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• W2079868601 abstract "We establish local well-posedness results for the Initial Value Problem associated to the Schrödinger-Debye system in dimensions <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N equals 2 comma 3> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>N=2, 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for data in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript s Baseline times upper H Superscript script l> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>s</mml:mi> </mml:msup> <mml:mo>×<!-- × --></mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ℓ<!-- ℓ --></mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>H^stimes H^{ell }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding=application/x-tex>s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script l> <mml:semantics> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:annotation encoding=application/x-tex>ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfying <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=max left-brace right-brace comma 0 comma minus minus s 1 less-than-or-equal-to script l less-than-or-equal-to min left-brace right-brace comma times times 2 s comma plus plus s 1> <mml:semantics> <mml:mrow> <mml:mo movablelimits=true form=prefix>max</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mo movablelimits=true form=prefix>min</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mn>2</mml:mn> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>max {0, s-1} le ell le min {2s, s+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, these include the energy space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript 1 Baseline times upper L squared> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>×<!-- × --></mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>H^1times L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Our results improve the previous ones obtained by B. Bidégaray, and by A. J. Corcho and F. Linares. Moreover, in the critical case (<inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N equals 2> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>N=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) and for initial data in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript 1 Baseline times upper L squared> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>×<!-- × --></mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>H^1times L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we prove that solutions exist for all times, thus providing a negative answer to the open problem mentioned by G. Fibich and G. C. Papanicolau concerning the formation of singularities for these solutions." @default.
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• W2079868601 date "2013-06-18" @default.
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• W2079868601 title "Local and global well-posedness for the critical Schrödinger-Debye system" @default.
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• W2079868601 doi "https://doi.org/10.1090/s0002-9939-2013-11612-6" @default.
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