In their recent breakthrough works [Ann. of Math. (2), 196 (2022), 567–778; Invent. Math., 227 (2022), 247–413], Merle, Raphaël, Rodnianski, and Szeftel constructed finite-time blow-up solutions to energy supercritical defocusing nonlinear Schrödinger equations, with the leading order given by imploding solutions to isentropic compressible Euler equations. Motivated by these results, in this paper and its sequel [Shao, Wei and Zhang, Forum Math. Pi, 13 (2025), Paper No. e15, 59 pp], we find a new connection between supercritical defocusing nonlinear wave equations (NLW) and the isentropic relativistic Euler equations. More precisely, in this paper, we construct self-similar smooth imploding solutions of the isentropic relativistic Euler equations with an isothermal equation of state <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p=\frac{1}{\ell }\varrho $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mi>ℓ</mi> </mfrac> <mi>ϱ</mi> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\ell >1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> in physical space dimensions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$d=2,3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\ell >1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> close to 1 in higher dimensions. In our companion paper, we construct finite-time blow-up solutions for the complex-valued supercritical defocusing nonlinear wave equation, with the leading-order dynamics governed by the imploding solutions to the isentropic relativistic Euler equations constructed in the present paper.

Self-similar Imploding Solutions of the Relativistic Euler Equations

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发表日期：

2026年7月1日

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Feng Shao,
Dongyi Wei,
Zhifei Zhang

摘要

In their recent breakthrough works [Ann. of Math. (2), 196 (2022), 567–778; Invent. Math., 227 (2022), 247–413], Merle, Raphaël, Rodnianski, and Szeftel constructed finite-time blow-up solutions to energy supercritical defocusing nonlinear Schrödinger equations, with the leading order given by imploding solutions to isentropic compressible Euler equations. Motivated by these results, in this paper and its sequel [Shao, Wei and Zhang, Forum Math. Pi, 13 (2025), Paper No. e15, 59 pp], we find a new connection between supercritical defocusing nonlinear wave equations (NLW) and the isentropic relativistic Euler equations. More precisely, in this paper, we construct self-similar smooth imploding solutions of the isentropic relativistic Euler equations with an isothermal equation of state $p=\frac{1}{\ell }\varrho $ $p = \frac{1}{ℓ} ϱ$ for all $\ell >1$ $ℓ > 1$ in physical space dimensions $d=2,3$ $d = 2, 3$ and for $\ell >1$ $ℓ > 1$ close to 1 in higher dimensions. In our companion paper, we construct finite-time blow-up solutions for the complex-valued supercritical defocusing nonlinear wave equation, with the leading-order dynamics governed by the imploding solutions to the isentropic relativistic Euler equations constructed in the present paper.