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<p>The kinetic behavior of the physical entropy of viscous and heat-conductive fluids is an important and challenging problem, since the entropy equation possesses high degeneracy and singularity in the vacuum region. This article presents a conclusion that for the heat-conductive compressible nematic liquid system, the strong solution to the Cauchy problem is uniformly bounded in entropy and the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> regularities of the velocity and temperature can be preserved, provided the initial density vanishes in the far field is less than <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(O \left( \frac{1}{|x |^{2}}\right) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mfenced close=")" open="("> <mfrac> <mn>1</mn> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, which improved the previous work [<CitationRef CitationID="CR18">18</CitationRef>]. The proof relies on the singular weighted energy method and a modified De Giorgi type iterative technique.</p>

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Well-posedness of Entropy-Bounded Solutions to 3D Compressible Nematic Liquid Crystal Flows with Far Field Vacuum

  • 期刊论文

  • 发表日期：

    2025年12月13日

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  • Qiang Tao,
  • Yuxin Zhai

摘要

The kinetic behavior of the physical entropy of viscous and heat-conductive fluids is an important and challenging problem, since the entropy equation possesses high degeneracy and singularity in the vacuum region. This article presents a conclusion that for the heat-conductive compressible nematic liquid system, the strong solution to the Cauchy problem is uniformly bounded in entropy and the \(L^{2}\) L 2 regularities of the velocity and temperature can be preserved, provided the initial density vanishes in the far field is less than \(O \left( \frac{1}{|x |^{2}}\right) \) O 1 | x | 2 , which improved the previous work [18]. The proof relies on the singular weighted energy method and a modified De Giorgi type iterative technique.