Local Ill-posedness of the 1d Zakharov System
نویسنده
چکیده
Ginibre-Tsutsumi-Velo (1997) proved local well-posedness for the Zakharov system i∂tu +∆u = nu ∂ t n −∆n = ∆|u| u(x, 0) = u0(x) n(x, 0) = n0(x), ∂tn(x, 0) = n1(x) u = u(x, t) ∈ C n = n(x, t) ∈ R x ∈ R, t ∈ R for any dimension d, in the inhomogeneous Sobolev spaces (u, n) ∈ Hk(Rd)×Hs(Rd) for a range of exponents k, s depending on d. Here we restrict to dimension d = 1 and present a few results establishing local ill-posedness for exponent pairs (k, s) outside of the well-posedness regime. The techniques employed are rooted in the work of Bourgain (1993), Birnir-Kenig-Ponce-Svanstedt-Vega (1996), and ChristColliander-Tao (2003) applied to the nonlinear Schrödinger equation.
منابع مشابه
Well-posedness for the 1d Zakharov-rubenchik System
Local and global well-posedness results are established for the initial value problem associated to the 1D Zakharov-Rubenchik system. We show that our results are sharp in some situations by proving Ill-posedness results otherwise. The global results allow us to study the norm growth of solutions corresponding to the Schrödinger equation term. We use ideas recently introduced to study the class...
متن کاملLow Regularity Global Well-posedness for the Zakharov and Klein-gordon-schrödinger Systems
We prove low-regularity global well-posedness for the 1d Zakharov system and 3d Klein-Gordon-Schrödinger system, which are systems in two variables u : Rx × Rt → C and n : Rx × Rt → R. The Zakharov system is known to be locally well-posed in (u, n) ∈ L2×H−1/2 and the Klein-Gordon-Schrödinger system is known to be locally well-posed in (u, n) ∈ L × L. Here, we show that the Zakharov and Klein-Go...
متن کاملOn Ill-Posedness and Local Ill-Posedness of Operator Equations in Hilbert Spaces
In this paper, we study ill-posedness concepts of nonlinear and linear inverse problems in a Hilbert space setting. We deene local ill-posedness of a nonlinear operator equation F(x) = y 0 in a solution point x 0 and the interplay between the nonlinear problem and its linearization using the Fr echet derivative F 0 (x 0). To nd an appropriate ill-posedness concept for 1 the linearized equation ...
متن کاملWell-posedness results for the 3D Zakharov-Kuznetsov equation
We prove the local well-posedness of the three-dimensional Zakharov-Kuznetsov equation ∂tu+∆∂xu+u∂xu = 0 in the Sobolev spaces Hs(R3), s > 1, as well as in the Besov space B 2 (R 3). The proof is based on a sharp maximal function estimate in time-weighted spaces.
متن کاملSome characterizations and properties of the "distance to ill-posedness" and the condition measure of a conic linear system
A conic linear system is a system of the form P(d) : find x that solves b− Ax ∈ CY , x ∈ CX , where CX and CY are closed convex cones, and the data for the system is d = (A, b). This system is“wellposed” to the extent that (small) changes in the data (A, b) do not alter the status of the system (the system remains solvable or not). Renegar defined the “distance to ill-posedness”, ρ(d), to be th...
متن کامل