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<p>We investigate the vanishing viscosity limit of a parabolic-elliptic coupled system arising in radiation hydrodynamics. The limit is the solution to a hyperbolic-elliptic coupled system. We study the problem on both the whole line ℝ and the half line ℝ<sub>+</sub>. Two types of conditions are considered: (1) the initial data are sufficiently close to a given initial state with small wave strength, and (2) the initial data are monotonically increasing. Under these conditions, we establish uniform convergence rates for both Cauchy problems and initial-boundary value problems. Specifically, we demonstrate that the solutions to the parabolic-elliptic system converge to those of the hyperbolic-elliptic system as the viscosity coefficient <i>ε</i> approaches zero, with convergence rates of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_609_Article_IEq1.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(O({\varepsilon ^{{3 \over 4}}})\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>O</mi> <mo stretchy="false">(</mo> <mrow> <msup> <mi>ε</mi> <mrow> <mrow> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> for <i>u</i> and <i>O</i>(<i>ε</i>) for <i>q</i> in <i>L</i><sup>∞</sup>-norm. Additionally, we prove the global well-posedness of the parabolic-elliptic coupled system by the maximum principle and energy method. Our results extend previous work by providing explicit convergence rates and addressing both Cauchy problems and initial-boundary value problems under various conditions.</p>

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Vanishing viscosity limit of a parabolic-elliptic coupled system

  • 期刊论文

  • 发表日期：

    2025年10月29日

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  • Changjiang Zhu,
  • Qiaolong Zhu

摘要

We investigate the vanishing viscosity limit of a parabolic-elliptic coupled system arising in radiation hydrodynamics. The limit is the solution to a hyperbolic-elliptic coupled system. We study the problem on both the whole line ℝ and the half line ℝ+. Two types of conditions are considered: (1) the initial data are sufficiently close to a given initial state with small wave strength, and (2) the initial data are monotonically increasing. Under these conditions, we establish uniform convergence rates for both Cauchy problems and initial-boundary value problems. Specifically, we demonstrate that the solutions to the parabolic-elliptic system converge to those of the hyperbolic-elliptic system as the viscosity coefficient ε approaches zero, with convergence rates of \(O({\varepsilon ^{{3 \over 4}}})\) O ( ε 3 4 ) for u and O(ε) for q in L∞-norm. Additionally, we prove the global well-posedness of the parabolic-elliptic coupled system by the maximum principle and energy method. Our results extend previous work by providing explicit convergence rates and addressing both Cauchy problems and initial-boundary value problems under various conditions.