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• W1553898812 endingPage "5654" @default.
• W1553898812 startingPage "5621" @default.
• W1553898812 abstract "We construct a one parameter family of finite time blow-ups to the co-rotational wave maps problem from <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S squared times double-struck upper R> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>×<!-- × --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>S^2times mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S squared comma> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>S^2,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> parameterized by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=nu element-of left-parenthesis one half comma 1 right-bracket period> <mml:semantics> <mml:mrow> <mml:mi>ν<!-- ν --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>]</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>nu in (frac {1}{2},1].</mml:annotation> </mml:semantics> </mml:math> </inline-formula> The longitudinal function <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u left-parenthesis t comma alpha right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>u(t,alpha )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is the main object of study will be obtained as a perturbation of a rescaled harmonic map of rotation index one from <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R squared> <mml:semantics> <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:annotation encoding=application/x-tex>mathbb {R}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S squared period> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>S^2.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> The domain of this harmonic map is identified with a neighborhood of the north pole in the domain <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S squared> <mml:semantics> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>S^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> via the exponential coordinates <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis alpha comma theta right-parenthesis period> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>θ<!-- θ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(alpha ,theta ).</mml:annotation> </mml:semantics> </mml:math> </inline-formula> In these coordinates <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u left-parenthesis t comma alpha right-parenthesis equals upper Q left-parenthesis lamda left-parenthesis t right-parenthesis alpha right-parenthesis plus script upper R left-parenthesis t comma alpha right-parenthesis comma> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>Q</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>+</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>R</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>u(t,alpha )=Q(lambda (t)alpha )+mathcal {R}(t,alpha ),</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=upper Q left-parenthesis r right-parenthesis equals 2 arc tangent r> <mml:semantics> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mi>arctan</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>r</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>Q(r)=2arctan {r}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the standard co-rotational harmonic map to the sphere, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=lamda left-parenthesis t right-parenthesis equals t Superscript negative 1 minus nu Baseline comma> <mml:semantics> <mml:mrow> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mi>ν<!-- ν --></mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>lambda (t)=t^{-1-nu },</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 upper R left-parenthesis t comma alpha right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>R</mml:mi> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {R}(t,alpha )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the error with local energy going to zero as <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=t right-arrow 0 period> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mn>0.</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>trightarrow 0.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Blow-up will occur at <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis t comma alpha right-parenthesis equals left-parenthesis 0 comma 0 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(t,alpha )=(0,0)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> due to energy concentration, and up to this point the solution will have regularity <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript 1 plus nu minus Baseline period> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ν<!-- ν --></mml:mi> <mml:mo>−<!-- − --></mml:mo> </mml:mrow> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>H^{1+nu -}.</mml:annotation> </mml:semantics> </mml:math> </inline-formula>" @default.
• W1553898812 created "2016-06-24" @default.
• W1553898812 creator A5033394719 @default.
• W1553898812 date "2015-09-15" @default.
• W1553898812 modified "2023-09-25" @default.
• W1553898812 title "Renormalization and blow-up for wave maps from 𝑆²×ℝ to 𝕊²" @default.
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