[Paper Review] Pointwise decay in space and in time for incompressible viscous flow around a rigid body moving with constant velocity

Paul Deuring|arXiv (Cornell University)|Sep 5, 2017

Navier-Stokes equation solutions30 references4 citations

TL;DR

This paper establishes pointwise space-time decay estimates for the velocity field in incompressible viscous flows around a rigid body moving at constant velocity, using the Oseen and Navier-Stokes systems with Oseen terms. The key result is a refined decay estimate involving the factor $|x|\nu(x)$, where $\nu(x) = 1 + |x| - x_1$, which captures slower decay in the wake region behind the body, with explicit temporal decay rates depending on data integrability and regularity.

ABSTRACT

We present pointwise space-time decay estimates for the velocity part of solutions to the time-dependent Oseen system in 3D, with Dirichlet boundary conditions and vanishing velocity at infinity. In addition, similar estimates are derived for solutions to the time-dependent incompressible Navier-Stokes system with Oseen term, and for solutions to the stability problem associated with the stationary incompressible Navier-Stokes system with Oseen term.

Motivation & Objective

• To analyze the long-time and large-distance asymptotic behavior of the velocity field in incompressible viscous flows around a rigid body moving with constant velocity.
• To derive pointwise space-time decay estimates for solutions to the time-dependent Oseen and Navier-Stokes systems with Oseen terms in three dimensions.
• To account for the wake effect behind the body by incorporating the factor $\nu(x) = 1 + |x| - x_1$ in the decay estimates, which slows decay along the downstream direction.
• To establish decay rates that depend on the integrability and regularity of the forcing, initial, and boundary data, particularly in the context of the stability problem for the stationary Navier-Stokes system with Oseen term.

Proposed method

• Transform the original Navier-Stokes system with non-zero velocity at infinity into an equivalent system by subtracting the background flow $ (1,0,0) $, leading to a perturbation $ u = v - (1,0,0) $ with zero decay at infinity.
• Introduce a stationary solution $ U $ to the Oseen system to reduce the problem to a stability problem for the perturbation $ u $, where $ u \to 0 $ as $ |x| \to \infty $.
• Use the Oseen kernel and its properties to derive pointwise estimates for the solution $ u $ via integral representations involving the forcing $ f $, initial data $ a $, and boundary data $ b $.
• Apply weighted $ L^p $-estimates and interpolation techniques in time and space, particularly using the weight $ \nu(x) = 1 + |x| - x_1 $, to capture anisotropic decay in the wake region.
• Employ maximal $ L^p $-regularity theory and decay estimates for the Oseen operator to derive bounds on $ \partial_x^\alpha u $, with $ |\alpha| \leq 1 $, in terms of data norms and temporal decay factors.
• Derive decay estimates of the form $ |\partial_x^\alpha u(x,t)| \leq \mathfrak{C} \bigl( |x|\nu(x) \bigr)^{(-1 - |\alpha|/2)(1 - \epsilon)} X(t)^\epsilon $, where $ X(t) $ involves $ L^2 $-norms of $ \nabla_x u $ and temporal decay rates.

Experimental results

Research questions

• RQ1 How does the velocity field decay in space and time for viscous incompressible flow around a rigid body moving at constant velocity?
• RQ2 What is the precise spatial decay behavior in the wake region behind the body, particularly along the positive $ x_1 $-axis?
• RQ3 How do the decay rates of the velocity and its gradient depend on the integrability and regularity of the forcing, initial, and boundary data?
• RQ4 Can pointwise space-time decay estimates be established for the full time-dependent Navier-Stokes system with Oseen term, beyond the linear Oseen case?
• RQ5 What role does the factor $ \nu(x) = 1 + |x| - x_1 $ play in modulating the decay rate, especially in the downstream direction?

Key findings

• The paper establishes pointwise decay estimates for the velocity field $ u $ of the form $ |\partial_x^\alpha u(x,t)| \leq \mathfrak{C} \bigl( |x|\nu(x) \bigr)^{(-1 - |\alpha|/2)(1 - \epsilon)} X(t)^\epsilon $, where $ \nu(x) = 1 + |x| - x_1 $, capturing slower decay in the wake region.
• For $ |\alpha| \leq 1 $, the decay rate in space is modulated by $ (|x|\nu(x))^{-1 - |\alpha|/2} $, with the $ \nu(x) $ factor accounting for reduced decay along the downstream axis.
• Temporal decay rates are derived as $ (1+t)^{-\min\{3/(2q_1)-1, 3\kappa_1(1-1/\hat{q}_1), \kappa_1\}} $, depending on the $ L^{q_1} $-integrability of the forcing and the regularity parameter $ \kappa_1 $.
• When the forcing $ f $ and initial data $ a $ are bounded with compact support, the decay rate improves to $ (1+t)^{-\min\{3/2, \kappa_1\}} $, with $ \kappa_1 > 0 $.
• The estimate holds uniformly for $ x \in \mathbb{R}^3 $ with $ |x| \geq R $, and for all $ t \in (0,\infty) $, ensuring global space-time decay.
• The analysis extends to the nonlinear Navier-Stokes system with Oseen term and the stability problem for the stationary system, showing that the same decay structure persists under suitable data assumptions.