<p>We consider the <i>SO</i>(<i>d</i>)-equivariant Yang-Mills heat flow <Equation ID="Equ168"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1944_Article_Equ168.gif" Format="GIF" Height="38" Rendition="HTML" Resolution="72" Type="Linedraw" Width="383" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} \partial _t u-\partial _r^2 u-\frac{(d-3)}{r}\partial _r u+\frac{(d-2)}{r^2}u(1-u)(2-u)=0 \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>u</mi> <mo>-</mo> <msubsup> <mi>∂</mi> <mi>r</mi> <mn>2</mn> </msubsup> <mi>u</mi> <mo>-</mo> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>-</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mi>r</mi> </mfrac> <msub> <mi>∂</mi> <mi>r</mi> </msub> <mi>u</mi> <mo>+</mo> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>-</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> </mfrac> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>-</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>in dimensions <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1944_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">\(d&gt;10.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>&gt;</mo> <mn>10</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We construct a family of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1944_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathcal {C}}^{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">C</mi> </mrow> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> solutions which blow up in finite time via concentration of a universal profile <Equation ID="Equ169"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1944_Article_Equ169.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="148" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} u(t,r)\sim Q\left( \frac{r}{\lambda (t)}\right) , \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mo>∼</mo> <mi>Q</mi> <mfenced close=")" open="("> <mfrac> <mi>r</mi> <mrow> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mfenced> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <i>Q</i> is a stationary state of the equation and the blow-up rates are quantized by <Equation ID="Equ170"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1944_Article_Equ170.gif" Format="GIF" Height="72" Rendition="HTML" Resolution="72" Type="Linedraw" Width="318" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} &amp; \lambda (t)\sim c_{u}(T-t)^{\frac{l}{\gamma }},\,\,\,l\,\,\,\text {is any positive integer},\\ &amp; \gamma =\gamma (d)=\frac{d-4-\sqrt{(d-6)^2-12}}{2}. \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd /> <mtd columnalign="left"> <mrow> <mi>λ</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>∼</mo> <msub> <mi>c</mi> <mi>u</mi> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>-</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mfrac> <mi>l</mi> <mi>γ</mi> </mfrac> </msup> <mo>,</mo> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mi>l</mi> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mtext>is any positive integer</mtext> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow /> </mtd> <mtd columnalign="left"> <mrow> <mi>γ</mi> <mo>=</mo> <mi>γ</mi> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>d</mi> <mo>-</mo> <mn>4</mn> <mo>-</mo> <msqrt> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>-</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <mn>12</mn> </mrow> </msqrt> </mrow> <mn>2</mn> </mfrac> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>Moreover, such solutions are in fact <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1944_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\((l-1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>l</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-codimension stable under pertubation of the initial data.</p>

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Stable Blow-Up Solutions for the SO(d)-Equivariant Supercritical Yang-Mills Heat Flow

  • Yezhou Yi

摘要

We consider the SO(d)-equivariant Yang-Mills heat flow \(\begin{aligned} \partial _t u-\partial _r^2 u-\frac{(d-3)}{r}\partial _r u+\frac{(d-2)}{r^2}u(1-u)(2-u)=0 \end{aligned}\) t u - r 2 u - ( d - 3 ) r r u + ( d - 2 ) r 2 u ( 1 - u ) ( 2 - u ) = 0 in dimensions \(d>10.\) d > 10 . We construct a family of \({\mathcal {C}}^{\infty }\) C solutions which blow up in finite time via concentration of a universal profile \(\begin{aligned} u(t,r)\sim Q\left( \frac{r}{\lambda (t)}\right) , \end{aligned}\) u ( t , r ) Q r λ ( t ) , where Q is a stationary state of the equation and the blow-up rates are quantized by \(\begin{aligned} & \lambda (t)\sim c_{u}(T-t)^{\frac{l}{\gamma }},\,\,\,l\,\,\,\text {is any positive integer},\\ & \gamma =\gamma (d)=\frac{d-4-\sqrt{(d-6)^2-12}}{2}. \end{aligned}\) λ ( t ) c u ( T - t ) l γ , l is any positive integer , γ = γ ( d ) = d - 4 - ( d - 6 ) 2 - 12 2 . Moreover, such solutions are in fact \((l-1)\) ( l - 1 ) -codimension stable under pertubation of the initial data.