Portal de Periódicos da CAPES

Global well-posedness for the derivative nonlinear Schrödinger equation

2022; Springer Science+Business Media; Volume: 229; Issue: 2 Linguagem: Inglês

10.1007/s00222-022-01113-0

1432-1297

Hajer Bahouri, Galina Perelman,

Mathematical Analysis and Transform Methods

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 $$H^s({\mathop {{\mathbb {R}}}\nolimits })$$ 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 $$H^{\frac{1}{2}}({\mathop {{\mathbb {R}}}\nolimits })$$ with mass strictly less than $$4\pi $$ or general initial conditions in the weighted Sobolev space $$H^{2, 2}({\mathop {{\mathbb {R}}}\nolimits })$$ . In this article, we prove that the derivative nonlinear Schrödinger equation is globally well-posed for general Cauchy data in $$H^{\frac{1}{2}}({\mathop {{\mathbb {R}}}\nolimits })$$ and that furthermore the $$H^{\frac{1}{2}}$$ 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.