<p>The present paper studies a two-component mathematical model representing shallow-water wave propagation primarily in equatorial ocean regions, incorporating the effects of weak Coriolis force and equatorial undercurrent. We start with the Green–Naghdi type equations under the weak Coriolis and equatorial undercurrent effects from the Euler equations, then the two-component Camassa–Holm system with the two effects is derived by truncating asymptotic expansions of the quantities to the appropriate order. Analytically, we study the mathematical properties of the solutions to the two-component Camassa–Holm system including the ill-posedness of the solutions in Besov spaces <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_940_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="92" /> </InlineMediaObject> <EquationSource Format="TEX">$B^{s}_{p,\infty }\times B^{s-1}_{p,\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>B</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>∞</mi> </mrow> <mi>s</mi> </msubsup> <mo>×</mo> <msubsup> <mi>B</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>∞</mi> </mrow> <mrow> <mi>s</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_940_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$1\le p\le \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_940_Article_IEq3.gif" Format="GIF" Height="33" Rendition="HTML" Resolution="72" Type="Linedraw" Width="140" /> </InlineMediaObject> <EquationSource Format="TEX">$s>\max \left\{ 2+\frac{1}{p},\frac{5}{2}\right\} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mo movablelimits="true">max</mo> <mfenced close="}" open="{"> <mn>2</mn> <mo>+</mo> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> <mo>,</mo> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, the Hölder continuity of the data-to-solution map in Besov spaces <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_940_Article_IEq4.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="85" /> </InlineMediaObject> <EquationSource Format="TEX">$B^{s}_{p,r}\times B^{s-1}_{p,r}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>B</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>r</mi> </mrow> <mi>s</mi> </msubsup> <mo>×</mo> <msubsup> <mi>B</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>r</mi> </mrow> <mrow> <mi>s</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_940_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="99" /> </InlineMediaObject> <EquationSource Format="TEX">$1\le p,r\le \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>,</mo> <mi>r</mi> <mo>≤</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_940_Article_IEq3.gif" Format="GIF" Height="33" Rendition="HTML" Resolution="72" Type="Linedraw" Width="140" /> </InlineMediaObject> <EquationSource Format="TEX">$s>\max \left\{ 2+\frac{1}{p},\frac{5}{2}\right\} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mo movablelimits="true">max</mo> <mfenced close="}" open="{"> <mn>2</mn> <mo>+</mo> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> <mo>,</mo> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. We then investigate the Gevrey regularity and analyticity of the system in <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_940_Article_IEq7.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">${G_{\delta ,s}^{\gamma }}\times {G_{\delta ,s-1}^{\gamma }}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>G</mi> <mrow> <mi>δ</mi> <mo>,</mo> <mi>s</mi> </mrow> <mi>γ</mi> </msubsup> <mo>×</mo> <msubsup> <mi>G</mi> <mrow> <mi>δ</mi> <mo>,</mo> <mi>s</mi> <mo>-</mo> <mn>1</mn> </mrow> <mi>γ</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_940_Article_IEq8.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="126" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta \ge 1,\ \nu>\gamma >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>≥</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt" /> <mi>ν</mi> <mo>></mo> <mi>γ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_940_Article_IEq9.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$s>\frac{5}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. Finally, we provide the persistence properties and the spatial asymptotic profiles of the solutions in weighted spaces <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_940_Article_IEq10.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="125" /> </InlineMediaObject> <EquationSource Format="TEX">$L ^ p_{\phi }=L^p(\mathbb {R},\phi ^pdx)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>ϕ</mi> <mi>p</mi> </msubsup> <mo>=</mo> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> <msup> <mi>ϕ</mi> <mi>p</mi> </msup> <mi>d</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

On a Two-Component Shallow-Water Model with the Weak Coriolis and Equatorial Undercurrent Effects

期刊论文
发表日期：

2025年5月9日

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Lili Huang,
Yaojun Yang,
Shouming Zhou

摘要

The present paper studies a two-component mathematical model representing shallow-water wave propagation primarily in equatorial ocean regions, incorporating the effects of weak Coriolis force and equatorial undercurrent. We start with the Green–Naghdi type equations under the weak Coriolis and equatorial undercurrent effects from the Euler equations, then the two-component Camassa–Holm system with the two effects is derived by truncating asymptotic expansions of the quantities to the appropriate order. Analytically, we study the mathematical properties of the solutions to the two-component Camassa–Holm system including the ill-posedness of the solutions in Besov spaces $B^{s}_{p,\infty }\times B^{s-1}_{p,\infty }$ $B_{p, \infty}^{s} \times B_{p, \infty}^{s - 1}$ with $1\le p\le \infty $ $1 \leq p \leq \infty$ and $s>\max \left\{ 2+\frac{1}{p},\frac{5}{2}\right\} $ $s > max \{2 + \frac{1}{p}, \frac{5}{2}\}$ , the Hölder continuity of the data-to-solution map in Besov spaces $B^{s}_{p,r}\times B^{s-1}_{p,r}$ $B_{p, r}^{s} \times B_{p, r}^{s - 1}$ with $1\le p,r\le \infty $ $1 \leq p, r \leq \infty$ and $s>\max \left\{ 2+\frac{1}{p},\frac{5}{2}\right\} $ $s > max \{2 + \frac{1}{p}, \frac{5}{2}\}$ . We then investigate the Gevrey regularity and analyticity of the system in ${G_{\delta ,s}^{\gamma }}\times {G_{\delta ,s-1}^{\gamma }}$ $G_{δ, s}^{γ} \times G_{δ, s - 1}^{γ}$ with $\delta \ge 1,\ \nu>\gamma >0$ $δ \geq 1, ν > γ > 0$ and $s>\frac{5}{2}$ $s > \frac{5}{2}$ . Finally, we provide the persistence properties and the spatial asymptotic profiles of the solutions in weighted spaces $L ^ p_{\phi }=L^p(\mathbb {R},\phi ^pdx)$ $L_{ϕ}^{p} = L^{p} (R, ϕ^{p} d x)$ .