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Abstract Letfbe an$$L^2$$-normalized holomorphic newform of weightkon$$\Gamma _0(N) \backslash \mathbb {H}$$withNsquarefree or, more generally, on any hyperbolic surface$$\Gamma \backslash \mathbb {H}$$attached to an Eichler order of squarefree level in an indefinite quaternion algebra over$$\mathbb {Q}$$. Denote byVthe hyperbolic volume of said surface. We prove the sup-norm estimate$$\begin{align*}\| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_{\varepsilon} (k V)^{\frac{1}{4}+\varepsilon} \end{align*}$$ with absolute implied constant. For a cuspidal Maaß newform$$\varphi $$of eigenvalue$$\lambda $$on such a surface, we prove that$$\begin{align*}\|\varphi \|_{\infty} \ll_{\lambda,\varepsilon} V^{\frac{1}{4}+\varepsilon}. \end{align*}$$ We establish analogous estimates in the setting of definite quaternion algebras.
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This content will become publicly available on April 17, 2027
Global inviscid limit of 2D, stationary Navier-Stokes and stability of Prandtl expansions
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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- PAR ID:
- 10690761
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Forum of Mathematics, Pi
- Volume:
- 14
- ISSN:
- 2050-5086
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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