<p>The paper deals with the Cauchy problem of the Debye-Hückel system in homogeneous Besov-Morrey spaces. By using the smoothing effect of the heat semigroup and the Littlewood-Paley theory, we obtain the global existence of solutions for small initial data in the critical Besov-Morrey spaces <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2024_1181_Article_IEq1.gif" Format="GIF" Height="33" Rendition="HTML" Resolution="72" Type="Linedraw" Width="179" /> </InlineMediaObject> <EquationSource Format="TEX">$ \dot{\mathcal {N}}_{p,h,\infty }^{-2+\frac{d}{p}}(\mathbb {R}^d) \times \dot{\mathcal {N}}_{p,h,\infty }^{-2+\frac{d}{p}}(\mathbb {R}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mover accent="true"> <mi mathvariant="script">N</mi> <mo>˙</mo> </mover> <mrow> <mi>p</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>∞</mi> </mrow> <mrow> <mo>-</mo> <mn>2</mn> <mo>+</mo> <mfrac> <mi>d</mi> <mi>p</mi> </mfrac> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo>×</mo> <msubsup> <mover accent="true"> <mi mathvariant="script">N</mi> <mo>˙</mo> </mover> <mrow> <mi>p</mi> <mo>,</mo> <mi>h</mi> <mo>,</mo> <mi>∞</mi> </mrow> <mrow> <mo>-</mo> <mn>2</mn> <mo>+</mo> <mfrac> <mi>d</mi> <mi>p</mi> </mfrac> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2024_1181_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$\frac{d}{2}<p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mi>d</mi> <mn>2</mn> </mfrac> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12215_2024_1181_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$1\le h\le p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>h</mi> <mo>≤</mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation>. Moreover, we demonstrate that the solutions obtained are analytic in a Gevrey class. Finally, as a consequence of Gevrey analyticity result, we get time decay estimates of solutions in critical Besov-Morrey spaces.</p>

Global existence and Gevrey analyticity of the Debye-Hückel system in critical Besov-Morrey spaces

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发表日期：

2024年12月31日

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Ahmed El Idrissi,
Halima Srhiri,
Brahim El Boukari,
Jalila El Ghordaf

摘要

The paper deals with the Cauchy problem of the Debye-Hückel system in homogeneous Besov-Morrey spaces. By using the smoothing effect of the heat semigroup and the Littlewood-Paley theory, we obtain the global existence of solutions for small initial data in the critical Besov-Morrey spaces $ \dot{\mathcal {N}}_{p,h,\infty }^{-2+\frac{d}{p}}(\mathbb {R}^d) \times \dot{\mathcal {N}}_{p,h,\infty }^{-2+\frac{d}{p}}(\mathbb {R}^d)$ ${\dot{N}}_{p, h, \infty}^{- 2 + \frac{d}{p}} (R^{d}) \times {\dot{N}}_{p, h, \infty}^{- 2 + \frac{d}{p}} (R^{d})$ with $\frac{d}{2}<p<\infty $ $\frac{d}{2} < p < \infty$ and $1\le h\le p$ $1 \leq h \leq p$ . Moreover, we demonstrate that the solutions obtained are analytic in a Gevrey class. Finally, as a consequence of Gevrey analyticity result, we get time decay estimates of solutions in critical Besov-Morrey spaces.