Well-posedness and ill-posedness results for dissipative Benjamin-Ono equations
نویسنده
چکیده
We study the Cauchy problem for the dissipative Benjamin-Ono equations ut +Huxx + |D| αu+ uux = 0 with 0 ≤ α ≤ 2. When 0 ≤ α < 1, we show the ill-posedness in Hs(R), s ∈ R, in the sense that the flow map u0 7→ u (if it exists) fails to be C 2 at the origin. For 1 < α ≤ 2, we prove the global well-posedness in Hs(R), s > −α/4. It turns out that this index is optimal.
منابع مشابه
Ill-posedness for Periodic Nonlinear Dispersive Equations
In this article, we establish new results about the ill-posedness of the Cauchy problem for the modified Korteweg-de Vries and the defocusing modified Korteweg-de Vries equations, in the periodic case. The lack of local well-posedness is in the sense that the dependence of solutions upon initial data fails to be continuous. We also develop a method for obtaining ill-posedness results in the per...
متن کاملLocal Well-posedness for Dispersion Generalized Benjamin-ono Equations in Sobolev Spaces
We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation ∂tu+ |∂x| ∂xu+ uux = 0, u(x, 0) = u0(x), is locally well-posed in the Sobolev spaces H for s > 1 − α if 0 ≤ α ≤ 1. The new ingredient is that we develop the methods of Ionescu, Kenig and Tataru [13] to approach the problem in a less perturbative way, in spite of the ill-posedness results of Molinet, Saut and T...
متن کامل. A P ] 2 1 O ct 2 00 2 LOCAL WELL - POSEDNESS FOR DISPERSION GENERALIZED BENJAMIN - ONO EQUATIONS
In this paper we study local well-posedness in the energy space for a family of dispersive equations that can be seen as dispersive " interpolations " between the KdV and the Benjamin-Ono equation.
متن کاملSharp ill-posedness result for the periodic Benjamin-Ono equation
We prove the discontinuity for the weak L(T)-topology of the flowmap associated with the periodic Benjamin-Ono equation. This ensures that this equation is ill-posed in Hs(T) as soon as s < 0 and thus completes exactly the well-posedness result obtained in [12]. AMS Subject Classification : 35B20, 35Q53.
متن کاملOn the Cauchy problem for higher-order nonlinear dispersive equations
We study the higher-order nonlinear dispersive equation ∂tu+ ∂ 2j+1 x u = ∑ 0≤j1+j2≤2j aj1,j2∂ j1 x u∂ j2 x u, x, t ∈ R. where u is a real(or complex-) valued function. We show that the associated initial value problem is well posed in weighted Besov and Sobolev spaces for small initial data. We also prove ill-posedness results when a0,k 6= 0 for some k > j, in the sense that this equation cann...
متن کامل