Global solution of 2D hyperbolic liquid crystal system for small initial data

1. Problem Statement

The paper addresses the global regularity problem for the two-dimensional (2D) simplified Ericksen-Leslie hyperbolic liquid crystal model with small initial data. The system couples the incompressible Navier-Stokes equations (for fluid velocity $u$) with a wave map equation (for the director field $d$).

The specific system studied is:
$$\begin{cases} \partial_t u + u \cdot \nabla u + \nabla p = \Delta u - \nabla \cdot (\nabla d \otimes \nabla d), \\ \nabla \cdot u = 0, \\ D_t^2 d - \Delta d = (-|D_t d|^2 + |\nabla_x d|^2)d, \end{cases}$$
where $D_t = \partial_t + u \cdot \nabla_x$ and $d \in S^1$. By parametrizing $d = (\cos \phi, \sin \phi)$, the system reduces to a coupled system of a heat-type equation for velocity $u$ and a wave equation for the phase $\phi$.

The Core Challenge:
In 2D, the optimal decay rate for solutions to the linear wave equation is $t^{-1/2}$, whereas in 3D it is $t^{-1}$. This slower decay in 2D makes it difficult to control the quadratic nonlinearities (specifically the self-interaction of the wave component) using standard energy methods. Previous work by Huang-Jiang-Zhao [15] established almost global existence (solutions exist for time $T \sim \exp(1/\epsilon)$) using Alinhac's ghost weight method but could not prove global existence or scattering. The question of whether solutions blow up in finite time or exist globally remained open.

2. Methodology

The author employs a sophisticated combination of Fourier analysis and the vector field method, moving away from the physical space approach used in previous 2D studies.

A. Discovery of a Novel Null Structure

The central breakthrough is the identification of a "null structure for wave $\times$ wave $\to$ heat" interactions within the velocity equation.

• Mechanism: The pressure term $\nabla p$ and the stress term $\nabla \cdot (\nabla \phi \otimes \nabla \phi)$ interact in a way that creates a cancellation.
• Symbol Analysis: In Fourier space, the symbol of the bilinear operator governing the wave-wave interaction in the velocity equation vanishes to a specific order when the frequencies resonate (i.e., when the phase function is stationary). This vanishing compensates for the insufficient $t^{-1/2}$ decay rate in 2D.
• Decomposition: The velocity $u$ is decomposed via a normal form transformation into:
  1. A "heat part" $v$ (which behaves like a free heat solution).
  2. A "wave-type quadratic self-interaction" term $A_{\mu,\nu}(U_\mu, U_\nu)$, which is decoupled from the heat flow.

B. Normal Form Transformation

The author defines a new variable $v = u + \sum A_{\mu,\nu}(U_\mu, U_\nu)$, where $U$ represents the half-wave components of $\phi$.

• This transformation eliminates the dangerous quadratic wave-wave self-interaction from the equation for $v$.
• The resulting equation for $v$ is more amenable to analysis because the nonlinearity no longer contains the resonant quadratic terms that cause growth.
• Crucially, the symbol of the normal form transformation is non-singular because the null structure ensures the numerator vanishes at the same order as the denominator's potential singularity.

C. Analytical Tools

• Super-localized Decay Estimates: To handle High $\times$ High $\to$ Low frequency interactions (which typically cause summation issues in 2D), the author uses super-localized cutoff functions in Fourier space, inspired by Ionescu-Pausader's work on Einstein-Klein-Gordon systems.
• Z-norm Method: The proof utilizes the $Z$-norm (a weighted $L^\infty$ norm in Fourier space) to capture the sharp decay rates of the nonlinear solution.
• Modified Energy Method: To control the energy of the wave part without losing derivatives, the author constructs a modified energy functional that includes cubic correction terms to cancel out quasilinear terms.

3. Key Contributions

1. Global Existence: The paper proves that for sufficiently small initial data, the 2D hyperbolic liquid crystal system admits a unique global solution. This improves upon the "almost global" result of Huang-Jiang-Zhao [15].
2. Scattering: The nonlinear wave solution $\phi(t)$ scatters to a linear solution $\phi_\infty(t)$ as $t \to \infty$.
3. Sharp Decay Estimates: The author establishes sharp $L^\infty_x$ decay estimates for both the heat part and the wave part, matching the decay rates of the corresponding linear equations:
   • Velocity $u$: $\|u(t)\|_{L^\infty} \lesssim \langle t \rangle^{-1}$.
   • Wave component $\phi$: $\|\nabla_{t,x}\phi(t)\|_{L^\infty} \lesssim \langle t \rangle^{-1/2}$.
4. Novel Null Structure: The discovery of the specific null structure in the velocity equation that facilitates the $w \times w \to h$ (wave $\times$ wave $\to$ heat) interaction is a significant theoretical contribution, potentially applicable to other hyperbolic liquid crystal models.

4. Main Results

Theorem 1.1: Let $N_0 = 105$. There exists a small constant $\epsilon_0$ such that if the initial data $(u_0, \phi_0, \phi_1)$ satisfies a specific smallness condition in weighted Sobolev spaces (involving $H^{N_0}$ norms and weighted $L^1$ norms), then:

• The system has a unique global solution.
• The solution scatters to a linear state.
• The following sharp decay estimates hold:
  $$\sum_{|\alpha| \leq 3} \langle t \rangle^{1/2} \|\nabla_x^\alpha \nabla_{t,x} \phi\|_{L^\infty_x} + \langle t \rangle \|\nabla_x^\alpha u(t)\|_{L^\infty} \lesssim \epsilon_0.$$
• The energy of the heat component decays at a rate of $t^{-1/2+\delta}$, which is a stronger result than the energy growth allowed in previous works.

5. Significance

• Resolution of a Long-standing Open Problem: This work resolves the question of global regularity for the 2D simplified Ericksen-Leslie system, a problem that had remained open despite extensive research on nonlinear wave and heat equations.
• Overcoming 2D Decay Limitations: It demonstrates that global existence in 2D is possible even with the slow $t^{-1/2}$ decay of waves, provided a specific null structure exists to handle the resonant interactions.
• Methodological Advancement: The successful integration of the normal form transformation (typically used for dispersive equations) with the heat equation and the use of super-localized estimates in a coupled hyperbolic-parabolic system provides a new blueprint for tackling similar problems in mathematical physics.
• Physical Relevance: The results provide a rigorous mathematical foundation for the long-term stability of liquid crystal flows in 2D, confirming that small perturbations do not lead to finite-time blow-up.