Existence of Singular Self-Similar Solutions of the Three-Dimensional Euler Equations in a Bounded Domain

Abstract

A self-similar solution of the three-dimensional (3d) incompressible Euler equations is defined by % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiaacI % cacaWG4bGaaiilaiaadshacaGGPaGaeyypa0JaamyvaiaacIcacaWG % 5bGaaiykaiaac+cacaGGOaGaamiDamaaCaaaleqabaGaaiOkaaaaki % abgkHiTiaadshacaGGPaWaaWbaaSqabeaacqaHXoqyaaGccaGGSaGa % aGjbVlaadMhacqGH9aqpcaWG4bGaai4laiaacIcacaWG0bWaaWbaaS % qabeaacaGGQaaaaOGaeyOeI0IaamiDaiaacMcadaahaaWcbeqaaiab % ek7aIbaakiaacYcacqaHXoqycaGGSaGaeqOSdiMaeyOpa4JaaGimai % aacYcaaaa!5ABC! $$u(x,t) = U(y)/(t^* - t)^\alpha ,\;y = x/(t^* - t)^\beta ,\alpha ,\beta > 0,$$ where U(y) satisfies $$ \alpha U + \beta y \cdot \nabla U + U \cdot \nabla U + \nabla P = 0,\quad {\text{div }}U = 0. $$ For % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey % ypa0JaeqOSdiMaeyypa0JaaGymaiaac+cacaaIYaGaaiilaaaa!3E14! $$\alpha = \beta = 1/2,$$ which is the limiting case of Leray’s self-similar Navier–Stokes equations, we prove the existence of % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadw % facaGGSaGaamiuaiaacMcacqGHiiIZcaWGibWaaWbaaSqabeaacaaI % ZaaaaOGaaiikaiabfM6axjaacYcatuuDJXwAK1uy0HMmaeHbfv3ySL % gzG0uy0HgiuD3BaGqbaiab-1risnaaCaaaleqabaGaaG4maaaakiab % gEna0kab-1risjaacMcaaaa!4F5A! $$(U,P) \in H^3 (\Omega ,\mathbb{R}^3 \times \mathbb{R})$$ in a smooth bounded domain Ω, with the inflow boundary data of non-zero vorticity. This implies the possibility that solutions of the Euler equations blow up at a time % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabg2 % da9iaadshadaahaaWcbeqaaiaacQcaaaGccaGGSaGaamiDamaaCaaa % leqabaGaaiOkaaaakiabgYda8iabgUcaRiabg6HiLkaac6caaaa!4062! $$t = t^* ,t^* < + \infty .$$