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Sameer Iyer

Publications and source records attributed to Sameer Iyer.

At least 19 recordsLinked to original sources
arXiv2025

Linear decay of the beta-plane equation near Couette flow on the plane

We prove new time decay estimates for the linearized $\beta$-plane equation near the Couette flow on the plane that combine inviscid damping and the dispersion of Rossby waves. Specifically, we show that the profiles of the velocity field components (i.e. $u(t,x+ty,y)$) decay pointwise on any compact set with polynomial rates. While mixing dominates for streamwise frequencies that are $O(1)$, dispersive effects need to be extracted for low streamwise frequencies that appear along a critical ray in frequency space. Our proof entails the analysis of oscillatory integrals with homogeneous phase and multipliers that diverge in the infinite time limit. To handle this singular limit, we prove a Van der Corput type estimate, followed by two delicate asymptotic analyses of the phase and multipliers: one that is of ``boundary layer" type, featuring sharp gradients that grow in $t$ across the critical ray, and one that is of ``multi-scale" type, which extracts a governing analytic profile function for the phase.

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arXiv2025

Stability threshold of close-to-Couette shear flows with no-slip boundary conditions in 2D

In this paper, we develop a stability threshold theorem for the 2D incompressible Navier-Stokes equations on the channel, supplemented with the no-slip boundary condition. The initial datum is close to the Couette flow in the following sense: the shear component of the perturbation is small, but independent of the viscosity $\nu$. On the other hand, the $x$-dependent fluctuation is assumed small in a viscosity-dependent sense, namely, $O(\nu^{\frac12}|\log \nu|^{-2})$. Under this setup, we prove nonlinear enhanced dissipation of the vorticity and a time-integrated inviscid damping for the velocity. These stabilizing phenomena guarantee that the Navier-Stokes solution stays close to an evolving shear flow for all time. The analytical challenge stems from a time-dependent nonlocal term that appears in the associated linearized Navier-Stokes equations.

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arXiv2024

Pseudo-Gevrey Smoothing for the Passive Scalar Equations near Couette

In this article, we study the regularity theory for two linear equations that are important in fluid dynamics: the passive scalar equation for (time-varying) shear flows close to Couette in $\mathbb T \times [-1,1]$ with vanishing diffusivity $\nu \to 0$ and the Poisson equation with right-hand side behaving in similar function spaces to such a passive scalar. The primary motivation for this work is to develop some of the main technical tools required for our treatment of the (nonlinear) 2D Navier-Stokes equations, carried out in our companion work. Both equations are studied with homogeneous Dirichlet conditions (the analogue of a Navier slip-type boundary condition) and the initial condition is taken to be compactly supported away from the walls. We develop smoothing estimates with the following three features: [1] Uniform-in-$\nu$ regularity is with respect to $\partial_x$ and a time-dependent adapted vector-field $\Gamma$ which approximately commutes with the passive scalar equation (as opposed to `flat' derivatives), and a scaled gradient $\sqrt{\nu} \nabla$; [2] $(\partial_x, \Gamma)$-regularity estimates are performed in Gevrey spaces with regularity that depends on the spatial coordinate, $y$ (what we refer to as `pseudo-Gevrey'); [3] The regularity of these pseudo-Gevrey spaces degenerates to finite regularity near the center of the channel and hence standard Gevrey product rules and other amenable properties do not hold. Nonlinear analysis in such a delicate functional setting is one of the key ingredients to our companion paper, \cite{BHIW24a}, which proves the full nonlinear asymptotic stability of the Couette flow with slip boundary conditions. The present article introduces new estimates for the associated linear problems in these degenerate pseudo-Gevrey spaces, which is of independent interest.

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arXiv2024

Uniform Inviscid Damping and Inviscid Limit of the 2D Navier-Stokes equation with Navier Boundary Conditions

We consider the 2D, incompressible Navier-Stokes equations near the Couette flow, $\omega^{(NS)} = 1 + \epsilon \omega$, set on the channel $\mathbb{T} \times [-1, 1]$, supplemented with Navier boundary conditions on the perturbation, $\omega|_{y = \pm 1} = 0$. We are simultaneously interested in two asymptotic regimes that are classical in hydrodynamic stability: the long time, $t \rightarrow \infty$, stability of background shear flows, and the inviscid limit, $\nu \rightarrow 0$ in the presence of boundaries. Given small ($\epsilon \ll 1$, but independent of $\nu$) Gevrey 2- datum, $\omega_0^{(\nu)}(x, y)$, that is supported away from the boundaries $y = \pm 1$, we prove the following results: \begin{align*} & \|\omega^{(\nu)}(t) - \frac{1}{2\pi}\int \omega^{(\nu)}(t) dx \|_{L^2} \lesssim \epsilon e^{-\delta \nu^{1/3} t}, & \text{(Enhanced Dissipation)} \\ & \langle t \rangle \|u_1^{(\nu)}(t) - \frac{1}{2\pi} \int u_1^{(\nu)}(t) dx\|_{L^2} + \langle t \rangle^2 \|u_2^{(\nu)}(t)\|_{L^2} \lesssim \epsilon e^{-\delta \nu^{1/3} t}, & \text{(Inviscid Damping)} \\ &\| \omega^{(\nu)} - \omega^{(0)} \|_{L^\infty} \lesssim \epsilon \nu t^{3+\eta}, \quad\quad t \lesssim \nu^{-1/(3+\eta)} & \text{(Long-time Inviscid Limit)} \end{align*} This is the first nonlinear asymptotic stability result of its type, which combines three important physical phenomena at the nonlinear level: inviscid damping, enhanced dissipation, and long-time inviscid limit in the presence of boundaries. The techniques we develop represent a major departure from prior works on nonlinear inviscid damping as physical space techniques necessarily play a central role. In this paper, we focus on the primary nonlinear result, while tools for handling the linearized parabolic and elliptic equations are developed in our separate, companion work.

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arXiv2024

Local Rigidity of the Couette Flow for the Stationary Triple-Deck Equations

The Triple-Deck equations are a classical boundary layer model which describes the asymptotics of a viscous flow near the separation point, and the Couette flow is an exact stationary solution to the Triple-Deck equations. In this paper we prove the local rigidity of the Couette flow in the sense that there are no other stationary solutions near the Couette flow in a scale invariant space. This provides a stark contrast to the well-studied stationary Prandtl counterpart, and in particular offers a first result towards the rigidity question raised by R. E. Meyer in 1983.

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arXiv2024

Stability of the Favorable Falkner-Skan Profiles for the Stationary Prandtl Equations

The (favorable) Falkner-Skan boundary layer profiles are a one parameter ($\beta \in [0,2]$) family of self-similar solutions to the stationary Prandtl system which describes the flow over a wedge with angle $\beta \frac{\pi}{2}$. The most famous member of this family is the endpoint Blasius profile, $\beta = 0$, which exhibits pressureless flow over a flat plate. In contrast, the $\beta > 0$ profiles are physically expected to exhibit a \textit{favorable pressure gradient}, a common adage in the physics literature. In this work, we prove quantitative scattering estimates as $x \rightarrow \infty$ which precisely captures the effect of this favorable gradient through the presence of new ``CK" (Cauchy-Kovalevskaya) terms that appear in a quasilinear energy cascade.

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arXiv2023

Stability threshold of nearly-Couette shear flows with Navier boundary conditions in 2D

In this work, we prove a threshold theorem for the 2D Navier-Stokes equations posed on the periodic channel, $\mathbb{T} \times [-1,1]$, supplemented with Navier boundary conditions $\omega|_{y = \pm 1} = 0$. Initial datum is taken to be a perturbation of Couette in the following sense: the shear component of the perturbation is assumed small (in an appropriate Sobolev space) but importantly is independent of $\nu$. On the other hand, the nonzero modes are assumed size $O(\nu^{\frac12})$ in an anisotropic Sobolev space. For such datum, we prove nonlinear enhanced dissipation and inviscid damping for the resulting solution. The principal innovation is to capture quantitatively the \textit{inviscid damping}, for which we introduce a new Singular Integral Operator which is a physical space analogue of the usual Fourier multipliers which are used to prove damping. We then include this SIO in the context of a nonlinear hypocoercivity framework.

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arXiv2023

The Feynman-Lagerstrom criterion for boundary layers

We study the boundary layer theory for slightly viscous stationary flows forced by an imposed slip velocity at the boundary. According to the theory of Prandtl (1904) and Batchelor (1956), any Euler solution arising in this limit and consisting of a single ``eddy" must have constant vorticity. Feynman and Lagerstrom (1956) gave a procedure to select the value of this vorticity by demanding a \textit{necessary} condition for the existence of a periodic Prandtl boundary layer description. In the case of the disc, the choice -- known to Batchelor (1956) and Wood (1957) -- is explicit in terms of the slip forcing. For domains with non-constant curvature, Feynman and Lagerstrom give an approximate formula for the choice which is in fact only implicitly defined and must be determined together with the boundary layer profile. We show that this condition is also sufficient for the existence of a periodic boundary layer described by the Prandtl equations. Due to the quasilinear coupling between the solution and the selected vorticity, we devise a delicate iteration scheme coupled with a high-order energy method that captures and controls the implicit selection mechanism.

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arXiv2022

Higher Regularity Theory for a Mixed-Type Parabolic Equation

In this paper, we study the higher regularity theory of a mixed-type parabolic problem. We extend the recent work of \cite{DMR} to construct solutions that have an arbitrary number of derivatives in Sobolev spaces. To achieve this, we introduce a counting argument based on a quantity called the "degree". In the second part of this paper, we apply this existence theory to the Prandtl system near the classical Falkner-Skan self-similar profiles in order to supplement the stability analysis of \cite{IM22} with a rigorous construction argument.

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arXiv2022

Improved well-posedness for the Triple-Deck and related models via concavity

We establish linearized well-posedness of the Triple-Deck system in Gevrey-$\frac32$ regularity in the tangential variable, under concavity assumptions on the background flow. Due to the recent result \cite{DietertGV}, one cannot expect a generic improvement of the result of \cite{IyerVicol} to a weaker regularity class than real analyticity. Our approach exploits two ingredients, through an analysis of space-time modes on the Fourier-Laplace side: i) stability estimates at the vorticity level, that involve the concavity assumption and a subtle iterative scheme adapted from \cite{GVMM} ii) smoothing properties of the Benjamin-Ono like equation satisfied by the Triple-Deck flow at infinity. Interestingly, our treatment of the vorticity equation also adapts to the so-called hydrostatic Navier-Stokes equations: we show for this system a similar Gevrey-$\frac32$ linear well-posedness result for concave data, improving at the linear level the recent work \cite{MR4149066}.

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arXiv2022

Reversal in the Stationary Prandtl Equations

We demonstrate the existence of an open set of data which exhibits \textit{reversal} and \textit{recirculation} for the stationary Prandtl equations (data is taken in an appropriately defined product space due to the simultaneous forward and backward causality in the problem). Reversal describes the development of the solution beyond the Goldstein singularity, and is characterized by the presence of (spatio-temporal) regions in which $u > 0$ and $u < 0$. The classical point of view of regarding the system as an evolution in the tangential direction completely breaks down past the Goldstein singularity. Instead, to describe the development, we view the problem as a \textit{mixed-type, non-local, quasilinear, free-boundary} problem across the curve $\{ u = 0 \}$. In a well-chosen nonlinear and self-similar coordinate system, we extract a coupled system for the bulk solution and several modulation variables describing the free boundary. Our work combines and introduces techniques from mixed-type problems, free-boundary problems, modulation theory, harmonic analysis, and spectral theory. As a byproduct, we obtain several new cancellations in the Prandtl equations, and develop several new estimates tailored to singular integral operators with Airy type kernels.

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arXiv2021

Formation of unstable shocks for 2D isentropic compressible Euler

In this paper we construct unstable shocks in the context of 2D isentropic compressible Euler in azimuthal symmetry. More specifically, we construct initial data that when viewed in self-similar coordinates, converges asymptotically to the unstable $C^{\frac15}$ self-similar solution to the Burgers' equation. Moreover, we show the behavior is stable in $C^8$ modulo a two dimensional linear subspace. Under the azimuthal symmetry assumption, one cannot impose additional symmetry assumptions in order to isolate the corresponding manifold of initial data leading to stability: rather, we rely on modulation variable techniques in conjunction with a Newton scheme.

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arXiv2021

Boundary Layer Expansions for the Stationary Navier-Stokes Equations

This is the first part of a two paper sequence in which we prove the global-in-x stability of the classical Prandtl boundary layer for the 2D, stationary Navier-Stokes equations. In this part, we provide a construction of an approximate Navier-Stokes solution, obtained by a classical Euler- Prandtl asymptotic expansion. We develop here sharp decay estimates on these quantities. Of independent interest, we establish \textit{without} using the classical von-Mise change of coordinates, proofs of global in x regularity of the Prandtl system. The results of this paper are used in the second part of this sequence, [IM20] (arXiv:2008.12347), to prove the asymptotic stability of the boundary layer as $\eps \rightarrow 0$ and $x \rightarrow \infty$.

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arXiv2021

Global-in-$x$ Stability of Steady Prandtl Expansions for 2D Navier-Stokes Flows

In this work, we establish the convergence of 2D, stationary Navier-Stokes flows, $(u^ε, v^ε)$ to the classical Prandtl boundary layer, $(\bar{u}_p, \bar{v}_p)$, posed on the domain $(0, \infty) \times (0, \infty)$: \begin{equation*} \| u^ε - \bar{u}_p \|_{L^\infty_y} \lesssim \sqrtε \langle x \rangle^{- \frac 1 4 + δ}, \qquad \| v^ε - \sqrtε \bar{v}_p \|_{L^\infty_y} \lesssim \sqrtε \langle x \rangle^{- \frac 1 2}. \end{equation*} This validates Prandtl's boundary layer theory \textit{globally} in the $x$-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 $ε\rightarrow 0$ 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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arXiv2020

Regularity and Expansion for Steady Prandtl Equations

Due to degeneracy near the boundary, the question of high regularity for solutions to the steady Prandtl equations has been a longstanding open question since the celebrated work of Olenick. We settle this open question in affirmative in the absence of an external pressure. Our method is based on energy estimates for the quotient, $q = \frac{v}{\bar{u}}$, $\bar{u}$ being the classical Prandtl solution, via the linear Derivative Prandtl Equation (LDP). As a consequence, our regularity result leads to the construction of Prandtl layer expansion up to any order.

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arXiv2019

Real analytic local well-posedness for the Triple Deck

The Triple Deck model is a classical high order boundary layer model that has been proposed to describe flow regimes where the Prandtl theory is expected to fail. At first sight the model appears to lose two derivatives through the pressure-displacement relation which links pressure to the tangential slip. In order to overcome this, we split the Triple Deck system into two coupled equations: a Prandtl type system on $\mathbb{H}$ and a Benjamin-Ono type equation on $\mathbb{R}$. This splitting enables us to extract a crucial leading order cancellation at the top of the lower deck. We develop a functional framework to subsequently extend this cancellation into the interior of the lower deck, which enables us to prove the local well-posedness of the model in tangentially real analytic spaces.

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arXiv2018

On Global-in-$x$ Stability of Blasius Profiles

We characterize the well known self-similar Blasius profiles, $[\bar{u}, \bar{v}]$, as downstream attractors to solutions $[u,v]$ to the 2D, stationary Prandtl system. It was established in \cite{Serrin} that $\| u - \bar{u}\|_{L^\infty_y} \rightarrow 0$ as $x \rightarrow \infty$. Our result furthers \cite{Serrin} in the case of localized data near Blasius by establishing convergence in stronger norms and by characterizing the decay rates. Central to our analysis is a "division estimate", in turn based on the introduction of a new quantity, $Ω$, which is globally nonnegative precisely for Blasius solutions. Coupled with an energy cascade and a new weighted Nash-type inequality, these ingredients yield convergence of $u - \bar{u}$ and $v - \bar{v}$ at the essentially the sharpest expected rates in $W^{k,p}$ norms.

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