[Paper Review] Global solutions of two dimensional incompressible viscoelastic flows with discontinuous initial data

Xianpeng Hu, Fanghua Lin|arXiv (Cornell University)|Dec 24, 2013

Navier-Stokes equation solutions11 references4 citations

TL;DR

This paper establishes the global existence of weak solutions for two-dimensional incompressible viscoelastic flows with discontinuous initial data, under the condition that the initial deformation gradient is close to the identity in $L^2 \cap L^∞$ and the initial velocity is small in $L^2$ and bounded in $L^p$ for some $p>2$. The key innovation lies in proving the regularity of the effective viscous flux, which enables pointwise estimates of the deformation gradient without relying on $L^\infty$ bounds on velocity gradients.

ABSTRACT

The global existence of weak solutions of the incompressible viscoelastic flows in two spatial dimensions has been a long standing open problem, and it is studied in this paper. We show the global existence if the initial deformation gradient is close to the identity matrix in $L^2\cap L^\infty$, and the initial velocity is small in $L^2$ and bounded in $L^p$, for some $p>2$. While the assumption on the initial deformation gradient is automatically satisfied for the classical Oldroyd-B model, the additional assumption on the initial velocity being bounded in $L^p$ for some $p>2$ may due to techniques we employed. The smallness assumption on the $L^2$ norm of the initial velocity is, however, natural for the global well-posedness . One of the key observations in the paper is that the velocity and the extquotedblleft effective viscous flux extquotedblright $\mathcal{G}$ are sufficiently regular for positive time. The regularity of $\mathcal{G}$ leads to a new approach for the pointwise estimate for the deformation gradient without using $L^\infty$ bounds on the velocity gradients in spatial variables.

Motivation & Objective

• To resolve the long-standing open problem of global existence for weak solutions of 2D incompressible viscoelastic flows with discontinuous initial data.
• To establish well-posedness under minimal regularity assumptions consistent with the energy law, particularly for initial data in $L^2$ and $L^\infty$ for the deformation gradient.
• To overcome the breakdown of classical methods when initial data lack smoothness, especially the failure of $L^\infty$ bounds on velocity gradients.
• To develop a new analytical framework based on the regularity of the effective viscous flux to control the deformation gradient in the absence of classical $L^\infty$ control.

Proposed method

• Introduce the effective viscous flux $\mathcal{G}$ as a key quantity to analyze the regularity of velocity and deformation gradient.
• Use a Galerkin approximation scheme to construct approximate solutions $({\bf u}^n, {\mathtt{F}}^n)$ satisfying the viscoelastic system in a weak sense.
• Establish uniform bounds in $L^2$ and $L^\infty$ for the deformation gradient and velocity, leveraging energy estimates and the structure of the Oldroyd-B model.
• Prove strong convergence of $({\mathtt{F}}^n - I)$ in $L^2$ using convexity and weak convergence, ensuring preservation of constraints $\det{\mathtt{F}} = 1$ and $\partial_{x_l} {\mathtt{F}}_{ij} = \partial_{x_l} {\mathtt{F}}_{ik}$ under the limit.
• Employ a matrix factorization technique to represent $\nabla{\bf u} + (\nabla{\bf u})^\top = \mathcal{S}\mathcal{S}^\top$, enabling control of quadratic terms in the energy estimates.
• Use the strong convergence of $\mathtt{F}^n$ and the structure of the stress tensor to pass to the limit and verify that the weak solution satisfies the system in the distributional sense.

Experimental results

Research questions

• RQ1 Can global weak solutions be constructed for 2D incompressible viscoelastic flows when the initial deformation gradient is discontinuous but bounded in $L^2 \cap L^\infty$?
• RQ2 Does the absence of $L^\infty$ bounds on velocity gradients prevent the construction of global solutions, and if so, can this be circumvented?
• RQ3 Can the effective viscous flux $\mathcal{G}$ serve as a regularizing mechanism to control the deformation gradient without relying on $L^\infty$ bounds on velocity gradients?
• RQ4 Is it possible to preserve the structural constraints $\det{\mathtt{F}} = 1$ and the compatibility condition $\mathtt{F}_{lk}\partial_{x_l}\mathtt{F}_{ij} = \mathtt{F}_{lj}\partial_{x_l}\mathtt{F}_{ik}$ under weak convergence of approximate solutions?
• RQ5 What is the minimal regularity required for initial velocity to ensure global well-posedness in the weak sense for the 2D Oldroyd-B model?

Key findings

• Global weak solutions exist for the 2D incompressible viscoelastic system if the initial deformation gradient is close to the identity in $L^2 \cap L^\infty$ and the initial velocity is small in $L^2$ and bounded in $L^p$ for some $p > 2$.
• The effective viscous flux $\mathcal{G}$ is shown to be sufficiently regular for positive time, enabling a new approach to pointwise estimates of the deformation gradient without requiring $L^\infty$ bounds on velocity gradients.
• Strong convergence of $\mathtt{F}^n - I$ in $L^2$ is established for almost every $t > 0$, which ensures the limit solution preserves the structural constraints $\det{\mathtt{F}} = 1$ and the compatibility condition $\mathtt{F}_{lk}\partial_{x_l}\mathtt{F}_{ij} = \mathtt{F}_{lj}\partial_{x_l}\mathtt{F}_{ik}$.
• The energy law is preserved in the limit, with $\frac{1}{2}\|{\bf u}(t)\|_{L^2}^2 + \frac{1}{2}\|{\mathtt{F}}(t)\|_{L^2}^2 + \mu \int_0^t \|\nabla{\bf u}(s)\|_{L^2}^2 ds = \frac{1}{2}\|{\bf u}_0\|_{L^2}^2 + \frac{1}{2}\|{\mathtt{F}}_0\|_{L^2}^2$, confirming energy dissipation.
• The smallness of the initial velocity in $L^2$ is essential for global well-posedness, and the additional $L^p$ boundedness for $p > 2$ is a technical requirement tied to the method, not a fundamental obstruction.
• The result provides the first global existence result for weak solutions of the classical Oldroyd-B model in 2D with discontinuous initial data, filling a critical gap in the theory of viscoelastic fluids.