Piotr Zgliczynski
Using ODE-methods and trapping regions derived by Mattingly and Sinai we give a new proof of the existence and uniqueness of solutions to Navier-Stokes equations with periodic boundary conditions on the plane.
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In 1999, J.C. Mattingly and Ya. G. Sinai used elementary methods to prove the existence and uniqueness of smooth solutions to the 2D Navier-Stokes equations with periodic boundary conditions. And they were almost successful in proving the existence and uniqueness of smooth solutions to the 3D Navier-Stokes equations using the same strategy. In this paper, we modify their technique to obtain a simpler proof of one of their results. We also argue that there is no logical reason why the 3D Navier-Stokes equations must always have solutions, even when the initial velocity vector field is smooth; i…
Gravity driven traveling bore wave solutions to the free boundary incompressible Navier-Stokes equations
本文首次构造了单层有限深度常密度流体自由边界不可压缩 Navier-Stokes 方程的二维行进孔波解，通过严格论证浅水极限将 PDE 简化为 ODE，并利用异宿轨道与不动点论证构建了充分浅层的 PDE 解。
A geometric trapping approach to global regularity for 2D Navier-Stokes on manifolds
针对无边界二维黎曼流形上Navier-Stokes方程的全局存在性与正则性问题，受Mattingly和Sinai的平坦周期边界方法启发，结合频率投影、多线性估计与微局部分析思想，利用频率分解技术给出了直接证明。
Ma and Wang derived an equation linking the separation location and times for the boundary layer separation of incompressible fluid flows. The equation gave a necessary condition for the separation (bifurcation) point. The purpose of this paper is to generalize the equation to other geometries, and to phrase it as a simple ODE. Moreover we consider the Navier-Stokes equation with the Coriolis effect, which is related to the presence of trade winds on Earth.
In 1944 L.D.Landau calculated a very interesting family of explicit solutions of the steady-state 3d Navier-Stokes equations. The solutions are derived under certain assumptions of symmetry, which reduce the Navier-Stokes equations to a system of ODEs. We investigate what happens when some of the symmetry conditions are dropped (and we have to deal with PDEs). Implications of these calculations for more general classes of solutions are also discussed. We also discuss the situation for general dimension.
These notes describe some applications of the analysis of ordinary differential equations to the study of the viscous approximation of conservation laws in one space dimension. The exposition mostly focuses on the analysis of invariant manifolds like the center manifold and the stable manifold. The last section addresses a more specific issue and describes a possible way of extending the notions of center and stable manifold to some classes of singular ordinary differential equations arising in the study of the Navier-Stokes equation in one space variable.
The determining modes for the two-dimensional incompressible Navier-Stokes equations (NSE) are shown to satisfy an ordinary differential equation of the form $dv/dt=F(v)$, in the Banach space, $X$, of all bounded continuous functions of the variable $s\in\mathbb{R}$ with values in certain finite-dimensional linear space. This new evolution ODE, named {\it determining form}, induces an infinite-dimensional dynamical system in the space $X$ which is noteworthy for two reasons. One is that $F$ is globally Lipschitz from $X$ into itself. The other is that the long-term dynamics of the determining …
In this paper we consider smooth solutions of the Navier--Stokes equations with a linear dependence on the spatial variable. We reduce the evolution of these solutions to a matrix ODE, and show that there are such solutions that blowup in finite-time. Note that because these solutions have linear growth at infinity, this blowup is not a counterexample disproving the global regularity of strong solutions of the Navier--Stokes equations, as strong solutions must have sufficient decay at infinity. This paper does not resolve the Millennium Problem. Nonetheless, these solutions do exhibit several …
We introduce a global scheme on the n-torus of a controlled incompressible Navier-Stokes equation in terms of a coupled controlled infinite ODE-system of Fourier-modes with smooth data. We construct a scheme of global approximations related to linear partial integrodifferential equations in dual space which are uniformly bounded in dual Sobolev spaces with polynomially decaying modes. The scheme is based on some infinite matrix algebra related to weakly singular integrals, and a Trotter-product formula for dissipative operators which leads to rigorous existence results and a uniform bound for …
We construct a solution to the spatially periodic $d$-dimensional Navier-Stokes equations with a given distribution of the initial data. The solution takes values in the Sobolev space $H^α$, where the index $α\in R$ is fixed arbitrary. The distribution of the initial value is a Gaussian measure on $H^α$ whose parameters depend on $α$. The Navier-Stokes solution is then a stochastic process verifying the Navier-Stokes equations almost surely. It is obtained as a limit in distribution of solutions to finite-dimensional ODEs which are Galerkin-type approximations for the Navier-Stokes equations. …
Based on numerical data and a-posteriori analysis we verify rigorously the uniqueness and smoothness of global solutions to a scalar surface growth model with striking similarities to the 3D Navier--Stokes equations, for certain initial data for which analytical approaches fail. The key point is the derivation of a scalar ODE controlling the norm of the solution, whose coefficients depend on the numerical data. Instead of solving this ODE explicitly, we explore three different numerical methods that provide rigorous upper bounds for its solution
Existence of Large Boundary Layer Solutions to Inflow Problem of 1D Full Compressible Navier-Stokes Equations
本文研究了一维完全可压缩Navier-Stokes方程入流问题的大边界层解，利用基于常微分方程定性理论的全局相平面分析，分别在超声速、跨声速和亚声速区域给出了大边界层解存在与不存在的充分必要条件。
This paper is concerned with geometric regularity criteria for the Navier-Stokes equations in $\mathbb{R}^3_{+}\times (0,T)$ with no-slip boundary condition, with the assumption that the solution satisfies the `ODE blow-up rate' Type I condition. More precisely, we prove that if the vorticity direction is uniformly continuous on subsets of $$\bigcup_{t\in(T-1,T)} \big(B(0,R)\cap\mathbb{R}^3_{+}\big)\times {\{t\}},\,\,\,\,\,\, R=O(\sqrt{T-t})$$ where the vorticity has large magnitude, then $(0,T)$ is a regular point. This result is inspired by and improves the regularity criteria given by Giga,…
Building on work of Barker, Humpherys, Lafitte, Rudd, and Zumbrun in the shock wave case, we study stability of compressive, or "shock-like", boundary layers of the isentropic compressible Navier-Stokes equations with gamma-law pressure by a combination of asymptotic ODE estimates and numerical Evans function computations. Our results indicate stability for gamma in the interval [1, 3] for all compressive boundary-layers, independent of amplitude, save for inflow layers in the characteristic limit (not treated). Expansive inflow boundary-layers have been shown to be stable for all amplitudes b…
A dyadic shell model for the Navier-Stokes equations is studied in the context of turbulence. The model is an infinite nonlinearly coupled system of ODEs. It is proved that the unique fixed point is a global attractor, which converges to the global attractor of the inviscid system as viscosity goes to zero. This implies that the average dissipation rate for the viscous system converges to the anomalous dissipation rate for the inviscid system (which is positive) as viscosity goes to zero. This phenomenon is called the dissipation anomaly predicted by Kolmogorov's theory for the actual Navier-S…
Shock profiles for the Navier--Stokes--Fourier--Poisson system under the Boltzmann relation
本文证明 Boltzmann 关系下 Navier--Stokes--Fourier--Poisson 系统存在唯一且轨道稳定的 Lax 型小振幅粘性激波剖面，采用中心流形约简和带平移的 $a$-收缩方法。
Extending results of Humpherys-Lyng-Zumbrun in the one-dimensional case, we use a combination of asymptotic ODE estimates and numerical Evans-function computations to examine the multidimensional stability of planar Navier--Stokes shocks across the full range of shock amplitudes, including the infinite-amplitude limit, for monatomic or diatomic ideal gas equations of state and viscosity and heat conduction coefficients $μ$, $μ+η$, and $ν=κ/c_v$ in the physical ratios predicted by statistical mechanics, with Mach number $M>1.035$. Our results indicate unconditional stability within the paramete…
The Navier-Stokes equation on the Euclidean space $\mathbf{R}^3$ can be expressed in the form $\partial_t u = Δu + B(u,u)$, where $B$ is a certain bilinear operator on divergence-free vector fields $u$ obeying the cancellation property $\langle B(u,u), u\rangle=0$ (which is equivalent to the energy identity for the Navier-Stokes equation). In this paper, we consider a modification $\partial_t u = Δu + \tilde B(u,u)$ of this equation, where $\tilde B$ is an averaged version of the bilinear operator $B$ (where the average involves rotations and Fourier multipliers of order zero), and which also …
Self-Similar Solutions to the steady Navier-Stokes Equations in a two-dimensional sector
本文针对二维扇形区域内满足无滑移边界条件的定常Navier-Stokes方程组，给出了特定类型自相似解存在的充要条件，研究了其唯一性与非唯一性，补充并修正了文献[Rosenhead40]的相关结论。
Starting from the classic contraction mapping principle, we establish a general, flexible, variational setting that turns out to be applicable to many situations of existence in Differential Equations. We show its potentiality with some selected examples including initial-value, Cauchy problems for ODEs; non-linear, monotone PDEs; linear and non-linear hyperbolic problems; and steady Navier-Stokes systems.
We study the long-time behavior of solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation (NLS/GP) with a symmetric double-well potential, continuing work of the 2nd and 3rd authors. NLS/GP governs nearly-monochromatic guided optical beams in weakly coupled waveguides with both linear and nonlinear (Kerr) refractive indices and zero absorption. The optical power ($L^2$ norm) is conserved with propagation distance. At low optical power, the beam energy executes beating oscillations between the two waveguides. There is an optical power threshold above which the set of guided mode sol…
The classical Schrödinger equation with a harmonic trap potential $V(x)=|x|^2$, describing the quantum harmonic oscillator, has been studied quite extensively in the last twenty years. Its ground states are bell-shaped and unique, among localized positive solutions. In addition, they have been shown to be non-degenerate and (strongly) orbitally stable. All of these results, produced over the course of many publications and multiple authors, rely on ODE methods specifically designed for the Laplacian and the power function potential. In this article, we provide a wide generalization of these re…
The adjoint Rayleigh and Orr-Sommerfeld equations: Green function and eigenmodes
本文研究剪切流线性化方程导出的Rayleigh与Orr-Sommerfeld方程的伴随问题，在依赖粘性的Navier边界条件下严格建立特征值、特征模态和Green函数的存在性与渐近性质，并借此刻画原算子的核与像。
Let the viscosity $\varepsilon \rightarrow 0$ for the 2D steady Navier-Stokes equations in the region $0\leq x\leq L$ and $0\leq y<\infty$ with no slip boundary conditions at $y=0$. For $L<<1$, we justify the validity of the steady Prandtl layer expansion for scaled Prandtl layers, including the celebrated Blasius boundary layer. Our uniform estimates in $\varepsilon$ are achieved through a fixed-point scheme: \begin{equation*} [u^{0}, v^0] \overset{\text{DNS}^{-1}}{\longrightarrow }v\overset{\mathcal{L}^{-1}}{ \longrightarrow }[u^{0}, v^0] \label{fixedpoint} \end{equation*} for solving the Na…
We consider a coupled PDE-ODE system describing the motion of the rigid body in a container filled with the incompressible, viscous fluid. The fluid and the rigid body are coupled via Navier slip boundary condition. We prove that the local in time strong solution is unique in the larger class of weak solutions on the interval of its existence. This is the first weak-strong uniqueness result in the area of fluid-structure interaction with a moving boundary.
Global Well-Posedness of the 3D Navier-Stokes Equations in the Limiting Case: Infinitely Nested Logarithmic Improvements
本文针对三维不可压缩Navier-Stokes方程，建立无限嵌套对数改进的正则性准则框架，证明临界阈值处的整体适定性，推导奇异集维数界并提供解决整体正则性问题的路径。
In this paper, we are interested in the nonlinear Rayleigh-Taylor instability for the gravity-driven incompressible Navier-Stokes equations with Navier-slip boundary conditions around a smooth increasing density profile $ρ_0(x_2)$ in a slab domain $2πL\mathbb{T} \times (-1,1)$ ($L>0$, $\mathbb{T}$ is the usual 1D torus). The linear instability study of the viscous Rayleigh-Taylor model amounts to the study of the following ODE on the finite interval $(-1,1)$, \begin{equation}\label{EqMain} λ^2 ( ρ_0 k^2 ϕ- (ρ_0 ϕ')')+ λμ(ϕ^{(4)} - 2k^2 ϕ'' + k^4 ϕ) = gk^2 ρ_0'ϕ, \end{equation} with the boundar…
We study the singular ordinary differential equation $$ \frac{d U}{d t} = f (U) / z (U) + g (U), $$ where $U \in R^N$, the functions $f \in R^N $ and $g \in R^N $ are of class $C^2$ and $z $ is a real valued $C^2$ function. The equation is singular in the sense that $z (U)$ can attain the value 0. We focus on the solutions of the singular ODE that belong to a small neighborhood of a point $\bar U$ such that $f (\bar U) = g (\bar U) = \vec 0$, $z (\bar U) =0$. We investigate the existence of manifolds that are locally invariant for the singular ODE and that contain orbits with a suitable prescr…
This manuscript derives an evolution equation for the symmetric part of the gradient of the velocity (the strain tensor) in the incompressible Navier-Stokes equation on $\mathbb{R}^3$, and proves the existence of $L^2$ mild solutions to this equation. We use this equation to obtain a simplified identity for the growth of enstrophy for mild solutions that depends only on the strain tensor, not on the nonlocal interaction of the strain tensor with the vorticity. The resulting identity allows us to prove a new family of scale-critical, necessary and sufficient conditions for the blow-up of a solu…
This paper is concerned with a blood flow problem coupled with a slow plaque growth at the artery wall. In the model, the micro (fast) system is the Navier-Stokes equation with a periodically applied force and the macro (slow) system is a fractional reaction equation, which is used to describe the plaque growth with memory effect. We construct an auxiliary temporal periodic problem and an effective time-average equation to approximate the original problem and analyze the approximation error of the corresponding linearized PDE (Stokes) system, where the simple front-tracking technique is used t…
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