Global solutions of the 2D inhomogeneous incompressible viscoelastic system

1. Problem Statement

The paper investigates the global existence of strong solutions for the two-dimensional (2D) inhomogeneous incompressible viscoelastic system on the whole space $\mathbb{R}^2$. The system is described by the following equations:

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

where $\tilde{\rho}$ is the fluid density, $u$ is the velocity field, $F$ is the deformation gradient, $p$ is the pressure, and $\mu, c$ are positive constants. The initial data approaches $(1, 0, I)$ as $|x| \to \infty$.

Key Challenges:

• Inhomogeneity: The density $\tilde{\rho}$ is not constant, making the velocity equation a parabolic equation with variable coefficients. This prevents the direct elimination of the pressure term $\nabla p$ via the standard Helmholtz projection operator.
• Absence of "Div-Curl" Structure: Previous global existence results for viscoelastic fluids often rely on specific structural identities (the "div-curl" structure, e.g., $\text{div}(\tilde{\rho}F^\top)=0$ or specific curl relations) to control nonlinear terms. This paper aims to prove global well-posedness without relying on these structural identities.
• Dimensionality and Criticality: While methods exist for the 3D case (using time-weighted energies), they cannot be directly applied to the 2D case due to the criticality of the time-weight. The decay rates and integrability conditions in 2D are more restrictive.

2. Methodology

The authors develop a novel analytical framework combining several advanced techniques:

A. Reformulation via Effective Tensor

To handle the coupling between density and deformation, the authors introduce an effective tensor $G$:
$$G := \tilde{\rho} F F^\top - I$$
This transforms the original system into a new dissipative system (Equation 2.4) where the evolution of $G$ is coupled with the velocity $u$. This reformulation helps isolate the dissipative mechanisms.

B. Transformation of the Pressure Term

Since $\tilde{\rho}$ is variable, the term $\frac{1}{\tilde{\rho}}\nabla p$ cannot be projected out directly. The authors define a new pressure variable $\tilde{p}$ such that:
$$\nabla \tilde{p} := \frac{1}{\tilde{\rho} + 1} \nabla p$$
This transformation allows the pressure term to be treated within the energy estimates without requiring the Helmholtz projection to eliminate it entirely, overcoming a major obstacle in the inhomogeneous case.

C. Fractional Time-Weighted Energy Framework

The core of the proof lies in constructing a fractional time-weighted energy framework. The authors define three distinct energy functionals:

1. Basic Energy ($E_0$): Controls low-frequency and high-frequency norms without time weights.
2. Fractional Time-Weighted Energy ($E_1$): Controls higher-order derivatives with a time weight $(1+t)^{1+2s_1-s_0}$. This is crucial for capturing the decay of solutions.
3. Assistant Energy ($E_a$): Controls the density $\rho$ and deformation $F-I$ in specific Sobolev spaces.

Key Innovation in Weights:
The authors carefully select fractional indices $s_0$ and $s_1$ satisfying specific bounds (e.g., $\frac{\sqrt{7}-1}{3} < s_1 < 1$ and specific relations between $s_0$ and $s_1$). This selection ensures that the time-weighted integrals converge (i.e., $\int_0^\infty (1+t)^{-\gamma} dt < \infty$) and allows them to close the energy estimates for the "nonlinear terms" that usually cause blow-up.

D. Commutator and Bilinear Estimates

To handle the nonlinear terms arising from the variable density and the effective tensor, the authors employ:

• Commutator estimates: Specifically for fractional derivatives involving the transport term $u \cdot \nabla$.
• New Bilinear Estimates: A new bilinear estimate (Lemma A.5) is derived to control terms like $\|g(\rho) \text{div} G\|$. This extends previous results from homogeneous cases to the inhomogeneous setting.

3. Key Contributions

1. First Global Result without "Div-Curl" in 2D Inhomogeneous Case: This is the first work to establish the global well-posedness of the 2D inhomogeneous incompressible viscoelastic system in Sobolev spaces without utilizing the "div-curl" structural identities.
2. Overcoming the 2D Criticality: The paper resolves the difficulty of extending 3D time-weighted methods to 2D by introducing a refined selection of fractional time-weight indices ($s_0, s_1$) that satisfy the critical integrability conditions unique to 2D.
3. Handling Variable Density: The introduction of the effective tensor $G$ and the transformed pressure $\tilde{p}$ provides a robust mechanism to handle the variable coefficient in the momentum equation, which is a significant advancement over methods that assume constant density.
4. New Analytical Tools: The derivation of Lemma A.5 (a specific bilinear estimate for fractional negative derivatives) and the construction of the assistant energy $E_a(t)$ to bound the density in negative Sobolev spaces are novel technical contributions.

4. Main Results

Theorem 1.1:
Under the assumption that the initial data $(\tilde{\rho}_0, u_0, F_0)$ is sufficiently small in specific Sobolev norms (specifically involving $\dot{H}^{-s_1} \cap \dot{H}^2$ for velocity and deformation, and $H^2 \cap \dot{H}^{s_1-s_0}$ for density), the Cauchy problem admits a unique global strong solution $(\tilde{\rho}, u, F)$.

The solution satisfies uniform bounds on the total energy:
$$\sup_{0 \le t < \infty} \left( \|u\|_{\dot{H}^{-s_1} \cap \dot{H}^2}^2 + \|\tilde{\rho}-1\|_{H^2 \cap \dot{H}^{s_1-s_0}}^2 + \|F-I\|_{H^2}^2 + \|G\|_{\dot{H}^{-s_1} \cap \dot{H}^2}^2 \right) + \int_0^\infty (1+t)^{1+2s_1-s_0} \|\nabla u\|_{\dot{H}^{1+s_1-s_0} \cap \dot{H}^2}^2 dt \le \epsilon$$

5. Significance

• Theoretical Advancement: This work bridges a critical gap in the mathematical theory of non-Newtonian fluids. It demonstrates that global existence can be achieved for inhomogeneous viscoelastic flows in 2D without relying on the restrictive "div-curl" structures that were previously thought necessary.
• Methodological Impact: The "fractional time-weighted energy framework" developed here offers a new template for analyzing other 2D fluid systems with variable coefficients or lacking standard structural cancellations.
• Physical Relevance: By proving global existence for inhomogeneous fluids (where density varies), the results are more applicable to real-world industrial and biological fluids (e.g., polymer solutions, biofluids) where density variations are non-negligible, compared to previous results restricted to constant density models.

In summary, the paper successfully tackles the 2D inhomogeneous viscoelastic problem by combining a clever reformulation of the system with a highly refined, fractional time-weighted energy analysis, overcoming the criticality issues inherent in two dimensions.