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• W3004357466 abstract "We are concerned with a nonlocal transport 1D-model with supercritical dissipation <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma element-of left-parenthesis 0 comma 1 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>γ<!-- γ --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>gamma in (0,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in which the velocity is coupled via the Hilbert transform, namely the so-called CCF model. This model arises as a lower dimensional model for the well-known 2D dissipative quasi-geostrophic equation and in connection with vortex-sheet problems. It is known that its solutions can blow up in finite time when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma element-of left-parenthesis 0 comma 1 slash 2 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>γ<!-- γ --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>gamma in (0,1/2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. On the other hand, as stated by Kiselev (2010), in the supercritical subrange <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma element-of left-bracket 1 slash 2 comma 1 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>γ<!-- γ --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo fence=false stretchy=false>[</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>gamma in lbrack 1/2,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> it is an open problem to know whether its solutions are globally regular. We show global existence of nonnegative <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript 3 slash 2> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>3</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>H^{3/2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-strong solutions in a supercritical subrange (close to 1) that depends on the initial data norm. Then, for each arbitrary smooth nonnegative initial data, the model has a unique global smooth solution provided that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=gamma element-of left-bracket gamma 1 comma 1 right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>γ<!-- γ --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo fence=false stretchy=false>[</mml:mo> <mml:msub> <mml:mi>γ<!-- γ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>gamma in lbrack gamma _{1},1)</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=gamma 1> <mml:semantics> <mml:msub> <mml:mi>γ<!-- γ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>gamma _{1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> depends on the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript 3 slash 2> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>3</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>H^{3/2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-initial data norm. Our approach is inspired by that of Coti Zelati and Vicol (IUMJ, 2016)." @default.
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• W3004357466 date "2020-03-17" @default.
• W3004357466 modified "2023-10-01" @default.
• W3004357466 title "Global smoothness for a 1D supercritical transport model with nonlocal velocity" @default.
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