[Paper Review] Global solutions to the $n$-dimensional incompressible Oldroyd-B model without damping mechanism

Xiaoping Zhai|arXiv (Cornell University)|Oct 18, 2018

Navier-Stokes equation solutions18 references4 citations

TL;DR

This paper establishes the existence of global classical solutions to the n-dimensional incompressible Oldroyd-B model without a damping mechanism in $\mathbb{R}^n$ for $n=2,3$, using a decomposition of initial data into low and high frequencies in critical Besov spaces. The key result extends previous work by Zhu and Chen-Hao by allowing highly oscillatory initial velocity data under smallness conditions in mixed-norm Besov spaces, leveraging a continuous argument and energy estimates in a frequency-localization framework.

ABSTRACT

The present work is dedicated to the global solutions to the incompressible Oldroyd-B model without damping on the stress tensor in $\mathbb{R}^n(n=2,3)$. This result allows to construct global solutions for a class of highly oscillating initial velocity. The proof uses the special structure of the system. Moreover, our theorem extends the previous result by Zhu [19] and covers the recent result by Chen and Hao [4].

Motivation & Objective

• To establish global existence of classical solutions for the incompressible Oldroyd-B model in $\mathbb{R}^n$ ($n=2,3$) without a damping mechanism on the stress tensor.
• To extend previous results by Zhu [19] and Chen-Hao [4] by allowing highly oscillatory initial velocity data in the critical Besov space framework.
• To overcome the lack of damping in the stress equation by exploiting the special structure of the Oldroyd-B system and frequency localization techniques.
• To prove global well-posedness under smallness conditions on initial data in mixed-norm Besov spaces, including low-frequency data in $\dot{B}_{2,1}^{\frac{n}{2}-1}$ and high-frequency data in $\dot{B}_{p,1}^{\frac{n}{p}-1}$ and $\dot{B}_{p,1}^{\frac{n}{p}}$ for $2 \leq p \leq \min(4, \frac{2n}{n-2})$.

Proposed method

• Decompose initial data into low-frequency ($u_0^\ell, \tau_0^\ell$) and high-frequency ($u_0^h, \tau_0^h$) components in frequency space.
• Apply the frequency localization technique using Littlewood-Paley decomposition to analyze the system in critical Besov spaces.
• Use a continuous argument based on a time-dependent norm $X(t)$ that tracks the evolution of low- and high-frequency components.
• Establish energy estimates for the velocity and stress tensor in the framework of homogeneous Besov spaces, particularly focusing on $\widetilde{L}_t^\infty(\dot{B}^{\frac{n}{p}-1}_{p,1})$ and $L_t^1(\dot{B}^{\frac{n}{p}+1}_{p,1})$ norms.
• Employ the projector $\mathbb{P} = \mathcal{I} - \nabla \Delta^{-1} \mathrm{div}$ to handle the pressure term and maintain incompressibility.
• Apply Gronwall's inequality to the resulting integral inequality to close the a priori estimate and ensure global existence.

Experimental results

Research questions

• RQ1 Can global classical solutions be established for the incompressible Oldroyd-B model in $\mathbb{R}^n$ ($n=2,3$) without a damping mechanism on the stress tensor?
• RQ2 What is the minimal regularity and smallness condition on initial data that guarantees global existence in the absence of damping?
• RQ3 Can the framework of critical Besov spaces accommodate highly oscillatory initial velocity data in the global well-posedness theory for the Oldroyd-B model?
• RQ4 How does the frequency decomposition of initial data in mixed Besov norms affect the long-time behavior of solutions?
• RQ5 To what extent can the results of Zhu [19] and Chen-Hao [4] be unified and extended under a single smallness condition in mixed-norm Besov spaces?

Key findings

• The system admits a unique global classical solution for initial data satisfying $\|(u_0^\ell, \tau_0^\ell)\|_{\dot{B}_{2,1}^{\frac{n}{2}-1}} + \|u_0^h\|_{\dot{B}_{p,1}^{\frac{n}{p}-1}} + \|\tau_0^h\|_{\dot{B}_{p,1}^{\frac{n}{p}}} \leq c_0$ for $2 \leq p \leq \min(4, \frac{2n}{n-2})$ and $p \neq 4$ when $n=2$.
• The result covers the case of highly oscillatory initial velocity, which previous works relying on Sobolev or $H^s$ spaces could not handle effectively.
• The proof establishes global existence via a continuous argument in time, using a time-dependent norm $X(t)$ that controls both low- and high-frequency components of the solution.
• The method avoids the need for damping ($\beta = 0$) by exploiting the special structure of the Oldroyd-B system and careful energy estimates in Besov spaces.
• The result extends Zhu's [19] work on $\mathbb{R}^3$ to $\mathbb{R}^2$ and generalizes Chen and Hao's [4] result to a broader class of initial data in mixed Besov norms.
• The analysis confirms that the absence of damping does not prevent global well-posedness as long as the initial data are sufficiently small in the specified mixed-norm Besov framework.