We propose a novel approach to induce anomalous dissipation through advection driven by turbulent fluid flows. Specifically, we establish the existence of a velocity field $v$ satisfying randomly forced Navier-Stokes equations, leading to total dissipation of kinetic energy in finite time when advecting a passive scalar. This dissipation phenomenon is uniform across viscosity parameters and initial conditions, representing a case of anomalous dissipation. We further explore dissipation induced by individual realizations of $v$. Our results extend to scenarios where the passive scalar is replaced by solutions to two or three-dimensional deterministic Navier-Stokes equations advected by $v$.
In this article we show that three dimensional vector advection equation is self dual in certain sense defined below. As a consequence, we infer classical result of Serrin of existence of strong solution of Navier-Stokes equation. Also we deduce Feynman-Kac type formula for solution of the vector advection equation and show that the formula is not unique i.e. there exist flows which differ from standard flow along which vorticity is conserved.
We construct solutions to the randomly-forced Navier--Stokes--Poisson system in periodic three-dimensional domains or in the whole three-dimensional Euclidean space. These solutions are weak in the sense of PDEs and also weak in the sense of probability. As such, they satisfy the system in the sense of distributions and the underlying probability space and the stochastic driving force are also unknowns of the problem. Additionally, these solutions dissipate energy, satisfies a relative energy inequality in the sense of [4] and satisfy a renormalized form of the continuity equation in the sense…
In this paper, we consider the forced incompressible Navier-Stokes equations with vanishing viscosity on the three-dimensional torus. We show that there are (classical) solutions for which the dissipation rate of the kinetic energy is bounded away from zero, uniformly in the viscosity parameter, while the body forces are uniformly bounded in some reasonable regularity class.
We answer positively to [BDL22, Question 2.4] by building new examples of solutions to the forced 3d-Navier-Stokes equations with vanishing viscosity, which exhibit anomalous dissipation and which enjoy uniform bounds in the space $L_t^3 C_x^{1/3 - \varepsilon}$, for any fixed $\varepsilon >0$. Our construction combines ideas of [BDL22] and [CCS22].
The paper is devoted to studying the asymptotics of the family $(μ^\varepsilon)$ of stationary measures of the Markov process generated by the flow of stochastic 2D Navier-Stokes equation with smooth white noise. By using the large deviations techniques, we prove that this family is exponentially tight in $H^{1-γ}(D)$ for any $γ>0$ and vanishes exponentially outside any neighborhood of the set $\cal O$ of $ω$-limit points of the deterministic equation. In particular, any of its weak limits is concentrated on the closure $\bar{\cal O}$. A key ingredient of the proof is a new formula that allows…
The ergodic properties of the randomly forced Navier-Stokes system have been extensively studied in the literature during the last two decades. The problem has always been considered in bounded domains, in order to have, for example, suitable spectral properties for the Stokes operator, to ensure some compactness properties for the resolving operator of the system and the associated functional spaces, etc. In the present paper, we consider the Navier-Stokes system in an unbounded domain satisfying the Poincaré inequality. Assuming that the system is perturbed by a bounded non-degenerate noise,…
In this paper we first show a blow-up criterion for solutions to the Navier-Stokes equations with a time-independent force by using the profile decomposition method. Based on the orthogonal properties related to the profiles, we give some examples of global solutions to the Navier-Stokes equations with a time-independent force, whose initial data are large.
In this article, we study optimal feedback control synthesis of stochastic 2D Navier-Stokes equations perturbed Levy type noise with distributed stochastic control process acting on the state equation. We use the dynamic programming approach to solve this control problem which involves the study of second order infinite dimensional Hamilton- Jacobi-Bellman (HJB) equation consisting of an integro-differential operator with Levy measure associated with the stochastic control problem. Using the regularizing properties of the transition semigroup corresponding to the stochastic 2D Navier-Stokes eq…
We study the mixing and dissipation properties of the advection-diffusion equation with diffusivity $0 < κ\ll 1$ and advection by a class of random velocity fields on $\mathbb T^d$, $d=\{2,3\}$, including solutions of the 2D Navier-Stokes equations forced by sufficiently regular-in-space, non-degenerate white-in-time noise. We prove that the solution almost surely mixes exponentially fast uniformly in the diffusivity $κ$. Namely, that there is a deterministic, exponential rate (independent of $κ$) such that all mean-zero $H^1$ initial data decays exponentially fast in $H^{-1}$ at this rate wit…
We prove that the any Markov solution to the 3D stochastic Navier-Stokes equations driven by a mildly degenerate noise (i.e.all but finitely many Fourier modes are forced) is uniquely ergodic. This follows by proving strong Feller regularity and irreducibility.
We prove the strong Feller property and exponential mixing for 3D stochastic Navier-Stokes equation driven by mildly degenerate noises (i.e. all but finitely many Fourier modes are forced) via Kolmogorov equation approach.
We study the asymptotic behavior of the forced linear Euler and nonlinear Navier-Stokes equations close to Couette flow in a periodic channel. As our main result we show that for smooth time-periodic forcing linear inviscid damping persists, i.e. the velocity field (weakly) asymptotically converges. However, stability and scattering to the transport problem fail in $H^{s}, s>-1$. We further show that this behavior is consistent with the nonlinear Euler equations and that a similar result also holds for the nonlinear Navier-Stokes equations. Hence, these results provide an indication that nonli…
In the seminal work [39], Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. We exhibit two distinct Leray solutions with zero initial velocity and identical body force. Our approach is to construct a `background' solution which is unstable for the Navier-Stokes dynamics in similarity variables; its similarity profile is a smooth, compactly supported vortex ring whose cross-section is a modification of the unstable two-dimensional vortex constructed by Vishik in [43,44]. The second solution is a trajectory on the unstable manifold asso…
Non-uniqueness of three-dimensional Euler equations and Navier-Stokes equations forced by random noise, path-wise and more recently even in law, have been proven by various authors. We prove non-uniqueness in law of the three-dimensional Navier-Stokes equations forced by random noise and diffused via a fractional Laplacian that has power between zero and one half. The solution we construct has H$\ddot{\mathrm{o}}$lder regularity with a small exponent rather than Sobolev regularity with a small exponent. For the power sufficiently small, the non-uniqueness in law holds at the level of Leray-Hop…
We study a singular limit of a scaled compressible Navier--Stokes--Coriolis system driven by both a deterministic and stochastic forcing terms in three dimensions. If the Mach number is comparable to the Froude number with both proportional to say $\varepsilon\ll 1$, whereas the Rossby number scales like $\varepsilon^m$ for $m>1$ large, then we show that any family of weak martingale solution to the $3$-D randomly forced rotating compressible equation (under the influence of a deterministic centrifugal force) converges in probability, as $\varepsilon\rightarrow0$, to the $2$-D incompressible N…
In this paper we study the Liouville-type properties for solutions to the steady incompressible Euler equations with forces in $\Bbb R^N$. If we assume "single signedness condition" on the force, then we can show that a $C^1 (\Bbb R^N)$ solution $(v,p)$ with $|v|^2+ |p|\in L^{\frac{q}{2}}(\Bbb R^N)$, $q\in (\frac{3N}{N-1}, \infty)$ is trivial, $v=0$. For the solution of of the steady Navier-Stokes equations, satisfying $v(x)\to 0$ as $|x|\to \infty$, the condition $\int_{\Bbb R^3} |Δv|^{\frac65} dx<\infty$, which is stronger than the important D-condition, $\int_{\Bbb R^3} |\nabla v|^2 dx <\in…
Stochastically forced Navier-Stokes equations interacting with an elastic structure
University of California at Berkeley（加州大学伯克利分校）；Technische Universität Darmstadt（达姆施塔特工业大学）；Basque Center for Applied Mathematics (BCAM)（巴斯克应用数学中心）；IKERBASQUE, Basque Foundation for Science（伊克尔巴斯基奎，巴斯克科学基金会）
We prove the existence and uniqueness of maximal solutions to the 3D SALT (Stochastic Advection by Lie Transport, [Holm arXiv:1410.8311]) Navier-Stokes Equation in velocity and vorticity form, on the torus and the bounded domain respectively. The current work partners the paper [Goodair et al, arXiv:2209.09137] as an application of the abstract framework presented there, justifying the results first announced in [Goodair, arXiv:2202.09242v2]. In particular this represents the first well-posedness result for a fluid equation perturbed by a general transport type noise on a bounded domain.
The paper is devoted to studying the image of probability measures on a Hilbert space under finite-dimensional analytic maps. We establish sufficient conditions under which the image of a measure has a density with respect to the Lebesgue measure and continuously depends on the map. The results obtained are applied to the 2D Navier--Stokes equations perturbed by various random forces of low dimension.
We consider three-dimensional stochastically forced Navier-Stokes equations subjected to white-in-time (colored-in-space) forcing in the absence of boundaries. Upper and lower bounds of the mean value of the time-averaged energy dissipation rate, $\mathbb{E} [\langle\varepsilon \rangle] $, are derived directly from the equations. First, we show that for a weak (martingale) solution to the stochastically forced Navier-Stokes equations, \[ \mathbb{E} [\langle\varepsilon \rangle] \leq G^2 + (2+ \frac{1}{Re})\frac{U^3}{L},\] where $G^2$ is the total energy rate supplied by the random force, $U$ is…
We give an overview of the ideas central to some recent developments in the ergodic theory of the stochastically forced Navier Stokes equations and other dissipative stochastic partial differential equations. Since our desire is to make the core ideas clear, we will mostly work with a specific example: the stochastically forced Navier Stokes equations. To further clarify ideas, we will also examine in detail a toy problem. A few general theorems are given. Spatial regularity, ergodicity, exponential mixing, coupling for a SPDE, and hypoellipticity are all discussed.
We show - in the framework of physical scales and $(K_1,K_2)$-averages - that Kolmogorov's dissipation law combined with the smallness condition on a Taylor length scale are sufficient to guarantee energy cascades in the forced Navier-Stokes equations. Moreover, in the periodic case we establish restrictive scaling laws - in terms of Grashof number - for kinetic energy, energy flux, and energy dissipation rate. These are used to improve our sufficient condition for forced cascades in physical scales.
Magnetohydrodynamics system consists of a coupling of the Navier-Stokes and Maxwell's equations and is most useful in studying the motion of electrically conducting fluids. We prove the existence of a unique invariant, and consequently ergodic, measure for the Galerkin approximation system of the three-dimensional magnetohydrodynamics system. The proof is inspired by those of \cite{EM01, R04} on the Navier-Stokes equations; however, computations involve significantly more complications due to the coupling of the velocity field equations with those of magnetic field that consists of four non-li…
We discuss the regularity of solutions of 2D incompressible Navier-Stokes equations forced by singular forces. The problem is motivated by the study of complex fluids modeled by the Navier-Stokes equations coupled to a nonlinear Fokker-Planck equation describing microscopic corpora embedded in the fluid. This leads naturally to bounded added stress and hence to $W^{-1,\infty}$ forcing of the Navier-Stokes equations.
We introduce a determining wavenumber for the forced 3D Navier-Stokes equations (NSE) defined for each individual solution. Even though this wavenumber blows up if the solution blows up, its time average is uniformly bounded for all solutions on the global attractor. The bound is compared to Kolmogorov's dissipation wavenumber and the Grashof constant.
In this paper, the system of particles coupled with fluid is considered. The particles are described by a Vlasov equation, and the fluid is governed by a forced Navier-Stokes equations. The interaction with fluid phase governed by Navier-Stokes equations is taken into account through a source term. The resulting system, namely Navier-Stokes-Vlasov equations, is shown to have global weak solutions in three spatial dimensions, and to have a unique global solution in two spatial dimensions.
We study the local and global wellposedness of a full system of Magneto-Hydro-Dynamic equations. The system is a coupling of the forced (Lorentz force) incompressible Navier-Stokes equations with the Maxwell equations through Ohm's law for the current. We show the local existence of mild solutions for arbitrarily large data in a space similar to the scale invariant spaces classically used for Navier-Stokes. These solutions are global if the initial data are small enough.
We investigate the long-time behavior of solutions to a stochastically forced one-dimensional Navier-Stokes system, describing the motion of a compressible viscous fluid, in the case of linear pressure law. We prove existence of an invariant measure for the Markov process generated by strong solutions. We overcome the difficulties of working with non-Feller Markov semigroups on non-complete metric spaces by generalizing the classical Krylov-Bogoliubov method, and by providing suitable polynomial and exponential moment bounds on the solution, together with pathwise estimates.
We investigate the properties of an abstract family of advection diffusion equations in the context of the fractional Laplacian. Two independent diffusion parameters enter the system, one via the constitutive law for the drift velocity and one as the prefactor of the fractional Laplacian. We obtain existence and convergence results in certain parameter regimes and limits. We study the long time behaviour of solutions to the general problem and prove the existence of a unique global attractor. We apply results to two particular active scalar equations arising in geophysical fluid dynamics, name…
