On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations
نویسندگان
چکیده
منابع مشابه
Well-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.
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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 کاملLocal and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation
We show that the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation ut −D α xux + uxyy = uux, (t, x, y) ∈ R , 1 ≤ α ≤ 2, is locally well-posed in the spaces Es, s > 2 α − 3 4 , endowed with the norm ‖f‖Es = ‖〈|ξ| α + μ〉f̂‖L2(R2). As a consequence, we get the global wellposedness in the energy space E1/2 as soon as α > 8 5 . The proof is based ...
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ژورنال
عنوان ژورنال: Mathematical Research Letters
سال: 2003
ISSN: 1073-2780,1945-001X
DOI: 10.4310/mrl.2003.v10.n6.a13