Quantitative asymptotic stability of the quasi-linearly stratified densities in the IPM equation with the sharp decay rates Min Jun JoJunha Kim

2025-05-02 17 0 699.89KB 53 页 10玖币

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Quantitative asymptotic stability of the quasi-linearly stratiﬁed

densities in the IPM equation with the sharp decay rates

Min Jun Jo∗Junha Kim†

March 12, 2024

Abstract

We analyze the asymptotic stability of the quasi-linearly stratiﬁed densities in the 2D inviscid

incompressible porous medium equation on R2with respect to the buoyancy frequency N. Our

target density of stratiﬁcation is the sum of the large background linear proﬁle with its slope N

and the small perturbation that could be both non-linear and non-monotone. Quantiﬁcation in

Nwill be performed not only on how large the initial density disturbance is allowed to be but

also on how much the target densities can deviate from the purely linear density stratiﬁcation

without losing their stability.

For the purely linear density stratiﬁcation, our method robustly applies to the three fun-

damental domains R2,T2,and T×[−1,1], improving both the previous result by Elgindi (On

the asymptotic stability of stationary solutions of the inviscid incompressible porous medium

equation, Archive for Rational Mechanics and Analysis, 225(2), 573-599, 2017) on R2and T2,

and the study by Castro-C´ordoba-Lear (Global existence of quasi-stratiﬁed solutions for the

conﬁned IPM equation. Archive for Rational Mechanics and Analysis, 232(1), 437-471, 2019)

on T×[−1,1]. The obtained temporal decay rates to the stratiﬁed density on R2and to the

newly found asymptotic density proﬁles on T2and T×[−1,1] are all sharp, fully realizing the

level of the linearized system. We require the initial disturbance to be small in Hmfor any

integer m≥4, which we even relax to any positive number m > 3 via a suitable anisotropic

commutator estimate.

Contents

1 Introduction 2

1.1 Hydrodynamicstability .................................. 2

1.2 TheinviscidIPMequation................................. 3

1.2.1 Thegoalofthisstudy ............................... 4

1.3 Darcy’slaw ......................................... 5

1.4 Density stratiﬁcation as stablizing mechanism . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Stationary solutions to the IPM equation . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6 Previousresults....................................... 7

1.7 Summaryofmainresults ................................. 8

Key words: quantitative stability, inviscid damping, stratiﬁcation, porous medium equation,

2020 AMS Mathematics Subject Classiﬁcation: 76B70, 35Q35

∗Mathematics Department, Duke University, Durham NC 27708 USA. E-mail: minjun.jo@duke.edu

†Department of Mathematics, Ajou University, Suwon 16499, Republic of Korea. E-mail: junha02@ajou.ac.kr

arXiv:2210.11437v2 [math.AP] 11 Mar 2024

2 Main results 9

2.1 Notations .......................................... 9

2.2 Statements ......................................... 9

2.3 Strategyforproofs ..................................... 12

3 Preliminaries 13

3.1 Existence of local-in-time solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Proof of Theorem A 15

4.1 Energyinequality...................................... 15

4.2 Proof of the global existence of solutions . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.3 Proof of the convergence of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.4 Proof of L2normdecayestimates............................. 27

5 Proof of Theorem B 30

5.1 Energyinequality...................................... 32

5.2 Proof of the global existence of solutions . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.3 Proof of the decay estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6 Sketch of proof of Theorem C 40

6.1 Properties of Xmand Ym................................. 41

6.2 Energyinequality...................................... 42

6.3 Proof of the global existence of solutions . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.4 Proof of the decay estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

A Appendix 50

A.1 Acalculuslemma...................................... 50

A.2 ProofofProposition4.2 .................................. 50

A.3 Additional properties of Xmand Ym........................... 53

1 Introduction

1.1 Hydrodynamic stability

Fluid instabilities are ubiquitous and they are observed on exhaustively various scales of space

and time. Due to the absence of any complete theory for such turbulent nature of ﬂuid ﬂows,

simpler problems have been investigated as starting points. For instance, it is hoped that a deeper

understanding of transition from a stable state, which is possibly laminar, to an unstable state

would give a more rigorous explanation for the mechanism of spontaneous instability. Hydrodynamic

stability, which is known as one of the oldest branches of ﬂuid mechanics, regards the robustness

of ﬂuid ﬂows against small perturbation around a steady state, serving as a measure of intrinsic

aﬃnity of the governing ﬂuid dynamics for turbulence.

One fundamental notion of hydrodynamic stability is Lyapunov stability. Consider, as a toy

model, the typical initial value problem

∂tw=f(w)

w(0) = w0

equipped with a stationary solution wssatisfying f(ws) = 0. The abstract mapping fcould

be nonlinear. We say that the stationary state wsis Lyapunov stable (or nonlinearly stable,

equivalently) if the following holds: given a Banach spaces X, for every ε > 0 there exists δ > 0

such that

∥w0−ws∥X< δ =⇒ ∥w−ws∥e

X< ε, (1.1)

where wis the corresponding solution. The space e

Xusually takes the form of L∞

tXdue to the

evolutionary nature of the problem. Lyapunov stability at least tells us that the solution will stay

close to the stationary state once it is initially placed suﬃciently close to the stationary state.

However, such an approximate notion of Lyapunov stability does not fully explain the dynamics

of was t→ ∞.This gives rise to the stronger notion called asymptotic stability. The stationary

solution wsis asymptotically stable if not only wsis Lyapunov stable but also there holds

lim

t→∞ ∥w(t)−ws∥Y= 0 (1.2)

for a spatially normed space Y. This gives us the information on the explicit ﬁnal destination of the

solution, excluding the possibility of permanent orbiting/wandering around the stationary state.

The necessity of an even stronger notion of stability, compared to the mere asymptotic stability,

stood out in Kelvin’s suggestion [29] that the governing dynamics could become increasingly sen-

sitive to perturbation as Reynolds number gets larger, while certain equilibria are still Lyapunov

stable for any ﬁnite Reynolds number. Quantiﬁcation of δin the notion of Lyapunov stability, see

(1.1), is desired to capture the information on the sensitivity of the ﬂuid system against pertur-

bation in terms of Reynolds number. A quantitatively aysmptotic stability result takes the form

of the following statement. Let Nbe any physical constant (e.g. ﬂuid viscosity) one wants to

quantitatively relate to the asymptotic stability. Then a stationary solution wsis quantitatively

asymptotically stable if for all ε > 0 there exists δ > 0 such that

∥w0−ws∥X≤δg(N) =⇒(∥(w−ws)(t)∥Y< ε

lim

t→∞ ∥(w−ws)(t)∥Y= 0 (1.3)

where g:R+→R+is a continuous function. For the results on quantitative asymptotic stability

with respect to Reynolds number, see [2, 3, 4, 5, 6, 38, 39, 40, 41]. For the Boussinesq equations,

see [1, 10].

This paper concerns the notion of quantitative asymptotic stability (1.3) for the two-dimensional

inviscid incompressible porous medium equation (IPM). Quantiﬁcation will be performed in terms

of the large-scale physical phenomenon called stratiﬁcation which works via buoyancy. The mo-

tivation is that stratiﬁcation tends to stabilize ﬂuid ﬂows by quenching the vertical motions and

such a tendency of stabilization is observed to increase as the intensity of stratiﬁcation grows, see

[27] and references therein.

1.2 The inviscid IPM equation

We study the 2D inviscid incompressible porous medium (IPM) equation











∂tρ+v· ∇ρ= 0,

v=−∇p+ (0, ρ),

div v= 0,

(IPM)

on the three diﬀerent domains R2,T2, and a conﬁned strip T×[−1,1]. The ﬁrst line in (IPM) is

the continuity equation expressing the density transport along the ﬂuid ﬂow, and the second part is

Darcy’s law introduced in (1.4). The condition div v= 0 is the incompressibility condition. For the

conﬁned scenario T×[−1,1], we supplement (IPM) with the no-penetration boundary condition

v|y=−1,1·n= 0 where ndenotes the outward normal unit vector on the boundary.

The above IPM equation falls into the category of active scalar equations, as the inviscid Surface

Quasi-Geostrophic (SQG) equation does. The SQG equation reads

(∂tθ+v· ∇θ= 0,

v=R⊥θ. (SQG)

where R= (R1, R2) is the vector of the 2D Riesz transforms with Fourier symbol FR(ξ) = −iξ

|ξ|.

The SQG equation is the ensuing equation on the boundary of R3

+for the 3D Quasi-Geostrophic

(QG) equations, and it has been introduced in [15] as the 2D model which has various features in

common with the 3D Euler equations. There have been numerous results on the SQG equation

during past several decades. For the global well-posedness type results when the dissipation term

(e.g. Ekman pumping) is present, see [14] and extensively various references therein. In the

opposite direction, a small scale creation result can be found in [24] and some ill-posedness results

are obtained in [16, 20, 26], for examples.

The Biot-Savart laws of (IPM) and (SQG), via which the active scalars determine the corre-

sponding velocities, are both singular operators of zero order; Darcy’s law in (IPM) can be rewritten

as v=R1R⊥ρwhile v=R⊥θin (SQG). Due to the similarities between (IPM) and (SQG), includ-

ing the same order of Biot-Savart laws mentioned above, it appears to be worth trying to establish

analogous results for (IPM).

However, it turns out that the structure of (IPM) is distinguishable; Kiselev and Yao in [32]

designed a new approach for (IPM) to prove small scale creation unlike the previous hyperbolic

scenarios used in [24, 31, 37] for the generalized SQG (gSQG) equation equipped with the extended

Biot-Savart law v=R⊥(−∆)−1−α

2θ. Here α= 0 and α= 1, respectively, correspond to the

2D Euler equations in vorticity formulation and the SQG equation. One notable diﬀerence is the

additional partial derivative ∂1in the Biot-Savart law v=R1R⊥ρfor (IPM), compared to v=

R⊥θfor (SQG). This enables the IPM equation to feature density stratiﬁcation as the stabilizing

mechanism in the presence of gravity. See Section 1.3 for the introduction of Darcy’s law, which

gives the relation v=R1R⊥ρ, and Section 1.4 for the background of density stratiﬁcation.

1.2.1 The goal of this study

The goal of this study is to quantitatively fathom the stabilizing eﬀect of stratiﬁcation on the ﬂuid

ﬂows moving through a porous medium, governed by (IPM). Speciﬁcally, the perturbation around

the stratiﬁed densities gives rise to the extra linear structure so that the resulting linearized system

becomes partially dissipative. Here we mean by partial dissipation that only certain directions of

motion are damped in the system. Leveraging properly the anisotropic damping of the linearized

system, we prove that the IPM solution that is initially disturbed around a steady state of certain

stratiﬁed density converges back to the equilibrium. Due to the lack of dissipation mechanism in

(IPM), we call such damping phenomenon inviscid damping.

1.3 Darcy’s law

In the 19th century, Henry Darcy designed a pipe system that provides water in Dijon, the city

in France, and that works without using any pumps. The ﬂows in the pipe are driven by gravity

only, see [17], as targeted in his past experiments. Speciﬁcally, Darcy found the linear relationship

between the unit ﬂux qand the hydraulic gradient dh

dxfor the ﬂuids ﬂowing through a porous

medium. The relationship reads

q=−kgρ

dx,

which is called Darcy’s law. In the above formulation, kis the intrinsic permeability of the porous

medium, gρ denotes the weight of the ﬂuid with gravity acceleration gand density ρ, and ηis the

dynamic viscosity. The function hdenotes the height from the lower plate to the upper plate of the

water pipe with corresponding constant water ﬂuxes. Then the pressure due to the weight of the

water in the pipe can be expressed as p=ρgh. This gives rise to a diﬀerent form of Darcy’s law

q=−k

dx.

Note that the derivatives, dh

dxand dp

dx, were taken only in a horizontal direction along which the pres-

sure drop happens, due to the conﬁnement of the original setting. Nevertheless, such a viewpoint

on the pressure drop can provide the extended Darcy’s law

q=−k

η∇p,

which applies to all the directions. If we assume for simplicity that the porosity of the medium

is equal to one, then the ﬂuid velocity vis exactly equal to q. In that case, the above version of

Darcy’s law reduces to the relationship for the ﬂuid velocity as

v=−k

η∇p. (1.4)

For brevity, we set k=η= 1 and so v=−∇p.

When one tries to describe the motion of the large-scale ﬂuid ﬂows more precisely, it is desirable

to take further into account the direct eﬀect of gravity on ﬂuid particles, other than the mere

gravitaional eﬀect via the hydraulic gradient. Then Darcy’s law can be rewritten as

v=−∇p−(0, gρ) (1.5)

which is the version we will use throughout this paper.

Experimentally, it has been observed that a ﬂuid ﬂow is Darcian only when the Reynolds

number is suﬃciently small, say smaller than 10. In principle, the Darcian ﬂows can be regarded as

the slow, laminar, and stationary ﬂows. If we apply such tendency to the Navier-Stokes equations,

we obtain, under the Boussinesq approximation, the Stokes equations

−∆v=−∇p−(0, gρ).

By using a formal averaging method, Neuman (1977) in [33] and Whitaker (1986) in [36] derived

from the classical Stokes equations that in a porous medium the resistive force of viscosity would

be linear in the velocity with its proper sign. In other words, the Stokes equations become

v=−∇p+ (0, ρ)

which is Darcy’s law revisited, taking g=−1 for brevity.

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Quantitativeasymptoticstabilityofthequasi-linearlystratifieddensitiesintheIPMequationwiththesharpdecayratesMinJunJo∗JunhaKim†March12,2024AbstractWeanalyzetheasymptoticstabilityofthequasi-linearlystratifieddensitiesinthe2DinviscidincompressibleporousmediumequationonR2withrespecttothebuoyancyfrequency...

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Quantitative asymptotic stability of the quasi-linearly stratified densities in the IPM equation with the sharp decay rates Min Jun JoJunha Kim

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