Local ill-posedness of the 1D Zakharov system
| dc.contributor.author | Holmer, Justin | |
| dc.date.accessioned | 2021-08-03T18:37:18Z | |
| dc.date.available | 2021-08-03T18:37:18Z | |
| dc.date.issued | 2007-02-12 | |
| dc.description.abstract | Ginibre-Tsutsumi-Velo (1997) proved local well-posedness for the Zakharov system i∂tu + Δu = nu ∂2tn - Δn = Δ|u|2 u(x, 0) = u0(x), n(x, 0) = n0(x), ∂tn(x, 0) = n1(x) where u = u(x, t) ∈ ℂ, n = n(x, t) ∈ ℝ, x ∈ ℝ, and t ∈ ℝ. The proof was made for any dimension d, in the inhomogeneous Sobolev spaces (u, n) ∈ Hk (ℝd) x 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 Christ-Colliander-Tao (2003) applied to the nonlinear Schrödinger equation. | |
| dc.description.department | Mathematics | |
| dc.format | Text | |
| dc.format.extent | 22 pages | |
| dc.format.medium | 1 file (.pdf) | |
| dc.identifier.citation | Holmer, J. (2007). Local ill-posedness of the 1D Zakharov system. <i>Electronic Journal of Differential Equations, 2007</i>(24), pp. 1-22. | |
| dc.identifier.issn | 1072-6691 | |
| dc.identifier.uri | https://hdl.handle.net/10877/14174 | |
| dc.language.iso | en | |
| dc.publisher | Texas State University-San Marcos, Department of Mathematics | |
| dc.rights | Attribution 4.0 International | |
| dc.rights.holder | This work is licensed under a Creative Commons Attribution 4.0 International License. | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.source | Electronic Journal of Differential Equations, 2007, San Marcos, Texas: Texas State University-San Marcos and University of North Texas. | |
| dc.subject | Zakharov system | |
| dc.subject | Cauchy problem | |
| dc.subject | Local well-posedness | |
| dc.subject | Local ill-posedness | |
| dc.title | Local ill-posedness of the 1D Zakharov system | |
| dc.type | Article |