Long time smooth solutions of 3D cubic quasilinear wave systems with small weakly decaying initial data

1. Problem Statement

The paper addresses the Cauchy problem for 3D cubic quasilinear wave systems with multiple propagation speeds. The system is given by:
$$\square_{c_i} u_i = G_i(u, \partial u, \partial^2 u), \quad (t, x) \in [0, \infty) \times \mathbb{R}^3, \quad i=1, \dots, m$$
where $\square_{c_i} = \partial_t^2 - c_i^2 \Delta$ and the nonlinearities $G_i$ are cubic in terms of $u$, $\partial u$, and $\partial^2 u$.

Key Challenges:

• Weakly Decaying Initial Data: Previous global existence results typically require initial data $(u_0, u_1)$ to be compactly supported or decay rapidly (e.g., with weights like $|x|^2$). This paper investigates data in lower-order weighted Sobolev spaces, specifically where the decay is governed by a weight $\langle x \rangle^\mu = (1+|x|^2)^{\mu/2}$ with $\mu \in (0, 1)$.
• Quasilinear Nature: The nonlinearity depends on the second derivatives $\partial^2 u$, making the system quasilinear. This prevents the direct application of standard semilinear techniques.
• Endpoint Strichartz Failure: In 3D, the endpoint Strichartz estimate ($L^2_t L^\infty_x$) fails for the linear wave equation. This usually obstructs the proof of global existence for cubic nonlinearities without strong null conditions or rapid decay.

2. Methodology

The authors develop a novel analytical framework to overcome the lack of rapid decay and the failure of endpoint estimates. The core methodology involves:

A. New Weighted Estimates based on Strong Huygens' Principle

Instead of relying on the standard Klainerman-Sobolev inequality or KSS (Klainerman-Sideris-Speck) estimates, which fail for $\mu < 1$, the authors establish a series of new weighted $L^\infty - L^2$ and Strichartz estimates.

• Strong Huygens' Principle: They exploit the fact that for 3D linear wave equations, the solution at $(t, x)$ depends only on data on the sphere $|y-x|=t$. This allows them to localize the interaction of waves.
• Localized $L^\infty - L^2$ Estimates: They prove that for functions with support in a ball $B(0, R)$, the decay rate of the solution is $\min\{1, (1+t)^{-1}R\}$ rather than the standard $(1+t)^{-1}R^{3/2}$. This improvement is crucial for handling weakly decaying data.
• Weighted Strichartz Estimates: They derive estimates of the form:
  $$\|(1+s+|x|)^{\frac{1}{p}\beta_1} P_k e^{\pm is|\nabla|} f\|_{L^p_t L^r_x} \lesssim 2^{\dots k} \|\langle x \rangle^{\frac{2}{p}\beta_2} f\|_{L^2}$$
  These estimates allow them to control the cubic nonlinearity $G \sim (\partial u)^3$ by distributing weights effectively across the terms.

B. Continuity Argument and Energy Estimates

• A Priori Estimates: The authors derive uniform energy estimates in high-order Sobolev spaces ($H^N$) and weighted norms.
• Nonlinearity Handling: They decompose the cubic nonlinearity using Littlewood-Paley projections. By carefully balancing the weights $(1+t+|x|)^\nu$ and the frequency localization, they show that the integral of the nonlinearity in the energy inequality remains bounded even for small $\mu$.
• Scattering: To prove global existence, they construct a "free" solution $u_\infty$ (solution to the linear wave equation) and show that the difference $u - u_\infty$ decays to zero in the $H^{N-1}$ norm as $t \to \infty$.

3. Key Contributions

1. Extension to Weakly Decaying Data: The paper significantly extends the lifespan results for 3D cubic quasilinear wave equations from compactly supported data to data with weak polynomial decay ($\langle x \rangle^\mu$ with $\mu \in (0, 1)$).
2. New Analytical Tools: The establishment of weighted Strichartz estimates and localized $L^\infty - L^2$ estimates specifically tailored for 3D wave equations with weak decay. These tools bypass the limitations of the endpoint Strichartz estimate failure.
3. Optimization of Lifespan:
   • General Case: For nonlinearities depending on $u$, the solution exists for an almost global time $T_\varepsilon \geq e^{C\varepsilon^{-1}}$.
   • Special Case: If the nonlinearity is independent of $u$ (i.e., $G=G(\partial u, \partial^2 u)$), the lifespan is improved to $T_\varepsilon \geq e^{C\varepsilon^{-2}}$.
4. Global Existence and Scattering: Under the assumption of small initial data in the weighted space $\sum_{|a|\leq 5} \|\langle x \rangle^\mu \partial^a_x (u_0, u_1)\|_{L^2} \leq \varepsilon$, the authors prove global existence and the scattering property of the solution to a free linear wave.

4. Main Results

Theorem 1.1 (Almost Global Existence):
If the initial data $(u_0, u_1)$ are small in $H^{N+1} \times H^N$ (with $N \geq 6$), then a unique smooth solution exists on $[0, T_\varepsilon]$ where:

• $T_\varepsilon \geq e^{C\varepsilon^{-1}}$ for general $G(u, \partial u, \partial^2 u)$.
• $T_\varepsilon \geq e^{C\varepsilon^{-2}}$ if $G$ is independent of $u$.

Theorem 1.2 (Global Existence and Scattering):
If the initial data satisfy the weighted condition:
$$\|u_0\|_{H^{N+1}} + \|u_1\|_{H^N} + \sum_{|a|\leq 5} \|\langle x \rangle^\mu \partial^a_x (u_0, u_1)\|_{L^2} \leq \varepsilon$$
for some fixed $\mu \in (0, 1)$ and sufficiently small $\varepsilon$, then:

1. The solution $u$ exists globally ($t \in [0, \infty)$).
2. The solution satisfies specific decay estimates: $(1+t)^{\mu-} (\|u\|_{L^\infty} + \|\partial u\|_{L^\infty}) \leq C\varepsilon$.
3. Scattering: There exists a free solution $u_\infty$ such that $\lim_{t\to\infty} \|\partial(u(t) - u_\infty(t))\|_{H^{N-1}} = 0$.

5. Significance

• Bridging the Gap: This work bridges the gap between the well-understood case of compactly supported data and the difficult case of slowly decaying data for quasilinear wave equations.
• Overcoming Dimensional Constraints: In 3D, the failure of the endpoint Strichartz estimate is a major bottleneck. The authors' use of the Strong Huygens' principle to derive weighted estimates provides a robust alternative to the vector field method, which often requires higher decay rates ($\mu \geq 1$).
• Physical Applications: The results apply to various physical models mentioned in the paper, including:
  • 3D relativistic membrane equations.
  • Nonlinear membrane equations.
  • Nonlinear wave equations from Maxwell's equations.
  • Scalar nematic liquid crystal equations.
  • Wave maps systems.
    The paper demonstrates that these systems admit global solutions even when the initial perturbations decay slowly in space, provided they are sufficiently small.

In summary, this paper represents a significant advancement in the theory of nonlinear wave equations by establishing global existence for 3D cubic quasilinear systems with weakly decaying initial data through the innovative use of strong Huygens' principle and new weighted dispersive estimates.