[Updated on July 12] Integration of CiNii Articles into CiNii Research from April 1, 2022

Ill-posedness for the nonlinear Schrödinger equation with quadratic non-linearity in low dimensions

Abstract

<p>We consider the ill-posedness issue for the nonlinear Schrödinger equation with a quadratic nonlinearity. We refine the Bejenaru-Tao result by constructing an example in the following sense. There exist a sequence of time <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript upper N Baseline right-arrow 0"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">T_N\to 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and solution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript upper N Baseline left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">u_N(t)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript upper N Baseline left-parenthesis upper T Subscript upper N Baseline right-parenthesis right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">u_N(T_N)\to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the Besov space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript 2 comma sigma Superscript negative 1 Baseline left-parenthesis double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>σ<!-- σ --></mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">B_{2,\sigma }^{-1}(\mathbb {R})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma greater-than 2"> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma >2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) for one space dimension. We also construct a similar ill-posed sequence of solutions in two space dimensions in the scaling critical Sobolev space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript negative 1 Baseline left-parenthesis double-struck upper R squared right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H^{-1}(\mathbb {R}^2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We systematically utilize the modulation space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript 2 comma 1 Superscript 0"> <mml:semantics> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>0</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">M_{2,1}^0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for one dimension and the scaled modulation space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper M Subscript 2 comma 1 Superscript 0 Baseline right-parenthesis Subscript upper N"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>0</mml:mn> </mml:msubsup> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mi>N</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">(M_{2,1}^0)_N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for two dimensions.</p>

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