In this paper, we consider the 2-dimensional non-viscous Oldroyd-B model with infinity Reynolds number (<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2024_2342_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu =a=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>=</mo> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>). It is a difficult case since the velocity field u(t, x) will no longer decay. Fortunately, observing the exponential decay of the stress tensor <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2024_2342_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau (t,x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we succeeded in proving the global existence for this system with some large initial data. Moreover, when <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2024_2342_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$a\rightarrow 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we investigate the behavior of the solutions in local time.

Global Well-Posedness for the Oldroyd-B Model with Infinity Reynolds Number

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发表日期：

2025年1月16日

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Zhi Chen,
Weikui Ye,
Zhaoyang Yin

摘要

In this paper, we consider the 2-dimensional non-viscous Oldroyd-B model with infinity Reynolds number ( $\nu =a=0$ $ν = a = 0$ ). It is a difficult case since the velocity field u(t, x) will no longer decay. Fortunately, observing the exponential decay of the stress tensor $\tau (t,x)$ $τ (t, x)$ , we succeeded in proving the global existence for this system with some large initial data. Moreover, when $a\rightarrow 0$ $a \to 0$ , we investigate the behavior of the solutions in local time.