Ill-posedness for nonlinear Schrodinger and wave equations

By Michael Christ, James E. Colliander, Terence Tao. 2003

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The nonlinear wave and Schrodinger equations on Euclidean space of any dimension, with general power nonlinearity and with both the focusing and defocusing signs, are proved to be ill-posed in the Sobolev space of index s whenever the exponent s is lower than that predicted by scaling or Galilean invariances, or when the regularity is too low to support distributional solutions. This extends previous work of the authors, which treated the one-dimensional cubic nonlinear Schrodinger equation. In the defocusing case soliton or blowup examples are unavailable, and a proof of ill-posedness requires the construction of other solutions. In earlier work this was achieved using certain long-time asymptotic behavior which occurs only for low power nonlinearities. Here we analyze instead a class of solutions for which the zero-dispersion limit provides a good approximation.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7a0b5da6cb9e354040084dcda721f11e3e37f01b

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In Applied Mathematics

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In Statistical and Nonlinear Physics

Citations in the corpus

Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data (2017, background): “500424. 1 2 B. DODSON, J. LÜHRMANN, AND D. MENDELSON based on Strichartz estimates break down. In fact, the Cauchy problem (1.1) is ill-behaved below the scaling critical regularity, see for instance [13], [26] and [23]. For initial data in the energy class, i.e. when s= 1, global well-posedness, sc”
Multilinear Eigenfunction Estimates And Global Existence For The Three Dimensional Nonlinear Schr (2004, methodology): “— The result of Theorem 6, in the particular case M = R3, endowed with the standard metric, can be found in [11].”
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On Fractional Schrodinger Equations in sobolev spaces (2015, methodology): “[9] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R3, Ann. of Math.”
Nonlinear wave interactions for the Benjamin-Ono equation. (2004, background): “For instance, in [9] failure of the uniform continuity of the map (1.”