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• W2019927399 abstract "We study the analytic property of the (generalized) quadratic derivative Ginzburg-Landau equation<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M1><mml:mo stretchy=false>(</mml:mo><mml:mrow><mml:mrow><mml:mn fontstyle=italic>1</mml:mn></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mn fontstyle=italic>2</mml:mn></mml:mrow></mml:mrow><mml:mo>⩽</mml:mo><mml:mi>α</mml:mi><mml:mo>⩽</mml:mo><mml:mn fontstyle=italic>1</mml:mn><mml:mo stretchy=false>)</mml:mo></mml:math>in any spatial dimension<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M2><mml:mi>n</mml:mi><mml:mo>⩾</mml:mo><mml:mn fontstyle=italic>1</mml:mn></mml:math>with rough initial data. For<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M3><mml:mrow><mml:mrow><mml:mn fontstyle=italic>1</mml:mn></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mn fontstyle=italic>2</mml:mn></mml:mrow></mml:mrow><mml:mo><</mml:mo><mml:mi>α</mml:mi><mml:mo>⩽</mml:mo><mml:mn fontstyle=italic>1</mml:mn></mml:math>, we prove the analyticity of local solutions to the (generalized) quadratic derivative Ginzburg-Landau equation with large rough initial data in modulation spaces<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M4><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mn fontstyle=italic>1</mml:mn></mml:mrow><mml:mrow><mml:mn fontstyle=italic>1</mml:mn><mml:mo>-</mml:mo><mml:mn fontstyle=italic>2</mml:mn><mml:mi>α</mml:mi></mml:mrow></mml:msubsup><mml:mo stretchy=false>(</mml:mo><mml:mn fontstyle=italic>1</mml:mn><mml:mo>⩽</mml:mo><mml:mi>p</mml:mi><mml:mo>⩽</mml:mo><mml:mi>∞</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:math>. For<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M5><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mn fontstyle=italic>1</mml:mn></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mn fontstyle=italic>2</mml:mn></mml:mrow></mml:mrow></mml:math>, we obtain the analytic regularity of global solutions to the fractional quadratic derivative Ginzburg-Landau equation with small initial data in<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M6><mml:mrow><mml:msubsup><mml:mover accent=true><mml:mi>B</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mrow><mml:mi>∞</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>0</mml:mn></mml:msubsup></mml:mrow></mml:math><mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M7><mml:mo stretchy=false>(</mml:mo><mml:msup><mml:mrow><mml:mi>ℝ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=false>)</mml:mo><mml:mo>∩</mml:mo><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>∞</mml:mi><mml:mo>,</mml:mo><mml:mn fontstyle=italic>1</mml:mn></mml:mrow><mml:mrow><mml:mn fontstyle=italic>0</mml:mn></mml:mrow></mml:msubsup><mml:mo stretchy=false>(</mml:mo><mml:msup><mml:mrow><mml:mi>ℝ</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=false>)</mml:mo></mml:math>. The strategy is to develop uniform and dyadic exponential decay estimates for the generalized Ginzburg-Landau semigroup<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M8><mml:mrow><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mfenced separators=|><mml:mrow><mml:mi>a</mml:mi><mml:mo>+</mml:mo><mml:mi>i</mml:mi></mml:mrow></mml:mfenced><mml:mi>t</mml:mi><mml:msup><mml:mrow><mml:mfenced separators=|><mml:mrow><mml:mo>-</mml:mo><mml:mo>Δ</mml:mo></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:mrow></mml:math>to overcome the derivative in the nonlinear term." @default.
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• W2019927399 date "2014-01-01" @default.
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• W2019927399 title "On the Analyticity for the Generalized Quadratic Derivative Complex Ginzburg-Landau Equation" @default.
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• W2019927399 doi "https://doi.org/10.1155/2014/607028" @default.
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