Global solutions of the 3D inhomogeneous incompressible viscoelastic system without structure assumptions

1. Problem Statement

The paper addresses the Cauchy problem for the three-dimensional (3D) inhomogeneous incompressible viscoelastic system on $\mathbb{R}^3$. The system describes the motion of a fluid where the density $\tilde{\rho}$ is not constant, coupled with a deformation gradient tensor $F$. The governing equations are:

$$\begin{cases} \tilde{\rho}_t + u \cdot \nabla \tilde{\rho} = 0, \\ \tilde{\rho}u_t + \tilde{\rho}u \cdot \nabla u + \nabla p = \mu \Delta u + c^2 \text{div}(\tilde{\rho} F F^\top), \\ F_t + u \cdot \nabla F = \nabla u F, \\ \text{div } u = 0, \end{cases}$$

subject to initial data $(\tilde{\rho}_0, u_0, F_0)$ approaching $(1, 0, I)$ as $|x| \to \infty$.

The Core Challenge:
Historically, global existence results for viscoelastic systems (both compressible and inhomogeneous incompressible) relied heavily on specific "div-curl" structural assumptions (e.g., $\text{div}(\tilde{\rho}F^\top)=0$ or specific curl-free conditions on $F$). These structures are often preserved by the flow and are crucial for closing energy estimates.

• Goal: Prove the global existence of strong solutions for the inhomogeneous case without assuming these structural identities on the initial data or the solution.
• Difficulty: In the inhomogeneous case, the density $\tilde{\rho}$ varies, making the velocity equation a parabolic equation with variable coefficients. Consequently, the pressure term $\nabla p$ cannot be eliminated directly via the standard Helmholtz projection operator, complicating the energy estimates significantly compared to the homogeneous case.

2. Methodology

The authors employ a combination of new transformation techniques, spectral analysis, and time-weighted energy methods.

A. Reformulation and Effective Tensors

1. Variable Transformation: They define the density perturbation $\rho = \tilde{\rho} - 1$ and introduce an effective tensor $G$:
   $$G := \tilde{\rho} F F^\top - I$$
   This allows the system to be rewritten as a coupled system for $(u, G)$ and $\rho$.
2. Effective Flux: To handle the coupling between velocity and deformation, they define an effective flux $\mathcal{G} = \Lambda^{-1} \text{div} G$ (where $\Lambda = (-\Delta)^{1/2}$), transforming the system into a form more compatible with standard dissipative analysis.
3. Pressure Handling: A critical innovation is the transformation of the pressure term. Since $\nabla p$ cannot be projected out directly, they introduce a modified pressure $\tilde{p}$ defined by $\nabla \tilde{p} := \frac{1}{\rho+1}\nabla p$. This allows the system to be cast into a form suitable for spectral analysis.

B. Spectral Analysis and Decay Rates

The authors utilize the Fourier transform to analyze the linearized system.

1. Eigenvalue Analysis: They derive the eigenvalues of the linearized operator matrix in Fourier space. The eigenvalues exhibit different behaviors in low-frequency ($S_1$) and high-frequency ($S_2$) regions.
2. Kernel Estimates: By splitting the frequency space, they derive explicit decay estimates for the Green's function kernels ($K_1, K_2, K_3, K_4$).
3. Enhanced Decay: Unlike previous works that relied on $H^{-1}$ conditions for initial data to get decay, this paper utilizes $L^1$ conditions on the initial data ($u_0, G_0$). This allows them to capture an enhanced time-decay rate for the velocity field $u$. Specifically, they prove that $\|u(t)\|_{L^2}$ decays at a rate of $(1+t)^{-3/4}$ (and higher for derivatives), which is faster than standard diffusion rates.

C. Time-Weighted Energy Estimates

To close the a priori estimates, the authors construct a time-weighted energy functional:
$$E_a(t) = \sup_{0 \le t' \le t} (1+t')^\sigma \| (u, G)(t') \|_{H^2}^2$$
where $\sigma \in (1, \sigma_0)$ with $1 < \sigma_0 < 3/2$.

• They combine the standard energy estimates with the spectral decay estimates.
• The $L^1$ condition on initial data ensures that the nonlinear terms can be controlled by the enhanced decay of the linear part, preventing blow-up.
• The uniform bounds on $\rho$ and $F-I$ are established by showing that the integral of the velocity gradient $\int \|\nabla u\| dt$ remains small, which is guaranteed by the enhanced decay rate.

3. Key Contributions

1. Removal of Structural Assumptions: The paper proves global well-posedness for the 3D inhomogeneous incompressible viscoelastic system without requiring the "div-curl" structural identities (1.6)–(1.7) that were previously considered necessary.
2. Relaxation of Initial Data Conditions: The authors replace the restrictive $H^{-1}$ smallness condition on initial data (used in previous works like Zhu et al.) with $L^1$ integrability conditions. Furthermore, the $L^1$ norm $M_0$ does not need to be small, only finite.
3. Novel Transformation for Pressure: They successfully handle the variable-coefficient pressure term in the inhomogeneous case by introducing a modified pressure variable and an effective tensor $G$, overcoming the inability to use the standard Helmholtz projection.
4. Optimal Decay Rates: They establish that the parameter $\sigma$ governing the time-decay weight can be chosen in the range $(1, 3/2)$, which is shown to be optimal for closing the estimates in 3D.

4. Main Results

Theorem 1.1:
Assume the initial data $(\tilde{\rho}_0, u_0, F_0)$ satisfies:

• $\text{div } u_0 = 0$.
• Smallness in $H^2$: $\|u_0\|_{H^2} + \|\tilde{\rho}_0 - 1\|_{H^2} + \|G_0\|_{H^2} \le \delta$ (for sufficiently small $\delta$).
• Finite $L^1$ norm: $\|u_0\|_{L^1} + \|G_0\|_{L^1} = M_0 < \infty$.

Then, the Cauchy problem admits a unique global strong solution $(\tilde{\rho}, u, F)$ satisfying:
$$\sup_{t \ge 0} \left[ \|( \tilde{\rho}-1, F-I )(t)\|_{H^2}^2 + (1+t)^\sigma \|u(t)\|_{H^2}^2 \right] + \int_0^\infty (1+t)^\sigma \|\nabla u(t)\|_{H^2}^2 dt \lesssim \|(u_0, \tilde{\rho}_0-1, F_0-I)\|_{H^2}^2 + M_0^2$$
where $\sigma \in (1, 3/2)$.

5. Significance

• Mathematical Rigor: This work resolves a significant open problem regarding the global existence of solutions for inhomogeneous viscoelastic fluids without relying on artificial structural constraints. It demonstrates that the intrinsic dissipation of the system is sufficient for global stability if the initial data has sufficient integrability ($L^1$).
• Methodological Advancement: The technique of combining spectral analysis (to get decay) with a specific transformation of the pressure and deformation variables provides a robust framework that could be applied to other complex fluid-structure interaction problems or variable-coefficient parabolic systems.
• Physical Relevance: By removing structural assumptions, the results apply to a broader class of physical scenarios where the initial deformation gradient may not satisfy specific topological constraints, making the model more applicable to real-world non-Newtonian fluid dynamics.