Well-posedness for equations of Benjamin-Ono type
نویسندگان
چکیده
منابع مشابه
Well-posedness and ill-posedness results for dissipative Benjamin-Ono equations
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.
متن کاملWell-posedness for a Higher-order Benjamin-ono Equation
In this paper we prove that the initial value problem associated to the following higher-order Benjamin-Ono equation ∂tv − bH∂ xv + a∂ xv = cv∂xv − d∂x(vH∂xv + H(v∂xv)), where x, t ∈ R, v is a real-valued function, H is the Hilbert transform, a ∈ R, b, c and d are positive constants, is locally well-posed for initial data v(0) = v0 ∈ H(R), s ≥ 2 or v0 ∈ H(R) ∩ L(R; xdx), k ∈ Z+, k ≥ 2.
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 کاملAn Improved Bilinear Estimate for Benjamin-ono Type Equations
A bilinear estimate in Fourier restriction norm spaces with applications to the Cauchy problem ut − |D| αux + uux = 0 in (−T, T ) × R u(0) = u0 is proved, for 1 < α < 2. As a consequence, local well-posedness in H(R) ∩ Ḣ(R) follows for s > − 3 4 (α − 1) and ω = 1/α − 1/2 This extends to global well-posedness for all s ≥ 0.
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.
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ژورنال
عنوان ژورنال: Illinois Journal of Mathematics
سال: 2007
ISSN: 0019-2082
DOI: 10.1215/ijm/1258131113