Global well-posedness for the derivative nonlinear Schrödinger equation

• Published in: Inventiones mathematicae Vol. 229; no. 2; pp. 639 - 688
• Main Authors: Bahouri, Hajer, Perelman, Galina
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2022; Springer Nature B.V; Springer Verlag
• Subjects: Cauchy problems; Initial conditions; Integral equations; Mathematics; Mathematics and Statistics; Schrodinger equation; Sobolev space; Well posed problems; Global well-posedness; Derivative nonlinear Schrödinger equation; Bäcklund transformation; Profile decompositions; Integrable systems; 43A30; Inverse scattering transform; 43A80; global well-posedness; profile decompositions AMS Subject Classification : 43A30; inverse scattering transform; integrable systems
• ISSN: 0020-9910; 1432-1297
• DOI: 10.1007/s00222-022-01113-0

This paper is dedicated to the study of the derivative nonlinear Schrödinger equation on the real line. The local well-posedness of this equation in the Sobolev spaces  H s ( R ) is well understood since a couple of decades, while the global well-posedness is not completely settled. For the latter issue, the best known results up-to-date concern either Cauchy data in H 1 2 ( R ) with mass strictly less than 4 π or general initial conditions in the weighted Sobolev space H 2 , 2 ( R ) . In this article, we prove that the derivative nonlinear Schrödinger equation is globally well-posed for general Cauchy data in H 1 2 ( R ) and that furthermore the H 1 2 norm of the solutions remains globally bounded in time. The proof is achieved by combining the profile decomposition techniques with the integrability structure of the equation.