\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 93, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/93\hfil Nonexistence of self-similar singularities] {Nonexistence of self-similar singularities in ideal viscoelastic flows} \author[A. Suen \hfil EJDE-2012/93\hfilneg] {Anthony Suen} \address{Anthony Suen \newline Department of Mathematics\\ Indiana University Bloomington, IN 47405, USA} \email{cksuen@indiana.edu} \thanks{Submitted November 16, 2011. Published June 7, 2012.} \subjclass[2000]{35A20} \keywords{Viscoelastic flow; self-similar singularities} \begin{abstract} We prove the nonexistence of finite time self-similar singularities in an ideal viscoelastic flow in $\mathbb{R}^3$. We exclude the occurrence of Leray-type self-similar singularities under suitable integrability conditions on velocity and deformation tensor. We also prove the nonexistence of asymptotically self-similar singularities in our system. The present work extends the results obtained by Chae in the case of magnetohydrodynamics (MHD). \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} We prove the nonexistence of finite time self-similar singularity in the Cauchy problem of the ideal viscoelastic flow in $\mathbb{R}^3$: \begin{equation} \label{1.1} \begin{gathered} u_t + (u\cdot\nabla)u=-\nabla p+\nabla\cdot FF^T,\\ F^k_t+(u\cdot\nabla)F^k=(F^k\cdot\nabla)u,\\ \operatorname{div}(u)=0,\\ u(x,0)=u_0, \quad F(x,0)=F_0(x). \end{gathered} \end{equation} Here $x\in\mathbb{R}^3$ is the spatial variable and $t\ge0$ is the time, $u=u(x,t)=(u^1,u^2 ,u^3)$ is the velocity of the flow, $P=P(x,t)$ is a prescribed scalar pressure, $F=F(x,t)\in\mathbb{R}^{3\times3}$ is the deformation tensor with $F^k$ being the k-th column of $F$. The initial data $(u_0,F_0)$ is assumed to satisfy the following condition: \begin{equation} \label{1.2} \operatorname{div}(u_0)=\operatorname{div}(F^k_0)=0. \end{equation} By taking divergence on the second equation in \eqref{1.1} and using $\operatorname{div}(u)=0$, \[ \operatorname{div}(F^k)_t+(u\cdot\nabla)(\operatorname{div}(F^k))=0, \] so $F$ is ``divergence free'' for later time. In view of the above, we can rewrite \eqref{1.1}-\eqref{1.2} into the following equivalent form: \begin{equation} \label{1.4} \begin{gathered} u_t + (u\cdot\nabla)u=-\nabla p+\sum(F^k\cdot\nabla)F^k,\\ F^k_t+(u\cdot\nabla)F^k=(F^k\cdot\nabla)u,\\ \operatorname{div}(u)=\operatorname{div}(F^k)=0,\\ u(x,0)=u_0,\quad F(x,0)=F_0(x). \end{gathered} \end{equation} We refer to \cite{bcah} or \cite{dafermos} for a more detailed derivation and physical discussion of the above system. The study of incompressible fluids can be dated back to 18th century when Euler \cite{euler} considered the well-known Euler's equations: \begin{equation} \label{1.3} \begin{gathered} u_t + (u\cdot\nabla)u=-\nabla p,\\ \operatorname{div}(u)=0. \end{gathered} \end{equation} The local-in-time existence of \eqref{1.3} can be found in Kato \cite{kato}, yet it is still interesting to know whether or not finite time singularities do occur in the above system. There seems to be no satisfactory result until Beale et al \cite{bkm} discovered that, if a solution $u$ for \eqref{1.3} possesses a singularity at a finite time $T>0$, then it is necessary to have \[ \int_0^T\|\nabla\times u(\cdot,t)\|_{L^\infty}dt=+\infty. \] The above blow-up criterion is later generalized to many other related systems, for example ideal magnetohydrodynamics (MHD) (Caflisch-Klapper-Steel \cite{cks}) and ideal viscoelastic flow (Hu and Hynd \cite{hu}). With a clever argument suggested by Chae in \cite{chae1}-\cite{chae2}, he applied the results in Caflisch et al \cite{cks} to exclude the possibility of a finite time apparition of self-similar singularities in both viscous and ideal MHD model. The goal of our present work is to extend Chae's results to ideal viscoelastic flow based on the blow-up criterion shown by Hu and Hynd \cite{hu}. We now give a precise formulation of our result. To begin with, we observe that if $(u,F,p)$ is a solution of \eqref{1.4}, then for any $\lambda>0$, $(u^{(\lambda)},F^{(\lambda)},p^{(\lambda)})$ is also a solution with the initial data $(u_0^{(\lambda)},F_0^{(\lambda)})$, where \begin{gather*} u^{(\lambda)}(x,t)=\lambda u(\lambda x,\lambda^2t),\\ F^{(\lambda)}(x,t)=\lambda F(\lambda x,\lambda^2t),\\ p^{(\lambda)}(x,t)=\lambda^2 p(\lambda x,\lambda^2t),\\ (u_0^{(\lambda)}(x),F_0^{(\lambda)}(x))=(\lambda u_0(\lambda x), \lambda F_0(\lambda x)). \end{gather*} Therefore, if there exists a self-similar blowing up solution $(u(x,t),F(x,t))$ of \eqref{1.4}, then it has to be of the form \begin{gather} \label{1.6} u(x,t)=\frac{1}{\sqrt{T_{*}-t}}U\Big(\frac{x}{\sqrt{T_{*}-t}}\Big),\\ \label{1.7} F^k(x,t)=\frac{1}{\sqrt{T_{*}-t}}E^k\Big(\frac{x}{\sqrt{T_{*}-t}}\Big), \\ \label{1.8} p(x,t)=\frac{1}{T_{*}-t}P\Big(\frac{x}{\sqrt{T_{*}-t}}\Big), \end{gather} when $t$ is being close to the possible blow-up time $T_*$. By substituting \eqref{1.6}-\eqref{1.8} into \eqref{1.4}, we find that $(U,E,P)=(U(y),E(y),P(y))$ satisfies of the stationary system \begin{equation} \label{1.9} \begin{gathered} \frac{1}{2}U+\frac{1}{2}(y\cdot\nabla)U+(U\cdot\nabla)U =-\nabla P+\sum_{k}(E^k\cdot\nabla)E^k,\\ \frac{1}{2}E^k+\frac{1}{2}(y\cdot\nabla)E^k+(U\cdot\nabla)E^k =(E^k\cdot\nabla)U,\\ \operatorname{div}(U)=\operatorname{div}(E^k)=0. \end{gathered} \end{equation} Conversely, if $(U,E,P)$ is a smooth solution of \eqref{1.9}, then $(u,F,p)$ as defined by \eqref{1.6}-\eqref{1.8} is a classical solution of \eqref{1.4} which blows up at $t=T_*$. Leray \cite{leray} was the first one to study self-similar singularities of the form similar to \eqref{1.6}-\eqref{1.8} in Navier-Stokes equations, while Chae \cite{chae1}-\cite{chae2} considered those in the case of (MHD). We apply those concepts to viscoelastic flow which will be given in later sections. The following theorems are the main results of this paper: \begin{theorem} \label{thm1.1} Suppose there exists $T_{*}>0$ such that we have a representation of a solution $(u,F)$ to \eqref{1.4} by \eqref{1.6}-\eqref{1.8} for all $t\in(0,T_*)$ with $(U,E)$ satisfying the following conditions: \begin{gather} \label{1.9b} (U,E)\in(C_0^1(\mathbb{R}^3))^2, \\ \label{1.10} \begin{gathered} \text{there exists $q>0$ such that $(\Omega,E)\in (L^{r}(\mathbb{R}^3))^2$ for all $r\in(0,q)$,}\\ \text{ where $\Omega=\nabla\times U$.} \end{gathered} \end{gather} Then we have $U=E=0$. \end{theorem} The above theorem can be proved in the same way as in Chae \cite[pp. 1014--1017]{chae2}, so we omit the proof here. By changing the decay conditions on $(U,E)$, we can derive similar results about the nonexistence of self-similar singularities. The following theorem is reminiscent of \cite[Theorem 1.2]{chae1}. \begin{theorem} \label{thm1.2} Suppose there exists $T_{*}>0$ such that we have a representation of a solution $(u,F)$ to \eqref{1.4} by \eqref{1.6}-\eqref{1.8} for all $t\in(0,T_*)$ with $(U,E)$ satisfying the following conditions: \begin{gather} \label{1.11} (U,E)\in(H^m(\mathbb{R}^3))^2\quad\text{for }m>\frac{3}{2}+1,\\ \label{1.12} \|\nabla U\|_{L^\infty}+\|\nabla E\|_{L^\infty}<\varepsilon, \end{gather} where $\varepsilon>0$ is a constant as chosen in Theorem \ref{thm2.1}. Then $U=E=0$. \end{theorem} Using Theorems \ref{thm1.1}-\ref{thm1.2}, we can obtain the following result about asymptotically self-similar singularity, which is reminiscent of \cite[Theorem 1.3]{chae1}. We refer to Chae \cite{chae2} (especially section~2) for more detailed descriptions of asymptotically self-similar singularities. \begin{theorem} \label{thm1.3} Given $T>0$ and $m>\frac{3}{2}+1$, let $(u,F)\in((C([0,T);H^m(\mathbb{R}^3)))^2$ be a classical solutions to \eqref{1.4}. Suppose there exists functions $U$, $E$ satisfying \eqref{1.8}--\eqref{1.9} as in Theorem \ref{thm1.1} such that \begin{equation} \label{1.12b} \begin{aligned} &\sup_{00$ such that $(u,F)$ can be extended to a solution of \eqref{1.4} in $[0,T+\delta]\times\mathbb{R}^3$ with $(u,F)\in C([0,T+\delta];H^m(\mathbb{R}^3))$. \end{theorem} This article is organized as follows. In section~2 we first prove Theorem~\ref{thm2.1} about a continuation principle for local solution of \eqref{1.4} with the help of an auxiliary lemma. We then begin the proofs of Theorems \ref{thm1.2}-\ref{thm1.3} in section~3 which basically follow the arguments as given in Chae \cite{chae1}-\cite{chae2}. \section{Continuation Principle for Local Solution} In this section we show the continuation principle for local solution of \eqref{1.4} with the help of an energy estimate as given in Lemma~\ref{lem2.2}. The result is reminiscent of \cite[Lemma 2.1]{chae1} except that we apply the blow-up criterion for singularities in ideal viscoelastic flow proved by Hu and Hynd \cite{hu}. We begin with the following theorem. \begin{theorem} \label{thm2.1} Given $T>0$ and $m>\frac{3}{2}+1$, let $(u,F)\in((C([0,T);H^m(\mathbb{R}^3)))^2$ be a classical solutions to \eqref{1.4}. There exists $\varepsilon>0$ and $\delta>0$ such that if \begin{equation} \label{2.1} \sup_{0\le t0$ is a generic constant which depends only on $m$. \end{lemma} The proof of the above lemma is exactly as in Hu and Hynd \cite[pp. 4--6]{hu}. \begin{proof}[Proof of Theorem \ref{thm2.1}] We follow the proof of \cite[Lemma 2.1]{chae1}. We claim \begin{equation} \label{2.3} \int_0^T(\|\nabla u\|^2_{L^\infty}+\sum_k\|\nabla F^k\|^2_{L^\infty})ds<\infty, \end{equation} and if \eqref{2.3} holds, then we have \[ \int_0^T(\|\nabla\times u\|_{L^\infty}+\sum_k\|\nabla\times F^k\|_{L^\infty})ds <\infty, \] and so by the blow-up criterion derived in \cite{hu}, there exists $\delta>0$ such that $(u,F)$ can be continuously extended to a solution of \eqref{1.4} in $[0,T+\delta]\times\mathbb{R}^3$ with $(u,F)\in (C([0,T+\delta];H^m(\mathbb{R}^3)))^2$. To prove \eqref{2.3}, we apply \eqref{2.2} as in Lemma~\ref{lem2.2} to obtain \begin{equation} \label{2.4} \begin{aligned} &\frac{d}{dt}\Big[(T-t)(\|u\|^2_{H^m}+\sum_k\|F^k\|^2_{H^m})\Big] +\|u\|^2_{H^m}+\sum_k\|F^k\|^2_{H^m}\\ & \le M(T-t)\Big(\|\nabla u\|_{L^\infty}+\sum_k\|F^k\|_{L^\infty}\Big) \Big(\|u\|^2_{H^m}+\sum_k\|F^k\|^2_{H^m}\Big). \end{aligned} \end{equation} If we choose $\varepsilon<1/(2M)$, then \eqref{2.1} and \eqref{2.4} imply \[ \frac{d}{dt}\Big[(T-t)(\|u\|^2_{H^m}+\sum_k\|F^k\|^2_{H^m})\Big] +\frac{1}{2}\Big(\|u\|^2_{H^m}+\sum_k\|F^k\|^2_{H^m}\Big)\le0. \] Upon integrating the above, \begin{equation} \label{2.5} \int_0^T(\|u\|^2_{H^m}+\sum_k\|F^k\|^2_{H^m})ds \le 2T(\|u_0\|^2_{H^m}+\sum_k\|F_0^k\|^2_{H^m})<\infty \end{equation} By the Sobolev embedding theorem, there exists $l\ge1$ and $\beta\in(0,1)$ such that $H^m(\mathbb{R}^3)\hookrightarrow C^{l,\beta}(\mathbb{R}^3)$, so we obtain from \eqref{2.5} that \begin{align*} \int_0^T(\|\nabla u\|_{L^\infty}+\sum_k\|\nabla F^k\|_{L^\infty})ds &\leq \int_0^T(\|u\|_{H^m}+\sum_k\|F^k\|_{H^m})ds\\ &\leq \sqrt{T}\Big[\int_{0}^T(\|u\|^2_{H^m}+\sum_k\|F^k\|^2_{H^m})ds\Big] ^{1/2}<\infty, \end{align*} and hence \eqref{2.3} follows. \end{proof} \section{Nonexistence of Asymptotically Self-similar Singularities} In this section we prove Theorems \ref{thm1.2}-\ref{thm1.3} as stated in section~1. These arguments are reminiscent of Chae \cite{chae1} and hence we omit some of those technical details. \begin{proof}[Proof of Theorem \ref{thm1.2}] By the definitions \eqref{1.6}--\eqref{1.7} for $t\in(0,T_{*})$, \begin{gather*} \|U(\cdot,t)\|_{L^\infty} =(T_{*}-t)\|\nabla u(\cdot,t)\|_{L^\infty},\\ \|\nabla E(\cdot,t)\|_{L^\infty} =(T_{*}-t)\|\nabla F(\cdot,t)\|_{L^\infty}. \end{gather*} So assumption \eqref{1.12} implies $$ \sup_{0\le t0$ where $B_R=\{x\in\mathbb{R}^3:|x|0$ such that $$ \sup_{s'\le s<\infty}\|\nabla\tilde u(\cdot,s)\|_{L^\infty} +\|\nabla\tilde F(\cdot,s)\|_{L^\infty}<\varepsilon, $$ where $\varepsilon>0$ is as chosen in Theorem \ref{thm2.1}. Let $t'=T(1-e^{-2s'})$, then $(u,F)$ satisfies $$ \sup_{t'\le t0$ such that $(u,F)$ can be extended to a solution of \eqref{1.4} in $[0,T+\delta]\times\mathbb{R}^3$ with $(u,F)\in C([0,T+\delta];H^m(\mathbb{R}^3))$. This completes the proof of Theorem \ref{thm1.3}. \end{proof} \subsection*{Acknowledgments} The author would like to thank Professor Dongho Chae for his comments and letting me to use his idea from his previous papers. \begin{thebibliography}{0} \bibitem{bkm} J. T. Beale, T. Kato and A. Majda; \emph{Remarks on the breakdown of smooth solutions for the 3-D Euler equations}, Comm. Math. Phys. 94 (1984), no. 1, 61--66. \bibitem{bcah} R. Byron Bird, C. F. Curtiss, R. C. Armstrong and O. Hassager; \emph{Dynamics of polymeric liquids, Kinetic Theory}, vol. 2, 2nd edn. Wiley Interscience, New York, 1987. \bibitem{cks} R. E. Caflisch, I. Klapper and G. Steele; \emph{Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD}, Comm. Math. Phys. 184 (1997), no. 2, 443--455. \bibitem{chae1} D. Chae; \emph{Nonexistence of self-similar singularities in the viscous magnetohydrodynamics with zero resistivity}, J. Funct. Anal. 254, no. 2, (2008), 441--453 \bibitem{chae2} D. Chae; \emph{Nonexistence of self-similar singularities in the ideal magnetohydrodynamics}, Arch. Rational Mech. Anal.194, no. 3, (2009), 1011--1027. \bibitem{dafermos} C. Dafermos; \emph{Hyperbolic Conservation Laws in Continuum Physics}, Springer, Heidelberg, 2000 \bibitem{euler} L. Euler; \emph{Principes g\'{e}n\'{e}raux de l'\'{e}tat d'\'{e}quilibre des fluides}, M\'{e}moires de l'acad\'{e}mie des sciences de Berlin, 11 (1755), 1757, pp. 217-273. Reprinted in Opera omnia Series II vol. 12, pp. 2-53. \bibitem{hu} X. Hu and R. Hynd; \emph{A blow-up criterion for ideal viscoelastic flow}, arXiv:1102.1113. \bibitem{leray} J. Leray; \emph{Essai sur le mouvement d\~oun fluide visqueux emplissant l\~0espace}, Acta Math. 63 (1934) 193--248. \bibitem{kato} T. Kato; \emph{Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations} (Proc. Sympos., Dundee, 1974; dedicated to Konrad J\"urgens), pp. 25Ð70. Lecture Notes in Math., Vol. 448, Springer, Berlin, 1975. \end{thebibliography} \end{document}