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Applying an excess entropy scaling formalism to the coarse-grained (CG) dynamics of liquids, we discovered that missing rotational motions during the CG process are responsible for artificially accelerated CG dynamics. In the context of the dynamic representability between the fine-grained (FG) and CG dynamics, this work introduces the well-known Stokes–Einstein and Stokes–Einstein–Debye relations to unravel the rotational dynamics underlying FG trajectories, thereby allowing for an indirect evaluation of the effective rotations based only on the translational information at the reduced CG resolution. Since the representability issue in CG modeling limits a direct evaluation of the shear stress appearing in the Stokes–Einstein and Stokes–Einstein–Debye relations, we introduce a translational relaxation time as a proxy to employ these relations, and we demonstrate that these relations hold for the ambient conditions studied in our series of work. Additional theoretical links to our previous work are also established. First, we demonstrate that the effective hard sphere radius determined by the classical perturbation theory can approximate the complex hydrodynamic radius value reasonably well. Furthermore, we present a simple derivation of an excess entropy scaling relationship for viscosity by estimating the elliptical integral of molecules. In turn, since the translational and rotational motions at the FG level are correlated to each other, we conclude that the "entropy-free" CG diffusion only depends on the shape of the reference molecule. Our results and analyses impart an alternative way of recovering the FG diffusion from the CG description by coupling the translational and rotational motions at the hydrodynamic level. [ABSTRACT FROM AUTHOR]

The Lattice Boltzmann Method (LBM) is an approach for modeling mesoscopic fluid flow and heat transfer, based on modeling distributions of particles moving and colliding on a lattice. Using a perturbative formulation of the Boltzmann equation, it scales to the macroscopic Navier–Stokes equation. We simulate natural thermal convection via LBM in a 2D rectangular box being heated from below and cooled from above, and use the results as training, testing, and generalization datasets to build a deep learning model. GoogLeNet, a convolutional neural network, is used to classify the simulation results based on two parameters: Rayleigh (R a) and Prandtl (P r) numbers, from a single snapshot of either the entire modeling field of resolution 1024 × 1024 , or a 224 × 224 crop. For each fixed P r in a range from 1 to 128, increasing by a factor of 2, we estimate R a with an accuracy varying from 40% to 90%, depending on the chosen augmentation strategy. For each fixed R a in the range from 10 5 to 10 9 , increasing of a factor 10 , the method predicts P r with a systematically lower accuracy ranging from 30% to 80%. This approach has great potential for industrial applications like being able to control the industrial flow or scientific research on geophysical ones including the transport of heat in the earth's interiors, ocean, and atmosphere. [ABSTRACT FROM AUTHOR]

In this paper, we address the lower bound and space-time decay rates for the compressible Hall-MHD equations under H 3 -framework in R 3. First of all, the lower bound of decay rate for the density, velocity and magnetic field converging to the constant equilibrium status in L 2 is (1 + t) − 3 4 ; the lower bound of decay rate for the first order spatial derivative of density and velocity converging to zero in L 2 is (1 + t) − 5 4 , and the k (∈ [ 1 , 3 ]) -th order spatial derivative of magnetic field converging to zero in L 2 is (1 + t) − 3 + 2 k 4 . Secondly, the lower bound of decay rate for time derivative of density and velocity converging to zero in L 2 is (1 + t) − 5 4 ; however, the lower bound of decay rate for time derivatives of magnetic field converging to zero in L 2 is (1 + t) − 7 4 . Finally, we address the decay rate of solution in weighted Sobolev space H γ 3. [ABSTRACT FROM AUTHOR]

In numerical time-integration with implicit-explicit (IMEX) methods, a within-step adaptable decomposition called residual balanced (RB) decomposition is introduced. This new decomposition maintains time-stepping accuracy even when the implicit equation is only roughly approximated. This novel property is possible because a suitable modification of the traditional IMEX algorithm allows the remaining residual to be seamlessly transferred to the explicit part of the decomposition. The RB decomposition allows an early termination of iterations while preserving time-step accuracy. It can gain computational efficiency by exploring the trade-off between the computational effort placed in the iterative solver and the numerically stable step size. We develop a rigorous theory showing that RB maintains the order of singly diagonally implicit schemes. In computational experiments we show that, in many cases, RB-IMEX reduces the number of iterations when compared with the traditional IMEX method. It is often more stable also. The stability of RB-IMEX is studied using a model containing diffusion and dispersion; in this way, one can visualize how the stability region changes as a function of the number of iterations. Here, computational experiments use ESDIRK schemes for a stiff reaction-advection-diffusion equation, for a Navier-Stokes simulation with acoustic stiffness, and for a semi-implicit implementation of Burguers equation. [ABSTRACT FROM AUTHOR]

In this paper, we deal with four problems. (i) Based on criteria for a continuous non-autonomous deterministic dynamical system (NDDS) and a continuous non-autonomous random dynamical system (NRDS), we construct an exponential attractor for a continuous NDDS and a family of random exponential attractors for a family of continuous non-autonomous random dynamical systems (NRDS), respectively. (ii) We prove that this family of random exponential attractors is continuous (or stable, robust, i.e., upper and lower semi-continuous) in the sense of the symmetric Hausdorff distance as the intensity of stochastic perturbations approaches zero. (iii) We prove that for two conjugate NRDS, if one has a random exponential attractor, then the other has a random exponential attractor, and that for two families of conjugate NRDS, if a family of random exponential attractors for one family is continuous, then a corresponding family of random exponential attractors for the other family is continuous. (iv) We apply our abstract result to study the existence and continuity of random exponential attractors for 3D globally modified non-autonomous Navier-Stokes equation with additive noise. [ABSTRACT FROM AUTHOR]

We present a variational approach for the construction of Leray–Hopf solutions to the non-Newtonian Navier–Stokes system. Inspired by the work [M. Ortiz, B. Schmidt and U. Stefanelli, A variational approach to Navier–Stokes, Nonlinearity 31 2018, 12, 5664–5682] on the corresponding Newtonian problem, we minimise certain stabilised Weighted Inertia-Dissipation-Energy (WIDE) functionals and pass to the limit of a vanishing parameter in order to recover a Leray–Hopf solution of the non-Newtonian Navier–Stokes equations. The investigation of the non-Newtonian Navier–Stokes system via this variational approach is particularly well suited to gain insights into weak, respectively strong convergence properties of approximating sequences for different flow-behaviour exponents. With this analysis we extend the results of [M. Bathory and U. Stefanelli, Variational resolution of outflow boundary conditions for incompressible Navier–Stokes, Nonlinearity 35 2022, 11, 5553–5592] to power-law exponents 2 ⁢ d d + 2 < p < 3 ⁢ d + 2 d + 2 {\tfrac{2d}{d+2}

We consider the steady Navier–Stokes equations with mixed boundary conditions, where a regularized directional do-nothing (RDDN) condition is defined on the Neumann boundary portion. An auxiliary Stokes reference flow, which also works as a lifting of the inhomogeneous Dirichlet boundary values, is used to define the RDDN condition. Our aim is to study the minimization of a velocity tracking cost functional with controls localized on a part of the boundary. We prove the existence of a solution for this optimal control problem and derive the corresponding first order necessary optimality conditions in terms of dual variables. All results are obtained under appropriate assumptions on the size of the data and the controls, which, however, are less restrictive when compared with the case of the classical do-nothing outflow condition. This is further confirmed by the numerical examples presented, which include scenarios where only noisy data is available. [ABSTRACT FROM AUTHOR]

In this work, we prove a threshold theorem for the 2D Navier-Stokes equations posed on the periodic channel, T × [ - 1 , 1 ] , supplemented with Navier boundary conditions ω | y = ± 1 = 0 . Initial datum is taken to be a perturbation of Couette in the following sense: the shear component of the perturbation is assumed small (in an appropriate Sobolev space) but importantly is independent of ν . On the other hand, the nonzero modes are assumed size O (ν 1 2) in an anisotropic Sobolev space. For such datum, we prove nonlinear enhanced dissipation and inviscid damping for the resulting solution. The principal innovation is to capture quantitatively the inviscid damping, for which we introduce a new Singular Integral Operator which is a physical space analogue of the usual Fourier multipliers which are used to prove damping. We then include this SIO in the context of a nonlinear hypocoercivity framework. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Keller–Segel system coupled with the Navier–Stokes equations in a bounded domain Ω ⊂ R 2 as follows n t + u · ∇ n = Δ n - ∇ · (n ∇ c) - κ n 2 , c t + u · ∇ c = Δ c - n c , v t + u · ∇ v = Δ v - γ v + n , u t + (u · ∇) u + ∇ π = Δ u - n f , ∇ · u = 0. We establish the global existence of classical solutions for the initial-boundary value problem in a bounded domain under mild assumption on the initial data. [ABSTRACT FROM AUTHOR]

The high-order gas-kinetic scheme (HGKS) with WENO spatial reconstruction method has been extensively validated through numerous numerical experiments, demonstrating its superior accuracy, efficiency and robustness. In comparison to WENO class schemes, TENO class schemes exhibit significantly improved robustness, low numerical dissipation and sharp discontinuity capturing. This paper introduces two types of fifth-order HGKS utilizing TENO class schemes. One approach involves replacing the WENO scheme with the TENO scheme in the conventional WENO GKS. Both WENO and TENO schemes provide non-equilibrium state values at the cell interface. The slopes of the non-equilibrium state, as well as the values and slopes of the equilibrium state, are obtained through additional linear reconstruction. Another type of HGKS scheme named TENO GKS with adaptive order (TENO-A GKS), is similar to most multi-resolution HGKS schemes. Following a robust scale-separation procedure, a tailored ENO-like stencil selection strategy is proposed to ensure that high-order accuracy is maintained in smooth regions by selecting the candidate reconstruction from a large stencil, while enforcing the ENO property near discontinuities by adopting the candidate reconstruction from small smooth stencils. These TENO schemes include TENO schemes with adaptive accuracy order and adaptive dissipation control (TENO-AA) and TENO schemes with dual ENO-like stencil selection (TENO-D). The HGKS scheme based on TENO-A reconstruction takes advantage of the large stencil to provide point values and slopes of the non-equilibrium states. By dynamically merging the reconstructed non-equilibrium slopes, the need for extra reconstruction of the equilibrium states at the beginning of each time step can be avoided. The simple TENO-A GKS scheme exhibits better robustness compared to the conventional WENO GKS or TENO GKS. [ABSTRACT FROM AUTHOR]

In this paper, we consider the incompressible 3D Euler and Navier–Stokes equations in a smooth bounded domain. First, we study the 3D Euler equations endowed with slip boundary conditions and we prove the same criteria for energy conservation involving the gradient, already known for the Navier–Stokes equations. Subsequently, we utilize this finding, which is based on a proper approximation of the velocity (and does not require any estimate or additional assumption on the pressure), to explore energy conservation for Beltrami flows. Finally, we explore Beltrami solutions to the Navier–Stokes equations and demonstrate that conditions leading to energy conservation are significantly distinct from those implying regularity. This remains true even when making use of the bootstrap regularity improvement, stemming from the solution being a Beltrami vector field. [ABSTRACT FROM AUTHOR]

Very recently, Barker (Proc. Amer. Math. Soc. Ser. B 11: 436-451, 2024), Barker and Wang (J Differ Equ 365:379–407, 2023), and, Beirao da Veiga and Yang (J. Geom. Anal. 35: no. 1, Paper No. 15, 14 pp, 2025) applied the higher integrability of weak solutions to study the singular set and energy equality of weak solutions to the incompressible Navier–Stokes equations with supercritical assumptions, respectively. In the spirit of this and, Leslie and Shvydkoy’s work (SIAM J Math Anal 50:870–890, 2018), we present some energy equality criteria for suitable weak solutions up to the first potential blow-up time in terms of pressure, its gradient or the direction of the velocity by L p bound of the incompressible Navier–Stokes equations. Furthermore, along the same lines, we establish some L p estimate of the isentropic compressible Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the fractional Navier-Stokes equations. We extend a previous non-uniqueness result due to Cheskidov and Luo, found in [5], from Navier-Stokes to the fractional case, and from $ L^1 $-in-time, $ W^{1, q} $-in-space solutions for every $ q>1 $ to $ L^s $-in-time, $ W^{\gamma, q} $-in-space solutions for appropriate ranges of $ s, q, \gamma, \theta $, which also extends the result of Li, Qu, Zeng, and Zhang found in [14]. [ABSTRACT FROM AUTHOR]

Computational fluid dynamics (CFD) is increasingly applied in the study of fishways. Applications and methodologies, however, vary substantially among scientific studies and no overview is currently available in the primary literature. Here we review published papers on CFD use in upstream fish passage solutions to identify and describe related spatial-temporal considerations, application fields, scopes and modeling procedures. Vertical slot was the most studied fishway type, followed by nature-like fishways and pool and weir fishways. Most often the CFD model was coupled with physical experiments, but only sometimes associated with actual fish behavior. Reynolds-Averaged Navier-Stokes equations (RANS) was the most frequently adopted set of equations, followed by Large Eddy Simulation (LES), but other promising approaches - scarcely applied so far - were also identified and suggested for future applications - e.g. Detached Eddy Simulation (DES). The use of commercial software was prevalent compared to open-source (notably, Ansys and FLOW-3D). The importance of model validation is highlighted, especially for merely numerical studies, together with the need for three-dimensional CFD to correctly represent turbulent flows. Overall, ecohydraulic studies on the interaction between fish movements and hydrodynamics are needed to complement the CFD-analysis and improve the design of more efficient fish passage solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the incompressible limit of all-time strong solutions to the isentropic compressible Navier-Stokes equations with ill-prepared initial data and slip boundary condition in three-dimensional bounded domains. The uniform estimates with respect to both the Mach number ϵ ∈ (0 , 1 ] and all time t ∈ [ 0 , + ∞) are derived by establishing a nonlinear integral inequality. In contrast to previous results for well-prepared initial data, the time derivatives of the velocity are unbounded which leads to the loss of strong convergence of the velocity. The novelties of this paper are to establish weighted energy estimates of new-type and to carefully combine the estimates for the fast variables and the slow variables, especially for the highest-order spatial derivatives of the fast variables. The convergence to the global strong solution of incompressible Navier-Stokes equations is shown by applying the Helmoltz decomposition and the strong convergence of the incompressible part of the velocity. [ABSTRACT FROM AUTHOR]

This paper is concerned with the nonlinear stability of planar Couette flow for the three‐dimensional NS‐NS equations, which are used to model the motion of two‐phase flow. Our result shows that the planar Couette flow is asymptotically stable for the initial perturbations sufficiently small in some Sobolev space if the background velocity and the Reynolds number are sufficiently small. Moreover, it is proved that the solution converges to the stationary state (ρ∗,us,n∗,ws)$$ \left({\rho}_{\ast },{u}_s,{n}_{\ast },{w}_s\right) $$ at an algebraic time decay rate, and we also give the decay rate of the relative velocity. [ABSTRACT FROM AUTHOR]

In this paper, we derive the optimal time-decay estimates for 2-D inhomogeneous Navier-Stokes equations. In particular, we prove that ‖ u (t) ‖ B ˙ p , 1 θ (R 2) = O (t 1 p − 3 2 − θ 2 ) as t → ∞ for any p ∈ [ 2 , ∞ [ , θ ∈ [ 0 , 2 ] if initially ρ 0 u 0 ∈ B ˙ 2 , ∞ − 2 (R 2). This is optimal even for the classical homogeneous Navier-Stokes equations. Different with Schonbek and Wiegner's Fourier splitting device, our method here seems more direct, and can adapt to many other equations as well. Moreover, our method allows us to work in the L p -based spaces. [ABSTRACT FROM AUTHOR]