Generalized Benjamin-Ono Equation Research Articles

Quadratic dispersive generalized Benjamin–Ono equation

This paper is concerned with quadratic dispersive generalized Benjamin–Ono equation (1.1)–(1.2). We prove sharp local well-posedness in H s α ( s α = − 1 + α 4 ) with a solution-map that is analytic from H s α ( R ) to C ( [ 0 , T ] ; H s α ( R ) ) . And the ill-posed result in H s ( R ) , as soon as s < s α , in the sense that the flow-map u 0 ↦ u ( t ) cannot be continuous.

Local well-posedness and blow-up in the energy space for a class of [formula omitted] critical dispersion generalized Benjamin–Ono equations

We consider a family of dispersion generalized Benjamin–Ono equations (dgBO)ut−∂x|D|αu+|u|2α∂xu=0,(t,x)∈R×R, where |D|αuˆ=|ξ|αuˆ and 1⩽α⩽2. These equations are critical with respect to the L2 norm and global existence and interpolate between the modified BO equation (α=1) and the critical gKdV equation (α=2).First, we prove local well-posedness in the energy space for 1<α<2, extending results in Kenig et al. (1991, 1993) [13,14] for the generalized KdV equations.Second, we address the blow-up problem in the spirit of Martel and Merle (2000) [19] and Merle (2001) [22] concerning the critical gKdV equation, by studying rigidity properties of the dgBO flow in a neighborhood of the solitons. We prove that for α close to 2, solutions of negative energy close to solitons blow up in finite or infinite time in the energy space Hα2.The blow-up proof requires both extensions to dgBO of monotonicity results for local L2 norms by pseudo-differential operator tools and perturbative arguments close to the gKdV case to obtain structural properties of the linearized flow around solitons.

Global Solutions for the Dirac–Klein–Gordon System in Two Space Dimensions

The Cauchy problem for the classical Dirac–Klein–Gordon system in two space dimensions is globally well-posed for L 2 Schrödinger data and wave data in . In the case of smooth data there exists a global smooth (classical) solution. The proof uses function spaces of Bourgain type based on Besov spaces – previously applied by Colliander, Kenig and Staffilani for generalized Benjamin-Ono equations and also by Bejenaru, Herr, Holmer and Tataru for the 2D Zakharov system – and the null structure of the system detected by d'Ancona, Foschi and Selberg, and a refined bilinear Strichartz estimate due to Selberg. The global existence proof uses an idea of Colliander, Holmer and Tzirakis for the 1D Zakharov system.

Well-posedness for the Generalized Benjamin–Ono Equations with Arbitrary Large Initial Data in the Critical Space

We prove the local well-posedness of the generalized Benjamin-Ono equations , k ≥ 4, in the scaling invariant spaces where s k = 1/2 − 1/ k. Our results also hold in the nonhomogeneous spaces . In the case k = 3, local well-posedness is obtained in , s > 1/3.

Sharp well-posedness results for the generalized Benjamin-Ono equation with high nonlinearity

We establish the local well posedness of the generalized Benjamin-Ono equation $\partial_tu+\mathcal{H}\partial_x^2u\pm u^k\partial_xu=0$ in $H^s({\mathbb{R}})$, $s > 1/2-1/k$ for $k\geq 12$ and without smallness assumption on the initial data. The condition $s > 1/2-1/k$ is known to be sharp since the solution map $u_0\mapsto u$ is not of class $\mathcal{C}^{k+1}$ on $H^s({\mathbb{R}})$ for $s < 1/2-1/k$. On the other hand, in the particular case of the cubic Benjamin-Ono equation, we prove the ill posedness in $H^s({\mathbb{R}})$, $s < 1/3$.

Well-posedness in $ H^1 $ for generalized Benjamin-Ono equations on the circle

We prove the local-well posedness of the generalized Benjamin-Ono equations in $ H^1(\T) $.

On the decay in time of solutions of some generalized regularized long waves equations

We consider the generalized Benjamin-Ono equation, regularized in the same manner that the Benjamin-Bona-Mahony equation is found from the Korteweg-de Vries equation [3], namely the equation $u_t + u_x +u^\rho u_x + H(u_{x t})=0,$ where $H$ is the Hilbert transform. In a second time, we consider the generalized Kadomtsev-Petviashvili-II equation, also regularized, namely the equation $u_t + u_x +u^\rho u_x - u_{x x t} +\partial_x^{-1}u_{y y} =0$. We are interested in dispersive properties of these equations for small initial data. We will show that, if the power $\rho$ of the nonlinearity is higher than $3$, the respective solution of these equations tends to zero when time rises with a decay rate of order close to $\frac{1}{2}$.

Smoothing and dispersive estimates for 1D Schrödinger equations with BV coefficients and applications

We prove smoothing estimates for Schrödinger equations i ∂ t ϕ + ∂ x ( a ( x ) ∂ x ϕ ) = 0 with a ( x ) ∈ BV , real and bounded from below. We then bootstrap these estimates to obtain optimal Strichartz and maximal function estimates, all of which turn out to be identical to the constant coefficient case. We also provide counterexamples showing a ∈ BV to be in a sense a minimal requirement. Finally, we provide an application to sharp well-posedness for a generalized Benjamin–Ono equation.

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).

Well-posedness results for the generalized Benjamin–Ono equation with small initial data

We study the generalized Benjamin–Ono equation ∂ tu+ H∂ 2 xu+u k∂ xu=0 , k⩾2. In the context of small initial data we prove local and global well-posedness results in Sobolev and in Besov spaces. Our results turn out to be almost sharp for k≠3.

On the Cauchy problem for the generalized Benjamin–Ono equation with small initial data

We prove global well-posedness results for small initial data in H s( R), s>s k , and in B ̇ s k,1 2( R) , s k =1/2−1/ k, for the generalized Benjamin–Ono equation ∂ tu+ H∂ 2 xu+∂ x(u k+1)=0, k⩾4 . We also consider the cases k=2,3. To cite this article: L. Molinet, F. Ribaud, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

On the existence, regularity and decay of solitary waves to a generalized Benjamin–Ono equation

in R2, where ; ? 0 and (− )1=2 is the operator de2ned by F((− )1=2u)( )=| |u( ). F or ˆ represent the Fourier transform. Eq. (1) describes the dynamics of three-dimensional, slightly nonlinear disturbances in boundary-layer shear ;ows (without the assumption of a di<erence in their scales along and across the ;ow), see [1,6]. The solitary waves of (1) are solutions of the form A(x; y; t) = v(x − ct; y) where c is the speed of the solitary wave. It seems that solitary waves play an important role in the evolution of (1). Such a solution must satisfy the equation

Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations

Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations

Large time asymptotics of solutions to the generalized Benjamin-Ono equation

We study the asymptotic behavior for large time of solutions to the Cauchy problem for the generalized Benjamin-Ono (BO) equation: u t + ( | u | ρ − 1 u ) x + H u x x = 0 u_{t} + (|u|^{\rho -1}u)_{x} + \mathcal {H} u_{xx} = 0 , where H \mathcal {H} is the Hilbert transform, x , t ∈ R x, t \in {\mathbf {R}} , when the initial data are small enough. If the power ρ \rho of the nonlinearity is greater than 3 3 , then the solution of the Cauchy problem has a quasilinear asymptotic behavior for large time. In the case ρ = 3 \rho =3 critical for the asymptotic behavior i.e. for the modified Benjamin-Ono equation, we prove that the solutions have the same L ∞ L^{\infty } time decay as in the corresponding linear BO equation. Also we find the asymptotics for large time of the solutions of the Cauchy problem for the BO equation in the critical and noncritical cases. For the critical case, we prove the existence of modified scattering states if the initial function is sufficiently small in certain weighted Sobolev spaces. These modified scattering states differ from the free scattering states by a rapidly oscillating factor.

Wave collapse in plasmas and fluids.

This lecture is a review of recent results (obtained mainly by the author and his co-authors) in the wave collapse theory with applications to plasma physics, fluid dynamics and nonlinear optics as well. The main attention in the review is paid to the qualitative reasons of the wave collapse and to the exact methods based on the integral estimations. Both approaches are applied to both the nonlinear Schrodinger equation and the two-dimensional generalized Benjamin-Ono equation which describes self-focusing of low-frequency oscillations in the boundary layer. (c) 1996 American Institute of Physics.

On the generalized Benjamin-Ono equation

We study well-posedness of the initial value problem for the generalized Benjamin-Ono equation∂tu+uk∂xu−∂xDxu=0{\partial _t}u + {u^k}{\partial _x}u - {\partial _x}{D_x}u = 0,k∈Z+k \in {\mathbb {Z}^ + }, in Sobolev spacesHs(R){H^s}(\mathbb {R}). For small data and higher nonlinearities(k≥2)(k \geq 2)new local and global (including scattering) results are established. Our method of proof is quite general. It combines several estimates concerning the associated linear problem with the contraction principle. Hence it applies to other dispersive models. In particular, it allows us to extend the results for the generalized Benjamin-Ono to nonlinear Schrödinger equations (or systems) of the form∂tu−i∂x2u+P(u,∂xu,u¯,∂xu¯)=0{\partial _t}u - i\partial _x^2u + P(u,{\partial _x}u,\bar u,{\partial _x}\bar u) = 0.

Smoothing properties and retarded estimates for some dispersive evolution equations

Smoothing properties, in the form of space-time integrability properties, play an important role in the study of dispersive evolution equations. A number of them follow from a combination of general arguments and specific estimates. We present a general formulation which makes the separation between the two types of ingredients as clear as possible, and we illustrate it with the examples of the Schrodinger equation, of the wave equation, and of a class of 1+1 dimensional equations related to the Benjamin-Ono equation. Of special interest for the Cauchy problem are retarded estimates expressed in terms of those properties. We derive a number of such estimates associated with the last example, and we mention briefly an application of those estimates to the Cauchy problem for the generalized Benjamin-Ono equation.

Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation

We study the Cauchy problem for the generalized Benjamin-Ono equation ∂ t u − D | D| 2 μ u = DV'( u), (∗) where D = d dx , U > 0 , and V ϵ E 1( R,R ) with V(0) = V′(0) = 0. Using commutator identities and estimates, we prove that the equation (∗) exhibits smoothing properties similar to those of the generalized Korteweg-de Vries equation (GKdV), which is the special case μ = 1 of (∗). If V satisfies suitable estimates at zero and at infinity, we prove that the Cauchy problem for the equation (∗) with initial data u(0) = u 0 has a solution u ϵ L ∞( R , l 2)∪ L 2 loc( R , H μ loc) if u 0 ϵ L 2, and has a solution u ϵ L ∞( R , H μ)∪ L 2 loc( R , H 2μ loc)if u 0 ϵ H μ (finite energy solution). In addition, we prove that the usual Benjamin-Ono equation ( μ = 1 2 , V′(u) = u 2 ) has a solution u ϵ L ∞( R , H 1)∪ L 2 loc( R , H 3 2 loc) for u 0 ϵ H 1. Those results are easily extended to related equations such as the intermediate long wave equation or the Smith equation.

Solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities

We study the nonlinear Schrödinger (NLS), generalized Korteweg-de Vries (GKdV), and generalized Benjamin-Ono equations (GBO) with critical power nonlinearities. The ground-state solitary wave is conditionally stable modulo the group of translations in space and phase, and dilations. This is a natural extension of the result for subcritical nonlinearities, where solitary waves are stable modulo translations alone, to a case where solutions may blow up. We also present results on global existence for GBO and the structure of singularities for certain solutions of GBO, which are analogues of previous results by the author for NLS and GKdV.