝ).

Remarks on the Mass Constraint for KP-Type Equations

For a rather general class of equations of Kadomtsev–Petviashvili type, we prove that the zero-mass (in x) constraint is satisfied at any nonzero time even if it is not satisfied at initial time zero. Our results are based on a precise analysis of the fundamental solution of the linear part and its anti-x-derivative.

Exponential decay of H1-localized solutions and stability of the train of N solitary waves for the Camassa–Holm equation

For the Camassa–Holm equation with κ≥0, we first prove that any global solution that is H1-localized and moves fast enough to the right decays exponentially in space uniformly with respect to time. We also prove that for κ>0, a train of N solitary waves, which are sufficiently decoupled, is orbitally stable in H1().

On the Cauchy Problem for the Generalized Korteweg-de Vries Equation

Abstract We consider the local and global Cauchy problem for the generalized Korteweg-de Vries equation , with initial data in homogeneous and nonhomogeneous Besov spaces. This allows us to slightly extend known results on this problem. Furthermore we prove existence and uniqueness of self-similar solutions.

The Global Cauchy Problem in Bourgain's-Type Spaces for a Dispersive Dissipative Semilinear Equation

We prove local and global well-posedness results for the Kadomtsev--Petviashvili--Burgers equations in Bourgains-type spaces. This approach is new for the study of semilinear evolution equations with a linear part which contains both dispersive and dissipative terms.

Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: The periodic case

We prove that the KdV-Burgers is globally well-posed in