Global well-posedness for the defocusing cubic nonlinear Schrödinger equation on 𝕋 ³

Abstract In this article, we investigate the global well-posedness for the defocusing, cubic nonlinear Schrödinger equation posed on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>𝕋</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> {\mathbb{T}^{3}} with intial data lying in its critical space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>𝕋</m:mi> <m:mn>3</m:mn> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {H^{\frac{1}{2}}(\mathbb{T}^{3})} . By establishing the linear profile decomposition, and applied this to the concentration-compactness/rigidity argument, we prove that if the solution remains bounded in the critical Sobolev space throughout the maximal lifespan, i.e., <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msubsup> <m:mi>L</m:mi> <m:mi>t</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:msup> <m:mi>H</m:mi> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>I</m:mi> <m:mo>×</m:mo> <m:msup> <m:mi>𝕋</m:mi> <m:mn>3</m:mn> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {u\in L_{t}^{\infty}{H}^{\frac{1}{2}}(I\times\mathbb{T}^{3})} , then u is global. As a comparison, our result is the periodic analogue of [C. E. Kenig and F. Merle, Scattering for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mover accent="true"> <m:mi>H</m:mi> <m:mo>˙</m:mo> </m:mover> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> \dot{H}^{1/2} bounded solutions to the cubic, defocusing NLS in 3 dimensions, Trans. Amer. Math. Soc. 362 2010, 4, 1937–1962].