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Abstract In this work, we establish the convergence of 2D, stationary Navier-Stokes flows with viscosity$$\varepsilon> 0$$,$$(u^\varepsilon , v^\varepsilon )$$to the classical Prandtl boundary layer,$$(\bar {u}_p, \bar {v}_p)$$, posed on the domain$$(0, \infty ) \times (0, \infty )$$:$$ \begin{align*} \| u^\varepsilon - \bar{u}_p \|_{L^\infty_y} \lesssim \sqrt{\varepsilon} \langle x \rangle^{- \frac 1 4 + \delta}, \qquad \| v^\varepsilon - \sqrt{\varepsilon} \bar{v}_p \|_{L^\infty_y} \lesssim \sqrt{\varepsilon} \langle x \rangle^{- \frac 1 2}. \end{align*} $$ This validates Prandtl’s boundary layer theorygloballyin thex-variable for a large class of boundary layers, including the entire one parameter family of the classical Blasius profiles, with sharp decay rates. The result demonstrates asymptotic stability in two senses simultaneously: (1) asymptotic as$$\varepsilon \rightarrow 0$$and (2) asymptotic as$$x \rightarrow \infty $$. In particular, our result provides the first rigorous confirmation for the Navier-Stokes equations that the boundary layer cannot “separate” in these stable regimes, which is very important for physical and engineering applications.
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This content will become publicly available on February 25, 2027
$L^2$-Contraction and Asymptotic Stability of Large Shock for Scalar Viscous Conservation Laws
We investigate $L^2$-contraction and time-asymptotic stability of large shock for scalar viscous conservation laws with polynomial flux. For the flux $f(u) = u^p (2 ≤ p ≤ 4)$ in the regime of its strict convexity, we can prove $L^2$-contraction and time-asymptotic stability of arbitrarily large viscous shock profile in $H^1$-framework by using $$a$$-contraction method with time-dependent shift and suitable weight function, which answers a question in [Blochas and Cheng, arXiv2501.01537, 2025]. Additionally, if the initial perturbation belongs to $L^1$ , then $L^2$ time-asymptotic decay rate $$t^{−1/4}$$ can be obtained.
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- PAR ID:
- 10688270
- Publisher / Repository:
- Global Science Press
- Date Published:
- Journal Name:
- Communications in Mathematical Analysis and Applications
- Volume:
- 5
- Issue:
- 1
- ISSN:
- 2790-1920
- Page Range / eLocation ID:
- 35 to 64
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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