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This article is cited in 1 scientific paper (total in 1 paper) Brief Communications Weak solvability of motion models for a viscoelastic fluid with a higher-order rheological relation V. G. Zvyagin, V. P. Orlov Voronezh State University
Bibliographic databases:
     We consider the motion on the interval of time $[0,T]$ of an incompressible viscoelastic fluid of constant density, which fills a bounded domain $\Omega\subset\mathbb{R}^N$, $N=2,3$, with locally Lipschitz boundary $\partial\Omega$ and satisfies the rheological relation
$$
\begin{equation}
\biggl(1+\sum_{k=1}^{m}p_{k}D_t^{a_k}\biggr)\sigma= \nu\biggl(1+\nu^{-1}\sum_{k=1}^{n}q_{k}D_t^{b_k}\biggr) \mathcal{E}(v), \qquad \nu>0,
\end{equation}
\tag{1}
$$
which connects the stress tensor deviator $\sigma(t,x)$ with the deformation velocity tensor $\mathcal{E}(v)(t,x)$ of the velocity field $v(t,x)$. Here $m,n \in \mathbb{N}$, $\nu>0$, $a_k\in [k,k+ 1)$ for $k=1,\dots,m$, $b_k\in [k,k+1)$ for $k=1,\dots,n$, and $D_t^{r}$ is the fractional Riemann–Liouville derivative of order $r$. We have a Maxwell fluid for $m<n$, an Oldroyd fluid for $m=n$, and a Kelvin–Voigt fluid for $m>n$ (see [1]). We use high-order models because they have higher accuracy in describing the motion of real media.
For integer ($a_k,b_j \in \mathbb{Z}$) fluid models (1) of high order the solvability and the properties of solutions of the corresponding initial boundary-value problems in classes of sufficiently smooth functions were established in [1]–[3]. We are interested in the fractional Oldroyd fluid model ($m=n$, $a_m=b_m$, $a_m,b_m\in (m,m+1)$, $p_m,q_m>0$) of high order. In this case it follows from (1) that
$$
\begin{equation}
\sigma(t,x)=\mu_0\mathcal{E}(v)(t,x)+\int_0^tG(t-s)\mathcal{E}(v)(s,x)\,ds
\end{equation}
\tag{2}
$$
up to the initial data for $\sigma$ and $\mathcal{E}(v)$, where $\mu_0=p_{m}^{-1}q_m$, $G(s)=s^{\gamma_1-1}G_0(s)$, $\gamma_1=a_m-b_{m-1}<1$, and $G_0(s)$ is a smooth function.
The presence of the integral term in (2) implies long-term memory with respect to the spatial variables. Models taking account of the state of the medium along integral curves of the velocity field $v$ are of great interest as more realistic ones from various points of view (see, for example, [4]). Models of this type of order at most 2 (integer and fractional ones) were studied in [5]–[10]. Substituting the expression for $\sigma(t,x)$ into the equation of motion in the Cauchy form $\partial v/\partial t+\sum_{i=1}^Nv_i\, \partial v/\partial x_i+ \nabla p-\operatorname{Div}\sigma=f$ with allowance for memory along fluid motion trajectories leads to the initial boundary-value problem
$$
\begin{equation}
\frac{\partial v}{\partial t}+ \sum_{i=1}^nv_i\, \frac{\partial v}{\partial x_i}- \mu_0\Delta v-\operatorname{Div}\int_{0}^tG(t-s) \mathcal{E}(v)(s,z(\tau;t,x))\,ds+\nabla p=f,
\end{equation}
\tag{3}
$$
$$
\begin{equation}
z(\tau;t,x)=x+\int_t^\tau v(s,z(s;t,x))\,ds, \qquad 0\leqslant t,\tau\leqslant T, \quad x\in\overline{\Omega},
\end{equation}
\tag{4}
$$
$$
\begin{equation}
\operatorname{div}v(t,x)=0, \qquad (t,x)\in Q_T=[0,T]\times \Omega,
\end{equation}
\tag{5}
$$
$$
\begin{equation}
v(0,x)=v^0(x),\quad x\in \Omega, \qquad v(t,x)=0, \quad (t,x)\in [0,T]\times \partial\Omega
\end{equation}
\tag{6}
$$
(for an $N\times N$ matrix function $A$ with rows $a_i$, $\operatorname{Div}A:=\operatorname{div}a_1,\dots,\operatorname{div}a_N)$).
Below we study the weak solvability of problem (3)–(6) in the space $W_1\equiv \{v\colon v\in L_2(0,T;V)\cap L_{\infty}(0,T;H)$, $v'\in L_1(0,T;V^{-1})\}$. Here $H$ and $V$ are the closures of the set of solenoidal functions $C^\infty_0(\Omega)^N$ with respect to the norms in $L_2(\Omega)^N$ and $W_2^1(\Omega)^N$, respectively (see [11]). In the case when $v\in W_1$, the existence of a classical solution of the Cauchy problem (4) is not guaranteed and its solvability is established in the class of regular Lagrangian flows, which generalize the notion of a classical solution of a system of ODEs. Recall that a regular Lagrangian flow generated by a function $v$ such that $\operatorname{div}v=0$ is a function $z(\tau;t,x)$, $(\tau,t,x)\in [0,T]\times [0,T] \times\overline{\Omega}$, satisfying the following conditions: 1) for almost all $x$ and all $t\in [0,T]$ the function $\gamma(\tau)=z(\tau;t,x)$ is absolutely continuous and satisfies (4) and the condition $z(t;t,x)=x$; 2) $m(z(\tau;t,B))=m(B)$ for all $t,\tau \in[0,T]$; 3) $z(t_3;t_1,x)=z(t_3;t_2,z(t_2;t_1,x))$ for all $t_1,t_2,t_3\in[0, T]$ and almost all $x\in \overline{\Omega}$. Here $B\subset\overline{\Omega}$ is an arbitrary Lebesgue-measurable set and $m$ is the Lebesgue measure. If $v\in L_1(0,T;W_{p}^1(\Omega)^N)$, $1\leqslant p\leqslant \infty$, $\operatorname{div}v(t,x)=0$, and $v(t,x)\big|_{\partial\Omega}=0$, then there exists a unique regular Lagrangian flow $z$ generated by $v$. See, for example, [12] and [13] for facts about regular Lagrangian flows. Definition. A weak solution of the problem (3)–(6) is a function $v\in W_1$ satisfying the identity
$$
\begin{equation*}
\begin{aligned} \, &\frac{d(v,\varphi)}{dt}- \sum_{i=1}^N\biggl(v_iv,\frac{\partial\varphi}{\partial x_i}\biggr)+ \mu_0(\mathcal{E}(v),\mathcal{E}(\varphi)) \\ &\qquad+\biggl(\,\int_{0}^tG(t-s)\mathcal{E}(v)(s,z(s;t,x))\,ds, \mathcal{E}(\varphi)\biggr)=\langle f,\varphi\rangle \end{aligned}
\end{equation*}
\notag
$$
for all $\varphi\in V$ and almost all $t\in[0,T]$ and the initial condition in (6) (here $z(s;t,x)$ is a regular Lagrangian flow generated by $v$). 
Citation in format AMSBIB
\Bibitem{ZvyOrl22}
This publication is cited in the following 1 articles:
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