<p>We prove finite-time vorticity blowup in the compressible Euler equations in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb{R}^d\)</EquationSource> </InlineEquation> for any <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(d \geq 3\)</EquationSource> </InlineEquation>, starting from smooth, localized, and non-vacuous initial data. This is achieved by lifting the vorticity blowup result from (Chen arXiv preprint arXiv: 2407.06455, 2024) in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb{R}^2\)</EquationSource> </InlineEquation> to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathbb{R}^d\)</EquationSource> </InlineEquation> and utilizing the axisymmetry in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathbb{R}^d\)</EquationSource> </InlineEquation>. At the time of the first singularity, both vorticity blowup and implosion occur on a sphere <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(S^{d-2}\)</EquationSource> </InlineEquation>. Additionally, the solution exhibits a non-radial implosion, accompanied by a stable swirl velocity that is sufficiently strong to initially dominate the non-radial components and to generate the vorticity blowup.</p>

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Vorticity Blowup in Compressible Euler Equations in \(\mathbb{R}^d, d \geq 3\)

  • Jiajie Chen

摘要

We prove finite-time vorticity blowup in the compressible Euler equations in \(\mathbb{R}^d\) for any \(d \geq 3\) , starting from smooth, localized, and non-vacuous initial data. This is achieved by lifting the vorticity blowup result from (Chen arXiv preprint arXiv: 2407.06455, 2024) in \(\mathbb{R}^2\) to \(\mathbb{R}^d\) and utilizing the axisymmetry in \(\mathbb{R}^d\) . At the time of the first singularity, both vorticity blowup and implosion occur on a sphere \(S^{d-2}\) . Additionally, the solution exhibits a non-radial implosion, accompanied by a stable swirl velocity that is sufficiently strong to initially dominate the non-radial components and to generate the vorticity blowup.