<p>We study the advection equation along vector fields singular at the initial time. More precisely, we prove that for divergence-free vector fields in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10231_2025_1543_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="322" /> </InlineMediaObject> <EquationSource Format="TEX">$L^1_{loc}((0,T];BV(\mathbb {T}^d;\mathbb {R}^d))\cap L^2((0,T) \times \mathbb {T}^d;\mathbb {R}^d))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mrow> <mi mathvariant="italic">loc</mi> </mrow> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mrow> <mo>;</mo> <mi>B</mi> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mi>d</mi> </msup> <mo>;</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>∩</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mi>d</mi> </msup> <mo>;</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, there exists a unique vanishing diffusivity solution. This class includes the vector field constructed by Depauw in [<CitationRef CitationID="CR13">13</CitationRef>], for which there are infinitely many distinct bounded solutions to the advection equation.</p>

On vanishing diffusivity selection for the advection equation

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2025年1月21日

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Giulia Mescolini,
Jules Pitcho,
Massimo Sorella

摘要

We study the advection equation along vector fields singular at the initial time. More precisely, we prove that for divergence-free vector fields in $L^1_{loc}((0,T];BV(\mathbb {T}^d;\mathbb {R}^d))\cap L^2((0,T) \times \mathbb {T}^d;\mathbb {R}^d))$ $L_{loc}^{1} ((0, T]; B V (T^{d}; R^{d})) \cap L^{2} ((0, T) \times T^{d}; R^{d}))$ , there exists a unique vanishing diffusivity solution. This class includes the vector field constructed by Depauw in [13], for which there are infinitely many distinct bounded solutions to the advection equation.