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In this article, we investigate the dynamics of self-organised suspensions formed by rod-like fd virus colloids. Two methods have been employed for analysing fluorescence microscopy movies: single particle tracking (SPT) in direct space and differential dynamic microscopy (DDM) in reciprocal space. We perform a quantitative analysis on this anisotropic system with complex diffusion across different self-assembled states, ranging from dilute and semi-dilute liquids to nematic and smectic organisations. By leveraging the complementary strengths of SPT and DDM, we provide new insights in the dynamics of viral colloidal rods, such as long time diffusion coefficients in the smectic phase. We further discuss the advantages and limitations of both methods for studying the intricate dynamics of anisotropic colloidal systems.

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

Differential dynamic microscopy (DDM) can be used to extract mean particle size from videos of suspensions. However, many suspensions have multimodal particle size distributions (PSDs), for which this is not a sufficient description. After clarifying how different particle sizes contribute to the signal in DDM, we show that standard DDM analysis can extract the mean sizes of two populations in a bimodal suspension given prior knowledge of the sample's bimodality. Further, the use of the CONTIN algorithm obviates the need for such prior knowledge. Finally, we show that by selectively analysing portions of the DDM images, we can size a trimodal suspension where the large particles would otherwise dominate the signal, again without prior knowledge of the trimodality., Comment: Main Body: 10 pages, 8 figures. Appendix: 5 pages, 6 figures. Manuscript accepted at Soft Matter with open access licence; for version of record see https://doi.org/10.1039/D3SM00593C. Relevant data available at https://doi.org/10.7488/ds/3851

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 characterize the full spatiotemporal gait of populations of swimming {\it Escherichia coli} using renewal processes to analyze the measurements of intermediate scattering functions. This allows us to demonstrate quantitatively how the persistence length of an engineered strain can be controlled by a chemical inducer and to report a controlled transition from perpetual tumbling to smooth swimming. For wild-type {\it E.~coli}, we measure simultaneously the microscopic motility parameters and the large-scale effective diffusivity, hence quantitatively bridging for the first time small-scale directed swimming and macroscopic diffusion.

We introduce a numerical method to extract the parameters of run-and-tumble dynamics from experimental measurements of the intermediate scattering function. We show that proceeding in Laplace space is unpractical and employ instead renewal processes to work directly in real time. We first validate our approach against data produced using agent-based simulations. This allows us to identify the length and time scales required for an accurate measurement of the motility parameters, including tumbling frequency and swim speed. We compare different models for the run-and-tumble dynamics by accounting for speed variability at the single-cell and population level, respectively. Finally, we apply our approach to experimental data on wild-type Escherichia coli obtained using differential dynamic microscopy., Comment: 10 pages, 5 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

A new backward compatible WiFi amendment is under development by the IEEE bd Task Group towards the so-called IEEE 802.11bd, which includes the possibility to transmit up to three repetitions of the same packet. This feature increases time diversity and enables the use of maximum ratio combining (MRC) at the receiver to improve the probability of correct decoding. In this work, we first investigate the packet repetition feature and analyze how it looses its efficacy increasing the traffic as an higher number of transmissions may augment the channel load and collision probability. Then, we propose two strategies for adaptively selecting the number of transmissions leveraging on an adapted version of the channel busy ratio (CBR), which is measured at the transmitter and is an indicator of the channel load. The proposed strategies are validated through network-level simulations that account for both the acquisition and decoding processes. Results show that the proposed strategies ensure that devices use optimal settings under variable traffic conditions.

We study a synthetic system of motile Escherichia coli bacteria encapsulated inside giant lipid vesicles. Forces exerted by the bacteria on the inner side of the membrane are sufficient to extrude membrane tubes filled with one or several bacteria. We show that a physical coupling between the membrane tube and the flagella of the enclosed cells transforms the tube into an effective helical flagellum propelling the vesicle. We develop a simple theoretical model to estimate the propulsive force from the speed of the vesicles, and demonstrate the good efficiency of this coupling mechanism. Together, these results point to design principles for conferring motility to synthetic cells., Comment: 22 pages, 12 figures

Recent advancements in V2X communications have greatly increased the flexibility of the physical and medium access control (MAC) layers. This increases the complexity when investigating the system from a network perspective to evaluate the performance of the supported applications. Such flexibility needs in fact to be taken into account through a cross-layer approach, which might lead to challenging evaluation processes. As an accurate simulation of the signals appears unfeasible, a typical solution is to rely on simple models for incorporating the physical layer of the supported technologies, based on off-line measurements or accurate link-level simulations. Such data is however limited to a subset of possible configurations and extending them to others is costly when not even impossible. The goal of this paper is to develop a new approach for modelling the physical layer of vehicle-to-everything (V2X) communications that can be extended to a wide range of configurations without leading to extensive measurement or simulation campaign at the link layer. In particular, given a scenario and starting from results in terms of packet error rate (PER) vs. signal-to-interference-plus-noise ratio (SINR) related to a subset of possible configurations, we derive one parameter, called implementation loss, that is then used to evaluate the network performance under any configuration in the same scenario. The proposed methodology, leading to a good trade-off among complexity, generality, and accuracy of the performance evaluation process, has been validated through extensive simulations with both IEEE 802.11p and LTE-V2X sidelink technologies in various scenarios.

Spectrum scarcity is one of the main challenges of future wireless technologies. When looking at vehicle-to-everything (V2X), this is amplified as spectrum sharing could impact road safety and traffic efficiency. It is therefore of particular importance to study solutions that allow the coexistence, in the same geographical area and in the same channels, of what are today the main V2X access technologies, namely IEEE 802.11p and sidelink LTE-V2X Mode 4. In this work, in addition to investigating the impact of the reciprocal interference, which we demonstrate to have a strong impact especially on the first and in congested channel conditions, a mitigation solution is extensively studied, which is based on the insertion of the IEEE 802.11p preamble at the beginning of the LTE-V2X sidelink transmission. The proposal, which is also under discussion within the standardization bodies, requires no modifications to the IEEE 802.11p protocol stack and minor changes to LTE-V2X sidelink. This solution is directly applicable to upcoming IEEE 802.11bd and extendable to NR-V2X sidelink. The paper shows, through analysis and simulations in free-flow and dense scenarios, that the proposal allows for a mitigation of collisions caused by co-channel coexistence under low to high-load channel conditions and that the improvement is also granted in congested cases when combined with additional countermeasures. Regarding the latter aspect, in particular, different approaches are compared, demonstrating that acting on the congestion control mechanisms is a simple but effective solution.

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

Microscopic dynamics reveal the origin of the bulk rheological response in complex fluids. In model systems particle motion can be tracked, but for industrially relevant samples this is often impossible. Here we adapt differential dynamic microscopy (DDM) to study flowing highly-concentrated samples without particle resolution. By combining an investigation of oscillatory flow, using a novel "echo-DDM" analysis, and steady shear, through flow-DDM, we characterise the yielding of a silicone oil emulsion on both the microscopic and bulk level. Through measuring the rate of shear-induced droplet rearrangements and the flow velocity, the transition from a solid-like to liquid-like state is shown to occur in two steps: with droplet mobilisation marking the limit of linear visco-elasticity, followed by the development of shear localisation and macroscopic yielding. Using this suite of techniques, such insight could be developed for a wide variety of challenging complex fluids., Comment: 11 pages, 8 figures

Motile bacteria are known to accumulate at surfaces, eventually leading to changes in bacterial motility and bio-film formation. We use a novel two-colour, three-dimensional Lagrangian tracking technique, to follow simultaneously the body and the flagella of a wild-type ${\it Escherichia~coli}$. We observe long surface residence times and surface escape corresponding mostly to immediately antecedent tumbling. A motility model accounting for a large behavioural variability in run-time duration, reproduces all experimental findings and gives new insights into surface trapping efficiency.

Understanding and controlling the rheology of polymeric complex fluids that are pushed out-of-equilibrium is a fundamental problem in both industry and biology. For example, to package, repair, and replicate DNA, cells use enzymes to constantly manipulate DNA topology, length, and structure. Inspired by this, here we engineer and study DNA-based complex fluids that undergo enzymatically-driven topological and architectural alterations via restriction endonuclease (RE) reactions. We show that these systems display time-dependent rheological properties that depend on the concentrations and properties of the comprising DNA and REs. Through time-resolved microrheology experiments and Brownian Dynamics simulations, we show that conversion of supercoiled to linear DNA topology leads to a monotonic increase in viscosity. On the other hand, the viscosity of entangled linear DNA undergoing fragmentation displays a universal decrease that we rationalize using living polymer theory. Finally, to showcase the tunability of these behaviours, we design a DNA fluid that exhibits a time-dependent increase, followed by a temporally-gated decrease, of its viscosity. Our results present a class of polymeric fluids that leverage naturally occurring enzymes to drive diverse time-varying rheology by performing architectural alterations to the constituents.

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

Particle size is a key variable in understanding the behaviour of the particulate products that underpin much of our modern lives. Typically obtained from suspensions at rest, measuring the particle size under flowing conditions would enable advances for in-line testing during manufacture and high-throughput testing during development. However, samples are often turbid, multiply scattering light and preventing the direct use of common sizing techniques. Differential dynamic microscopy (DDM) is a powerful technique for analysing video microscopy of such samples, measuring diffusion and hence particle size without the need to resolve individual particles while free of substantial user input. However, when applying DDM to a flowing sample, diffusive dynamics are rapidly dominated by flow effects, preventing particle sizing. Here, we develop "flow-DDM", a novel analysis scheme that combines optimised imaging conditions, a drift-velocity correction and modelling of the impact of flow. Flow-DDM allows a decoupling of flow from diffusive motion that facilitates successful particle size measurements at flow speeds an order of magnitude higher than for DDM. We demonstrate the generality of the technique by applying flow-DDM to two separate microscopy methods and flow geometries., Comment: 9 pages, 8 figures

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

Most V2X (Vehicle-to-Everything) applications rely on broadcasting awareness messages known as CAM (Cooperative Awareness Messages) in ETSI or BSM (Basic Safety Message) in SAE standards. A large number of studies have been devoted to guarantee their reliable transmission. However, to date, the studies are generally based on simplified data traffic models that generate awareness messages at periodic intervals or with a constant message size. These models do not accurately represent the real generation of CAM messages that follow specific mobility-based rules. Using simplified and unrealistic traffic models can significantly impact the results and validity of the studies, and hence accurate models for the generation of awareness messages are necessary. This paper proposes the first set of models that can realistically generate CAM messages. The models have been created from real traces collected by two car manufacturers in urban, sub-urban and highway test drives. The models are based on mth order Markov sources, and model the size of CAMs and the time interval between CAMs. The models are openly provided to the community and can be easily integrated into any simulator.

In the last few years, two technologies have been developed to enable direct exchange of information between vehicles. These technologies, currently seen as alternatives, are ITS-G5, as commonly referred in Europe, and sidelink LTE-vehicle-to-everything (LTE-V2X) (one of the solutions of the so-called cellular-V2X, C-V2X). For this reason, the attention has been mostly concentrated on comparing them and remarking their strengths and weaknesses to motivate a choice. Differently, in this work we focus on a scenario where both are used in the same area and using the same frequency channels, without the assistance from any infrastructure. Our results show that under co-channel coexistence the range of ITS-G5 is severely degraded, while impact on LTE-V2X is marginal. Additionally, a mitigation method where the CAM data generation is constrained to periodical intervals is shown to reduce the impact of co-channel coexistence, with less degradation on ITS-G5 performance and even improvement for LTE-V2X.

The experimental characterisation of the swimming statistics of populations of microorganisms or artificially propelled particles is essential for understanding the physics of active systems and their exploitation. Here, we construct a theoretical framework to extract information on the three-dimensional motion of micro-swimmers from the Intermediate Scattering Function (ISF) obtained from Differential Dynamic Microscopy (DDM). We derive theoretical expressions for the ISF of helical and oscillatory breaststroke swimmers, and test the theoretical framework by applying it to video sequences generated from simulated swimmers with precisely-controlled dynamics. We then discuss how our theory can be applied to the experimental study of helical swimmers, such as active Janus colloids or suspensions of motile microalgae. In particular, we show how fitting DDM data to a simple, non-helical ISF model can be used to derive three-dimensional helical motility parameters, which can therefore be obtained without specialised 3D microscopy equipment. Finally, we discus how our results aid the study of active matter and describe applications of biological and ecological importance., Comment: 26 pages, 9 figures

Self-propelled colloids constitute an important class of intrinsically non-equilibrium matter. Typically, such a particle moves ballistically at short times, but eventually changes its orientation, and displays random-walk behavior in the long-time limit. Theory predicts that if the velocity of non-interacting swimmers varies spatially in 1D, $v(x)$, then their density $\rho(x)$ satisfies $\rho(x) = \rho(0)v(0)/v(x)$, where $x = 0$ is an arbitrary reference point. Such a dependence of steady-state $\rho(x)$ on the particle dynamics, which was the qualitative basis of recent work demonstrating how to `paint' with bacteria, is forbidden in thermal equilibrium. We verify this prediction quantitatively by constructing bacteria that swim with an intensity-dependent speed when illuminated. A spatial light pattern therefore creates a speed profile, along which we find that, indeed, $\rho(x)v(x) = \mathrm{constant}$, provided that steady state is reached.

We use moving light patterns to control the motion of {\it Escherichia coli} bacteria whose motility is photo-activated. Varying the pattern speed controls the magnitude and direction of the bacterial flux, and therefore the accumulation of cells in up- and down-stream reservoirs. We validate our results with two-dimensional simulations and a 1-dimensional analytic model, and use these to explore parameter space. We find that cell accumulation is controlled by a competition between directed flux and undirected, stochastic transport. We articulate design principles for using moving light patterns and light-activated micro-swimmers to achieve particular experimental goals.

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

Unraveling bacterial strategies for spatial exploration is crucial for understanding the complexity in the organization of life. Bacterial motility determines the spatio-temporal structure of microbial communities, controls infection spreading and the microbiota organization in guts or in soils. Most theoretical approaches for modeling bacterial transport rely on their run-and-tumble motion. For Escherichia coli, the run time distribution was reported to follow a Poisson process with a single characteristic time related to the rotational switching of the flagellar motors. However, direct measurements on flagellar motors show heavy-tailed distributions of rotation times stemming from the intrinsic noise in the chemotactic mechanism. Currently, there is no direct experimental evidence that the stochasticity in the chemotactic machinery affect the macroscopic motility of bacteria. In stark contrast with the accepted vision of run-and-tumble, here we report a large behavioral variability of wild-type \emph{E. coli}, revealed in their three-dimensional trajectories. At short observation times, a large distribution of run times is measured on a population and attributed to the slow fluctuations of a signaling protein triggering the flagellar motor reversal. Over long times, individual bacteria undergo significant changes in motility. We demonstrate that such a large distribution of run times introduces measurement biases in most practical situations. Our results reconcile the notorious conundrum between run time observations and motor switching statistics. We finally propose that statistical modeling of transport properties currently undertaken in the emerging framework of active matter studies, should be reconsidered under the scope of this large variability of motility features., Comment: 12 pages, 7 figures, Supplementary information included

We demonstrate differential dynamic microscopy and particle tracking for the characterization of the spatiotemporal behavior of active Janus colloids in terms of the intermediate scattering function (ISF). We provide an analytical solution for the ISF of the paradigmatic active Brownian particle model and find striking agreement with experimental results from the smallest length scales, where translational diffusion and self-propulsion dominate, up to the largest ones, which probe effective diffusion due to rotational Brownian motion. At intermediate length scales, characteristic oscillations resolve the crossover between directed motion to orientational relaxation and allow us to discriminate active Brownian motion from other reorientation processes, e.g., run-and-tumble motion. A direct comparison to theoretical predictions reliably yields the rotational and translational diffusion coefficients of the particles, the mean and width of their speed distribution, and the temporal evolution of these parameters.

External control of the swimming speed of `active particles' can be used to self assemble designer structures in situ on the micrometer to millimeter scale. We demonstrate such reconfigurable templated active self assembly in a fluid environment using light powered strains of Escherichia coli. The physics and biology controlling the sharpness and formation speed of patterns is investigated using a bespoke fast-responding strain., Comment: 19 pages, 11 figures

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

Bacterial motility, and in particular repulsion or attraction towards specific chemicals, has been a subject of investigation for over 100 years, resulting in detailed understanding of bacterial chemotaxis and the corresponding sensory network in many bacterial species. For Escherichia coli most of the current understanding comes from the experiments with low levels of chemotactically-active ligands. However, chemotactically-inactive chemical species at concentrations found in the human gastrointestinal tract produce significant changes in E. coli's osmotic pressure, and have been shown to lead to taxis. To understand how these nonspecific physical signals influence motility, we look at the response of individual bacterial flagellar motors under step-wise changes in external osmolarity. We combine these measurements with a population swimming assay under the same conditions. Unlike for chemotactic response, a long-term increase in swimming/motor speeds is observed, and in the motor rotational bias, both of which scale with the osmotic shock magnitude. We discuss how the speed changes we observe can lead to steady state bacterial accumulation., Comment: 24 pages, 11 figures