Scattering for Defocusing Mass Sub-Critical NLS
Abstract
<p> In this paper, we consider the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript x Superscript 2"> <mml:semantics> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">L_x^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -scattering of defocusing mass sub-critical nonlinear Schrödinger equations with low weighted initial condition. </p> <p> In prior work, Tsutsumi and Yajima [Bull. Amer. Math. Soc. (N.S.) <bold>11</bold> (1984), no. 1, 186–188, MR741737] established the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -scattering result for initial data <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u 0 element-of upper H Superscript 1 intersection script upper F upper H Superscript 1"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo> ∩ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">F</mml:mi> </mml:mrow> </mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">u_0\in H^1 \cap {\mathcal {F}}H^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and this condition was later refined to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper F upper H Superscript 1"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">F</mml:mi> </mml:mrow> </mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathcal {F}}H^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by Hayashi and Ozawa [Ann. Inst. H. Poincaré Phys. Théor. <bold>48</bold> (1988), no. 1, 17–37, <ext-link xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="https://mathscinet.ams.org/mathscinet-getitem?mr=947158"> MR <bold>947158</bold> </ext-link> ]. Furthermore, Lee [Int. Math. Res. Not. IMRN <bold>5</bold> (2021), 3571–3596, <ext-link xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="https://mathscinet.ams.org/mathscinet-getitem?mr=4227579"> MR <bold>4227579</bold> </ext-link> ] proved that the inverse wave operator does not exhibit Hölder’s continuity solely within <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , indicating the necessity of introducing a weighted condition for scattering. The determination of how many levels of weighting are required becomes a natural inquiry. Moreover, for large <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper F upper H Superscript s"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">F</mml:mi> </mml:mrow> </mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathcal {F}}H^s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -data with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">s>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , there only exists the wave operator result, but scattering results are lacking. </p> <p> In the first portion of our result, we establish the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -scattering within the framework of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper F upper H Superscript s"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">F</mml:mi> </mml:mrow> </mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathcal {F}}H^s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s greater-than 1"> <mml:semantics>