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We study a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporates the effects of memory while being subjected to random perturbations via an additive Gaussian noise. We show that for a broad class of non-linear potentials, the system always admits invariant probability measures. However, the presence of memory effects precludes access to compactness in a typical fashion. In this paper, this obstacle is overcome by introducing functional spaces adapted to the memory kernels, thereby allowing one to recover compactness. Under the assumption of sufficiently smooth noise, it is then shown that the statistically stationary states possess higher-order regularity properties dictated by the structure of the nonlinearity. This is established through a control argument that asymptotically transfers regularity onto the solution by exploiting the underlying Lyapunov structure of the system in a novel way., Comment: arXiv admin note: substantial text overlap with arXiv:2212.05646

We consider the incompressible Navier-Stokes and Euler equations in a bounded domain with non-characteristic boundary condition, and study the energy dissipation near the outflow boundary in the zero-viscosity limit. We show that in a general setting, the energy dissipation rate is proportional to $\bar U \bar V ^2$, where $\bar U$ is the strength of the suction and $\bar V$ is the tangential component of the difference between Euler and Navier-Stokes on the outflow boundary. Moreover, we show that the enstrophy within a layer of order $\nu / \bar U$ is comparable with the total enstrophy. The rate of enstrophy production near the boundary is inversely proportional to $\nu$., Comment: 25 pages, 3 figures

This article establishes estimates on the dimension of the global attractor of the two-dimensional rotating Navier-Stokes equation for viscous, incompressible fluids on the $\beta$-plane. Previous results in this setting by M.A.H. Al-Jaboori and D. Wirosoetisno (2011) had proved that the global attractor collapses to a single point that depends only the longitudinal coordinate, i.e., zonal flow, when the rotation is sufficiently fast. However, an explicit quantification of the complexity of the global attractor in terms of $\beta$ had remained open. In this paper, such estimates are established which are valid across a wide regime of rotation rates and are consistent with the dynamically degenerate regime previously identified. Additionally, a decomposition of solutions is established detailing the asymptotic behavior of the solutions in the limit of large rotation., Comment: 23 pages

This article studies two particular algorithms, a Relaxation Least Squares (RLS) algorithm and a Relaxation Newton Iteration (RNI) scheme , for reconstructing unknown parameters in dissipative dynamical systems. Both algorithms are based on a continuous data assimilation (CDA) algorithm for state reconstruction of A. Azouani, E. Olson, and E.S. Titi \cite{Azouani_Olson_Titi_2014}. Due to the CDA origins of these parameter recovery algorithms, these schemes provide on-the-fly reconstruction, that is, as data is collected, of unknown state and parameters simultaneously. It is shown how both algorithms give way to a robust general framework for simultaneous state and parameter estimation. In particular, we develop a general theory, applicable to a large class of dissipative dynamical systems, which identifies structural and algorithmic conditions under which the proposed algorithms achieve reconstruction of the true parameters. The algorithms are implemented on a high-dimensional two-layer Lorenz 96 model, where the theoretical conditions of the general framework are explicitly verifiable. They are also implemented on the two-dimensional Rayleigh-B\'enard convection system to demonstrate the applicability of the algorithms beyond the finite-dimensional setting. In each case, systematic numerical experiments are carried out probing the efficacy of the proposed algorithms, in addition to the apparent benefits and drawbacks between them.

This article studies the intimate relationship between two filtering algorithms for continuous data assimilation, the synchronization filter and the nudging filter, in the paradigmatic context of the two-dimensional (2D) Navier-Stokes equations (NSE) for incompressible fluids. In this setting, the nudging filter can formally be viewed as an affine perturbation of the 2D NSE. Thus, in the degenerate limit of zero nudging parameter, the nudging filter converges to the solution of the 2D NSE. However, when the nudging parameter of the nudging filter is large, the perturbation becomes singular. It is shown that in the singular limit of infinite nudging parameter, the nudging filter converges to the synchronization filter. In establishing this result, the article fills a notable gap in the literature surrounding these algorithms. Numerical experiments are then presented that confirm the theoretical results and probes the issue of selecting a nudging strategy in the presence of observational noise. In this direction, an adaptive nudging strategy is proposed that leverages the insight gained from the relationship between the synchronization filter and the nudging filter that produces measurable improvement over the constant nudging strategy., Comment: 23 pages, 8 figures

This article studies the interrelation between the determining modes property in the two-dimensional (2D) Navier-Stokes equations (NSE) of incompressible fluids and the synchronization property of two filtering algorithms for continuous data assimilation applied to the 2D NSE. These two properties are realized as manifestations of a more general phenomenon of "self-synchronous intertwinement". It is shown that this concept is a logically stronger form of asymptotic enslavement, as characterized by the existence of finitely many determining modes in the 2D NSE. In particular, this stronger form is shown to imply convergence of the synchronization filter and the nudging filter from continuous data assimilation (CDA), and then subsequently invoked to show that convergence in these filters implies that the 2D NSE possesses finitely many determining modes. The main achievement of this article is to therefore identify a new concept, that of self-synchronous intertwinement, through which a rigorous relationship between the determining modes property and synchronization in these CDA filters is established and made decisively clear. The theoretical results are then complemented by numerical experiments that confirm the conclusions of the theorems., Comment: 46 pages, 6 figures

In this article, some elementary observations are made regarding the behavior of solutions to the two-dimensional curl-free Burgers equation which suggest the distinguished role played by the scalar divergence field in determining the dynamics of the solution. These observations inspire a new divergence-based regularity condition for the two-dimensional Kuramoto-Sivashinsky equation (KSE) that provides conceptual clarity to the nature of the potential blow-up mechanism for this system. The relation of this regularity criterion to the Ladyzhenskaya-Prodi-Serrin-type criterion for the KSE is also established, thus providing the basis for the development of an alternative framework of regularity criterion for this equation based solely on the low-mode behavior of its solutions. The article concludes by applying these ideas to identify a conceptually simple modification of KSE that yields globally regular solutions, as well as providing a straightforward verification of this regularity criterion to establish global regularity of solutions to the 2D Burgers-Sivashinsky equation. The proofs are direct, elementary, and concise., Comment: 24 pages, referee comments incorporated, accepted to J. Math. Fluid Mech

This paper considers a family of active scalar equations which modify the generalized surface quasi-geostrophic (gSQG) equations through its constitutive law or dissipative perturbations. These modifications are characteristically mild in the sense that they are logarithmic. The problem of well-posedness, in the sense of Hadamard, is then studied in a borderline setting of regularity in analog to the scaling-critical spaces of the gSQG equations. A novelty of the system considered is the nuanced form of smoothing provided by the proposed mild form of dissipation, which is able to support global well-posedness at the Euler endpoint, but in a setting where the inviscid counterpart is known to be ill-posed. A novelty of the analysis lies in the simultaneous treatment of modifications in the constitutive law, dissipative mechanism, and functional setting, which the existing literature has typically treated separately. A putatively sharp relation is identified between each of the distinct system-modifiers that is consistent with previous studies that considered these modifications in isolation. This unified perspective is afforded by the introduction of a linear model equation, referred to as the protean system, that successfully incorporates the more delicate commutator structure collectively possessed by the gSQG family and upon which each facet of well-posedness can be reduced to its study., Comment: 48 pages

Using limited observations of the velocity field of the two-dimensional Navier-Stokes equations, we successfully reconstruct the steady body force that drives the flow. The number of observed data points is less than 10\% of the number of modes that describes the full flow field, indicating that the method introduced here is capable of identifying complicated forcing mechanisms from a relatively small collection of observations. In addition to demonstrating the efficacy of this method on turbulent flow data generated by simulations of the two-dimensional Navier-Stokes equations, we also rigorously justify convergence of the derived algorithm. Beyond the practical applicability of such an algorithm, the reliance of this method on the dynamical evolution of the system yields physical insight into the turbulent cascade., Comment: 18 pages, 4 figures

We consider a class of semi-linear differential Volterra equations with polynomial-type potentials that incorporates the effects of memory while being subjected to random perturbations via an additive Gaussian noise. Our main study is the long time statistics of the system in the singular regime as the memory kernel collapses to a Dirac function. Specifically, we show that provided that sufficiently many directions in the phase space are stochastically forced, there is a unique invariant probability measure to which the system converges, with respect to a suitable Wasserstein-type topology, and at an exponential rate which is independent of the decay rate of the memory kernel. We then prove the convergence of this unique statistically steady state to the unique invariant probability measure of the classical stochastic reaction-diffusion equation in the zero-memory limit. Consequently, we establish the global-in-time validity of the short memory approximation.

This article is concerned with the problem of determining an unknown source of non-potential, external time-dependent perturbations of an incompressible fluid from large-scale observations on the flow field. A relaxation-based approach is proposed for accomplishing this, which leverages a nonlinear property of the equations of motions to asymptotically enslave small scales to large scales. In particular, an algorithm is introduced that systematically produces approximations of the flow field on the unobserved scales in order to generate an approximation to the unknown force; the process is then repeated to generate an improved approximation of the unobserved scales, and so on. A mathematical proof of convergence of this algorithm is established in the context of the two-dimensional Navier-Stokes equations with periodic boundary conditions under the assumption that the force belongs to the observational subspace of phase space; at each stage in the algorithm, it is shown that the model error, represented as the difference between the approximating and true force, asymptotically decrements to zero in a geometric fashion provided that sufficiently many scales are observed and certain parameters of the algorithm are appropriately tuned; the issue of the sharpness of the assumptions, among other practical considerations such as the transient periods between updates, are also discussed., Comment: 17 pages, introduction and appendix expanded, main results reworded for clarity, additional remarks and references added, accepted to Proc. R. Soc. Edinb. A: Math

In this paper, the convergence of an algorithm for recovering the unknown kinematic viscosity of a two-dimensional incompressible, viscous fluid is studied. The algorithm of interest is a recursive feedback control-based algorithm that leverages observations that are received continuously-in-time, then dynamically provides updated values of the viscosity at judicious moments. It is shown that in an idealized setup, convergence to the true value of the viscosity can indeed be achieved under a natural and practically verifiable non-degeneracy condition. This appears to be first such result of its kind for parameter estimation of nonlinear partial differential equations. Analysis for two parameter update rules is carried out: one which involves instantaneous evaluation in time and the other, averaging in time. The proofs of convergence for either rule exploits sensitivity-type bounds in higher-order Sobolev topologies, while the instantaneous version particularly requires delicate energy estimates involving the time-derivative of the sensitivity-type variable. Indeed, a crucial component in the analysis of the first update rule is the identification of a dissipative structure for the time-derivative of the sensitivity-type variable, which ultimately ensures a favorable dependence on the tuning parameter of the algorithm., Comment: 35 pages, results expanded to include alternative update formula, exposition adjusted accordingly, additional remarks included, to appear in Nonlinearity

Motivated by recent progress in data assimilation, we develop an algorithm to dynamically learn the parameters of a chaotic system from partial observations. Under reasonable assumptions, we rigorously establish the convergence of this algorithm to the correct parameters when the system in question is the classic three-dimensional Lorenz system. Computationally, we demonstrate the efficacy of this algorithm on the Lorenz system by recovering any proper subset of the three non-dimensional parameters of the system, so long as a corresponding subset of the state is observable. We also provide computational evidence that this algorithm works well beyond the hypotheses required in the rigorous analysis, including in the presence of noisy observations, stochastic forcing, and the case where the observations are discrete and sparse in time.

This paper considers a nudging-based scheme for data assimilation for the two-dimensional (2D) Navier-Stokes equations (NSE) with periodic boundary conditions and studies the synchronization of the signal produced by this algorithm with the true signal, to which the observations correspond, in all higher-order Sobolev topologies. This work complements previous results in the literature where conditions were identified under which synchronization is guaranteed either with respect to only the $H^1$--topology, in the case of general observables, or to the analytic Gevrey topology, in the case of spectral observables. To accommodate the property of synchronization in the stronger topologies, the framework of general interpolant observable operators, originally introduced by Azouani, Olson, and Titi, is expanded to a far richer class of operators. A significant effort is dedicated to the development of this more expanded framework, specifically, their basic approximation properties, the identification of subclasses of such operators relevant to obtaining synchronization, as well as the detailed relation between the structure of these operators and the system regarding the syncrhonization property. One of the main features of this framework is its "mesh-free" aspect, which allows the observational data itself to dictate the subdivision of the domain. Lastly, estimates for the radius of the absorbing ball of the 2D NSE in all higher-order Sobolev norms are obtained, thus properly generalizing previously known bounds; such estimates are required for establishing the synchronization property of the algorithm in the higher-order topologies., Comment: 36 pages, some definitions revised, results adjusted accordingly, title changed

This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar $\theta$ on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these regularized models in borderline Sobolev regularity have previously been studied by D. Chae and J. Wu when the velocity $u$ is of lower singularity, i.e., $u=-\nabla^{\perp}\Lambda^{\beta-2}p(\Lambda)\theta$, where $p$ is a logarithmic smoothing operator and $\beta \in [0,1]$. We complete this study by considering the more singular regime $\beta\in(1,2)$. The main tool is the identification of a suitable linearized system that preserves the underlying commutator structure for the original equation. We observe that this structure is ultimately crucial for obtaining continuity of the flow map. In particular, straightforward applications of previous methods for active transport equations fail to capture the more nuanced commutator structure of the equation in this more singular regime. The proposed linearized system nontrivially modifies the flux of the original system in such a way that it coincides with the original flux when evaluated along solutions of the original system. The requisite estimates are developed for this modified linear system to ensure its well-posedness., Comment: Minor revisions. Accepted to Communications on Pure and Applied Analysis. 18 pages

This paper considers the damped periodic Korteweg-de Vries (KdV) equation in the presence of a white-in-time and spatially smooth stochastic source term and studies the long-time behavior of solutions. We show that the integrals of motion for KdV can be exploited to prove regularity and ergodic properties of invariant measures for damped stochastic KdV. First, by considering non-trivial modifications of the integrals of motion, we establish Lyapunov structure by proving that moments of Sobolev norms of solutions at all orders of regularity are bounded globally-in-time; existence of invariant measures follows as an immediate consequence. Next, we prove a weak Foias-Prodi type estimate for damped stochastic KdV, for which the synchronization occurs in expected value. This estimate plays a crucial role throughout our subsequent analysis. As a first novel application, we combine the Foias-Prodi estimate with the Lyapunov structure to establish that invariant measures are supported on smooth functions provided that the external driving forces are smooth. We then establish ergodic properties of invariant measures, treating the regimes of arbitrary damping and large damping separately. For arbitrary damping, we demonstrate that the framework of `asymptotic coupling' can be implemented for a compact proof of uniqueness of the invariant measure provided that sufficiently many directions in phase space are stochastically forced. Our proof is paradigmatic for SPDEs for which a weak Foias-Prodi type property holds. Lastly, for large damping, we establish the existence of a spectral gap with respect to a Wasserstein-like distance, and exponential mixing and uniqueness of the invariant measure follows., Comment: 74 pages, minor typographical errors fixed, references updated, accepted to Trans. Amer. Math. Soc

This paper studies the dissipative generalized surface quasi-geostrophic equations in a supercritical regime where the order of the dissipation is small relative to order of the velocity, and the velocities are less regular than the advected scalar by up to one order of derivative. We also consider a non-degenerate modification of the endpoint case in which the velocity is less smooth than the advected scalar by slightly more than one order. The existence and uniqueness theory of these equations in the borderline Sobolev spaces is addressed, as well as the instantaneous smoothing effect of their corresponding solutions. In particular, it is shown that solutions emanating from initial data belonging to these Sobolev classes immediately enter a Gevrey class. Such results appear to be the first of its kind for a quasilinear parabolic equation whose coefficients are of higher order than its linear term; they rely on an approximation scheme which modifies the flux in such a way that preserves the underlying commutator structure lost by having to work in the critical space setting, as well as delicate adaptations of well-known commutator estimates to Gevrey classes., Comment: 34 pages and 1 figure

In this paper, we study the initial-boundary value problem and its asymptotic behavior for a repulsive chemotaxis model with logarithmic sensitivity and logistic growth. We establish global well-posedness of strong solutions for large initial data with Neumann boundary conditions and, moreover, establish the qualitative result that both the population density and chemical concentration asymptotically converge to constant states with the population density specifically converging to its carrying capacity. We additionally prove that the vanishing chemical diffusivity limit holds in this regime. Lastly, we provide numerical confirmation of the rigorous qualitative results, as well as numerical simulations that demonstrate a separation of scales phenomenon., Comment: 23 pages, 6 figures, 1 table

An intrinsic property of almost any physical measuring device is that it makes observations which are slightly blurred in time. We consider a nudging-based approach for data assimilation that constructs an approximate solution based on a feedback control mechanism that is designed to account for observations that have been blurred by a moving time average. Analysis of this nudging model in the context of the subcritical surface quasi-geostrophic equation shows, provided the time-averaging window is sufficiently small and the resolution of the observations sufficiently fine, that the approximating solution converges exponentially fast to the observed solution over time. In particular, we demonstrate that observational data with a small blur in time possess no significant obstructions to data assimilation provided that the nudging properly takes the time averaging into account. Two key ingredients in our analysis are additional boundedness properties for the relevant interpolant observation operators and a non-local Gronwall inequality., Comment: 44 pages

We study the long-time behavior of spatially periodic solutions of the Navier-Stokes equations in the three-dimensional space. The body force is assumed to possess an asymptotic expansion or, resp., finite asymptotic approximation, in either Sobolev or Gevrey spaces, as time tends to infinity, in terms of polynomial and decaying exponential functions of time. We establish an asymptotic expansion, or resp., finite asymptotic approximation, of the same type for the Leray-Hopf weak solutions. This extends the previous results, obtained in the case of potential forces, to the non-potential force case, where the body force may have different levels of regularity and asymptotic approximation. In fact, our analysis identifies precisely how the structure of the force influences the asymptotic behavior of the solutions., Comment: 30 pages, small notational inconsistencies in the statement of Proposition 3.2 are corrected, as well as a typo found in the constants defined there

We construct a determining form for the surface quasi-geostrophic (SQG) equation with subcritical dissipation. In particular, we show that the global attractor for this equation can be embedded in the long-time dynamics of an ordinary differential equation (ODE) called a determining form. Indeed, there is a one-to-one correspondence between the trajectories in the global attractor of the SQG equation and the steady state solutions of the determining form. The determining form is a true ODE in the sense that its vector field is Lipschitz. This is shown by combining De Giorgi techniques and elementary harmonic analysis. Finally, we provide elementary proofs of the existence of time-periodic solutions, steady state solutions, as well as the existence of finitely many determining parameters for the SQG equation., Comment: 34 pages

In this article, we prove that data assimilation by feedback nudging can be achieved for the three-dimensional quasi-geostrophic equation in a simplified scenario using only large spatial scale observables on the dynamical boundary. On this boundary, a scalar unknown (buoyancy or surface temperature of the fluid) satisfies the surface quasi-geostrophic equation. The feedback nudging is done on this two-dimensional model, yet ultimately synchronizes the streamfunction of the three-dimensional flow. The main analytical difficulties are due to the presence of a nonlocal dissipative operator in the surface quasi-geostrophic equation. This is overcome by exploiting a suitable partition of unity, the modulus of continuity characterization of Sobolev space norms, and the Littlewood-Paley decomposition to ultimately establish various boundedness and approximation-of-identity properties for the observation operators., Comment: 28 pages, referee comments incorporated, references added, abstract and introduction modified, main theorems cover full subcritical range of dissipation, certain boundedness properties of observation operators extended

In this paper, we study the asymptotic behavior of solutions to the three-dimensional incompressible Navier-Stokes equations (NSE) with periodic boundary conditions and potential body forces. In particular, we prove that the Foias-Saut asymptotic expansion for the regular solutions of the NSE in fact holds in {\textit{all Gevrey classes}}. This strengthens the previous result obtained in Sobolev spaces by Foias-Saut. By using the Gevrey-norm technique of Foias-Temam, the proof of our improved result simplifies the original argument of Foias-Saut, thereby, increasing its adaptability to other dissipative systems. Moreover, the expansion is extended to all Leray-Hopf weak solutions.

In this paper, a lower bound estimate on the uniform radius of spatial analyticity is established for solutions to the incompressible, forced Navier-Stokes system on an n-torus. This estimate improves or matches previously known estimates provided that certain bounds on the initial data are satisfied. It is argued that for 2D or 3D turbulent flows, the initial data is guaranteed to satisfy these hypothesized bounds on a significant portion of the 2D global attractor or the 3D weak attractor. In these scenarios, the estimate obtained for 3D generalizes and improves upon that of [Doering-Titi], while in 2D, the estimate matches the best known one found in [Kukavica]. A key feature in the approach taken here is the choice of the Wiener algebra as the phase space, i.e., the Banach algebra of functions with absolutely convergent Fourier series, whose structure is suitable for the use of the so-called Gevrey norms., Comment: 26 pages