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Flow reversal of the Stokes system with localized boundary data in the half space

Tongkeun Chang, Kyungkeun Kang, Chanhong Min

TL;DR

This work establishes that localized, time-dependent boundary influxes can generate flow reversal for the unsteady Stokes system in the half-space, by exploiting an explicit Green tensor representation and sharp kernel estimates. A detailed separation-and-reversal framework is developed: separation points arise for certain boundary-data-exponent regimes, and reversal points for tangential and normal velocity components are located with precise asymptotics that depend on the boundary-data parameter $a$ and the observation time. The analysis hinges on decomposing the solution into boundary- and interior-driven kernels, deriving pointwise bounds for the associated integral operators, and tracing how sign changes emerge from dominant algebraic and exponential terms in the kernels. The results illuminate how boundary-layer-like effects near $x_n=0$ couple with nonlocal Stokes dynamics to produce sign reversals, with implications for understanding boundary-layer separation phenomena in viscous flows. Overall, the paper provides a rigorous, quantitative account of flow reversal mechanisms in a linear model, offering precise criteria and asymptotics that could inform more complex, nonlinear settings.

Abstract

We consider the unsteady Stokes system in the half-space with zero initial data and nonzero, space-time localized boundary data. We show that there exist boundary influxes for which the induced flow exhibits flow reversal, in the sense that at least one component of the velocity field changes its sign in the half-space. This phenomenon is demonstrated by a careful analysis of the representation formula for the Stokes system in the half-space, including pointwise estimates, based on the Green tensor with nonzero boundary data. We construct solutions of the Stokes system such that the tangential components of the velocity field exhibit at least one sign change, while the normal component exhibits at least two sign changes. Moreover, the normal component of the constructed velocity field has the opposite sign to the tangential components near the boundary, whereas it has the same sign as the tangential components sufficiently far from the boundary.

Flow reversal of the Stokes system with localized boundary data in the half space

TL;DR

This work establishes that localized, time-dependent boundary influxes can generate flow reversal for the unsteady Stokes system in the half-space, by exploiting an explicit Green tensor representation and sharp kernel estimates. A detailed separation-and-reversal framework is developed: separation points arise for certain boundary-data-exponent regimes, and reversal points for tangential and normal velocity components are located with precise asymptotics that depend on the boundary-data parameter and the observation time. The analysis hinges on decomposing the solution into boundary- and interior-driven kernels, deriving pointwise bounds for the associated integral operators, and tracing how sign changes emerge from dominant algebraic and exponential terms in the kernels. The results illuminate how boundary-layer-like effects near couple with nonlocal Stokes dynamics to produce sign reversals, with implications for understanding boundary-layer separation phenomena in viscous flows. Overall, the paper provides a rigorous, quantitative account of flow reversal mechanisms in a linear model, offering precise criteria and asymptotics that could inform more complex, nonlinear settings.

Abstract

We consider the unsteady Stokes system in the half-space with zero initial data and nonzero, space-time localized boundary data. We show that there exist boundary influxes for which the induced flow exhibits flow reversal, in the sense that at least one component of the velocity field changes its sign in the half-space. This phenomenon is demonstrated by a careful analysis of the representation formula for the Stokes system in the half-space, including pointwise estimates, based on the Green tensor with nonzero boundary data. We construct solutions of the Stokes system such that the tangential components of the velocity field exhibit at least one sign change, while the normal component exhibits at least two sign changes. Moreover, the normal component of the constructed velocity field has the opposite sign to the tangential components near the boundary, whereas it has the same sign as the tangential components sufficiently far from the boundary.
Paper Structure (27 sections, 26 theorems, 397 equations, 1 figure)

This paper contains 27 sections, 26 theorems, 397 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Integral estimates
  5. Signs of velocity components and asymptotics of reversal points for
  6. Tangential components of the velocity field
  7. Estimates of
  8. Asymptotics of and sign of
  9. Normal component of the velocity field
  10. Estimates of
  11. Asymptotics of and sign of
  12. Signs of velocity components and asymptotics of reversal points for
  13. Tangential components of the velocity field
  14. Estimates of
  15. Asymptotics of and sign of
  16. ...and 12 more sections

Key Result

Theorem 1.6

Let $-1<a\leq -\frac{1}{2}$ and $w$ be the solution of the Stokes system StokesRn+ defined by rep-bvp-stokes-w with the boundary data $g=g_n{\bf e}_n$ given by 0502-6-boundarydataspecific. There exists a sufficiently large constant $N>1$ such that if $|x'|>N$, then $(x',0,1)$ is a separation point o

Figures (1)

  • Figure 1: Separation point

Theorems & Definitions (65)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Remark 1.5
  • Theorem 1.6: Existence of separation point
  • Remark 1.7
  • Theorem 1.8: Asymptotics of $x_{kn}^*$
  • Remark 1.9
  • Theorem 1.10: Existence of reversal point
  • ...and 55 more