Singularity formation for the general Poiseuille flow of nematic liquid crystals

The state of a nematic liquid crystal is characterized by its velocity field ${\bf u}$ for the flow and its director field ${\bf n}\in\mathbb{S}^{2}$ for the alignment of the rod-like feature. These two characteristics interact with each other so that any distortion of the director ${\bf n}$ causes a motion ${\bf u}$ and, likewise, any flow ${\bf u}$ affects the alignment ${\bf n}$. One famous model on nematic liquid crystal is the Ericksen-Leslie model for nematics that was first proposed by Ericksen [11] and Leslie [15] in the 1960’s.

In this paper, we consider the Poiseulle flow via the full Ericksen-Leslie model, when ${\bf u}$ and ${\bf n}$ take the form

where $(x,t)\in{\mathbb{R}}\times{\mathbb{R}}^{+}$ and the motion ${\bf u}$ is along the $z$-axis and the director ${\bf n}$ lies in the $(x,z)$-plane with angle $\theta$ made from the $z$-axis. Then the Ericksen-Leslie system can be written as

where the $C^{\infty}$ functions $c,$ $g,$ and $h$ are explicitly given by

Here, $K_{1}$, $K_{3}$ are positive elastic constants in the Oseen-Frank energy. The material coefficients $\gamma_{1}$ and $\gamma_{2}$ reflect the molecular shape and the slippery part between fluid and particles. The coefficients $\alpha_{i}^{\prime}s$ and coefficients $\gamma_{1}$, $\gamma_{2}$ satisfy the following physical relations:

The first two relations are compatibility conditions, while the third relation is called Parodi’s relation, derived from Onsager reciprocal relations expressing the equality of certain relations between flows and forces in thermodynamic systems out of equilibrium (cf. [17]). The coefficients also satisfy the following empirical relations (p.13, [16])

A detailed derivation of the Ericksen-Leslie system for Poiseulle flows can be found in [7], where without loss of generality, we choose density $\rho$ to be $1$, and the inertial coefficient $\nu$ of the director ${\bf n}$ to be $1$. Furthermore, for simplicity, the constant $a$ in the model in [7], which is the gradient of pressure along the flow direction, is set to be zero.

Due to the quasilinearity of the wave equation on $\theta$, finite time gradient blowup might happen even when the initial data are smooth. In [7], Chen, Huang and Liu established an example showing such kind of singularity formation phenomena, for a special case of (1.1)-(1.2) when $g=h=1$ and $\gamma_{1}=2$.

The construction of the singularity formation example relies on the framework in [13] by Glassy-Hunter-Zheng on the variational wave equation

The global well-posedness theories of Hölder continuous solutions for variational wave equations and systems related to nematic liquid crystals have been intensively studied in the last two decades [1, 2, 3, 4, 5, 6, 9, 10, 14, 18, 19, 20].

The proof of finite time singularity formation in [7] for the special case of (1.1)-(1.2) also relies on a crucial estimate on the $L^{\infty}$ bound of a new function $J$, where in the general case,

We note that functions $u_{x}$ and $\theta_{t}$ both blow up when singularity forms, however, their combination $J$ will be proved uniformly bounded. This estimate, first obtained in [7] for the special case (when $g=h=1$ and $\gamma_{1}=2$), is crucial for both singularity formation and global existence of Hölder continuous solution for the system (1.1)-(1.2).

Using (1.7), the wave equation (1.2) can be written as

From (1.3)-(1.5), we have the following bounds for $g$, $h$ and $c$

where $g_{L},g_{U},h_{L},h_{U},C_{L}$ and $C_{U}$ are constants such that $g_{L},g_{U},h_{U},C_{L}$ and $C_{U}$ are strictly positive. Furthermore, physical laws in (1.4) and (1.5) give that

for some positive constant $\overline{C}$, where the proof can be found in [7].

In this paper, we will first find a bound on $J$ in terms of the initial energy. So (1.8) is a damped variational wave equation adding a uniform bounded source term $hJ$. Then we can prove the singularity formation of cusp singularity using the methods in [13, 10, 7]. Note the source term $hu_{x}$ in the original equation (1.2) may be unbounded.

In another companion paper [8], we will show that $J$ is bounded under the norm of $L^{2}\cap L^{\infty}\cap C^{\alpha}$ for some positive constant $\alpha$, even for weak solutions including singularities. This is one of the key estimates for the global existence proof.

Except showing one example forming the cusp singularity, another motivation of this paper is to introduce the major mathematical idea why we can get better regularity on $J$ than $u_{x}$ or $\theta_{t}$, in the general case. For smooth solutions, we can explain the idea in a relatively easier manner than for weak solutions.

Now, let’s expain the main difficulty in controlling $J$ for the general case comparing to the special case. For the special case considered in [7] (when $g=h=1$ and $\gamma_{1}=2$), (1.1) is

so $u$ can be solved directly by $\theta_{tx}$. However, this is not true for the general case. The main difficulty we need to overcome is the varying coefficient $g(\theta)$ in the heat equation (1.1). Although $g(\theta)$ is strictly positive and uniformly bounded, it is only Hölder continuous on $x$ and $t$ at the blowup. Therefore, the derivatives of $g(\theta)$ on $x$ and $t$ blow up when singularity forms. This creates a lot of essential problems for finding a uniform bound on $J$, since we need to use both heat and wave equations to bound $J$. One of our key ideas is to consider the potential $A$ of $J$, with $A_{x}=J$.

Before we state the main result we define the following function $\phi(x)$ with $x\in\mathbb{R}$, that is used to design the initial data. Take $\phi\in C^{1}$ with

and

where $\theta^{*}$ is a constant such that $c^{\prime}(\theta^{*})>0$ and $C_{2},k_{0}$ are some positive constants.

Here is the main singularity formation result.

In the proof of the theorem we will show that the singularity happens in the following form: there exists a time $0<t_{0}<T$ such that

as $t\to t_{0}^{-}$ along the characteristic having the blowup phenomena. In this example, the initial energy is small, and initial data are smooth, but finite time singularity still forms. The singularity is a cusp type of singularity, combining with the existence result in [8], also see [7, 13].

The rest of the paper is divided into 3 sections as follows. In Section 2 we show the decay of the energy associated with a smooth solution. Section 3 contains the main estimate on the quantity $J$: we show the uniform bound of $J$ over a fixed time interval. In Section 4, we prove the singularity formation result in Theorem 1.

The energy of the system (1.1)-(1.2) is defined as

In this section we show the energy decay for smooth solutions.

Therefore, considering only smooth solutions, for any $T>0$, we have

For convenience, we just fix $T_{0}=1$ and give a uniform upper bound on $J$, defined in (1.7), when $0\leq t\leq T_{0}$, when solution is smooth. In fact, one can choose $T_{0}$ to be any positive constant, and still prove that $J$ is bounded.

Before our calculation, we recall the bounds in (1.9) and (1.10), and also note that $|h^{\prime}|$ and $|g^{\prime}|$ are bounded above by a constant because of (1.3).