<p>We establish a complete picture for well-posedness of parabolic Cauchy problems with time-independent, uniformly elliptic, bounded measurable complex coefficients. We exhibit a range of <i>p</i> for which tempered distributions in homogeneous Hardy–Sobolev spaces <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3149_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">$\dot{H}^{s,p}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mover accent="true"> <mi>H</mi> <mo>˙</mo> </mover> <mrow> <mi>s</mi> <mo>,</mo> <mi>p</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> with regularity index <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3149_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">$s \in (-1,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are initial data. Source terms of Lions’ type lie in weighted tent spaces, and weak solutions are built with their gradients in weighted tent spaces as well. A similar result can be achieved for initial data in homogeneous Besov spaces <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3149_Article_IEq3.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$\dot{B}^{s}_{p,p}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mover accent="true"> <mi>B</mi> <mo>˙</mo> </mover> <mrow> <mi>p</mi> <mo>,</mo> <mi>p</mi> </mrow> <mi>s</mi> </msubsup> </math></EquationSource> </InlineEquation>.</p>

On well-posedness for parabolic Cauchy problems of Lions type with rough initial data

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2025年5月26日

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Pascal Auscher,
Hedong Hou

摘要

We establish a complete picture for well-posedness of parabolic Cauchy problems with time-independent, uniformly elliptic, bounded measurable complex coefficients. We exhibit a range of p for which tempered distributions in homogeneous Hardy–Sobolev spaces $\dot{H}^{s,p}$ ${\dot{H}}^{s, p}$ with regularity index $s \in (-1,1)$ $s \in (- 1, 1)$ are initial data. Source terms of Lions’ type lie in weighted tent spaces, and weak solutions are built with their gradients in weighted tent spaces as well. A similar result can be achieved for initial data in homogeneous Besov spaces $\dot{B}^{s}_{p,p}$ ${\dot{B}}_{p, p}^{s}$ .