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We study motility-induced phase separation in symmetric and asymmetric active binary mixtures. We start with the coarse-grained run-and-tumble bacterial model that provides evolution equations for the density fields ρ i ( r ⃗ , t). Next, we study the phase separation dynamics by solving the evolution equations using the Euler discretization technique. We characterize the morphology of domains by calculating the equal-time correlation function C(r, t) and the structure factor S(k, t), both of which show dynamical scaling. The form of the scaling functions depends on the mixture composition and the relative activity of the species, Δ. For k → ∞, S(k, t) follows Porod's law: S(k, t) ∼ k−(d+1) and the average domain size L(t) shows a diffusive growth as L(t) ∼ t1/3 for all mixtures. [ABSTRACT FROM AUTHOR]

Local stay of the solutions to the Euler equations for an ideal incompressible fluid in the end‐point Triebel–Lizorkin space F1,∞sℝd$$ {F}_{1,\infty}^s\left({\mathbb{R}}^d\right) $$ with s≥d+1$$ s\ge d+1 $$ is clarified. [ABSTRACT FROM AUTHOR]

We investigate the well-posedness of \alpha-surface quasi-geostrophic (\alpha-SQG) equations in the half-plane, where \alpha =0 and \alpha =1 correspond to the 2D Euler and SQG equations respectively. For 0<\alpha \le 1/2, we prove local well-posedness in certain weighted anisotropic Hölder spaces. We also show that such a well-posedness result is sharp: for any 0<\alpha \le 1, we prove nonexistence of Hölder regular solutions (with the Hölder regularity depending on \alpha) for initial data smooth up to the boundary. [ABSTRACT FROM AUTHOR]

Ensuring safe flight is a fundamental prerequisite for developing hypersonic propulsion systems. A comprehensive investigation of the steady boundary associated with oblique detonation wave in a wide speed range was conducted, with the aim of exploring the feasibility of oblique detonation engine across a diverse array of flight conditions. In this study, the wedge angle applicable in a wide-speed range was acquired via the analysis of oblique detonation wave polar curve. The configuration of the internal injection oblique detonation engine was subsequently designed and established, considering the effect of fuel-air inhomogeneity and complex wave system interactions within a confined combustor. The compressible Euler equations coupled with a 9-species and 19-step chemical reaction mechanism are employed to simulate the oblique detonation process. Ultimately, the safe flight envelope of an air-breathing vehicle equipped with the internal injection oblique detonation engine is mapped across a broad range of Mach numbers, demonstrating the engine's capability to operate within the Mach 8 to 12 range. Furthermore, the findings reveal that decreasing either the flight Mach number or altitude results in unsteady oblique detonation wave within the internal injection oblique detonation engine combustor, however, reducing the equivalence ratio can stabilize the oblique detonation wave once again. This study provides valuable guidance for the design and wide-speed-range operation of an internal injection oblique detonation engine. • An Internal Injection Oblique Detonation Engine is proposed for flight Mach numbers from 8 to 12. • The safe flight envelope of an air-breathing vehicle equipped with the IIODE is obtained. • The steady boundary associated with oblique detonation wave in a wide speed range is studied. • The flow and combustion properties within the IIODE are discussed. • The impacts of flight Mach number, altitude height and equivalence ratio on the stability of ODW are analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the time-periodic solutions of one-dimensional compressible Euler equations with linear damping $ \alpha(x)\rho u $. The boundary conditions have no dissipative structure on one side, which involves some physical reality. By using a delicately designed iteration scheme, we establish the existence and stability of time-periodic solutions on the perturbation of a subsonic Fanno flow. Moreover, if the boundary has a higher regularity, we show that the time-periodic solution has the same higher regularity and stability. Our results are also valid for Euler equations without damping, in that case the steady subsonic background solution is a constant state. Both isentropic and isothermal polytropic gases are considered. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in comparing solutions to stochastic Volterra equations for the convex order on the space of continuous R d -valued paths and for the monotonic convex order when d = 1 . Even if these solutions are in general neither semimartingales nor Markov processes, we are able to exhibit conditions on their coefficients enabling the comparison. Our approach consists in first comparing their Euler schemes and then taking the limit as the time step vanishes. We consider two types of Euler schemes, depending on the way the Volterra kernels are discretised. The conditions ensuring the comparison are slightly weaker for the first scheme than for the second, and this is the other way around for convergence. Moreover, we weaken the integrability needed on the starting values in the existence and convergence results in the literature to be able to only assume finite first moments, which is the natural framework for convex ordering. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying structural stability of steady subsonic Euler flows in 2D finitely long nozzles. The reference flow is subsonic shear flows with general size of vorticity. The problem is described by the steady compressible Euler system. With admissible physical conditions and prescribed pressures at the entrances and the exits of the nozzles respectively, we establish unique existence and structural stability of this kind of subsonic shear flows. Due to the hyperbolic-elliptic coupled form of the Euler system in subsonic regions, the problem is reformulated via Lagrange transformation and then decoupled into an elliptic mode and two hyperbolic modes. The elliptic mode is a mixed type boundary value problem of second order quasilinear elliptic equation for the stream function. The hyperbolic modes are transport types to control the total energy and the entropy. Mathematically, the iteration scheme is executed in a weight Hölder space with low regularity. • 2-D subsonic Euler flow with physical boundary conditions. • Weighted Hölder space with optimally low regularity. • Reformulating 2-D compressible Euler system via Lagrange transformation. [ABSTRACT FROM AUTHOR]

We obtain the local existence and uniqueness for a system describing interaction of an incompressible inviscid fluid, modeled by the Euler equations, and an elastic plate, represented by the fourth-order hyperbolic PDE. We provide a priori estimates for the existence with the optimal regularity H r , for r > 2.5 , on the fluid initial data and construct a unique solution of the system for initial data u 0 ∈ H r for r ≥ 3. An important feature of the existence theorem is that the Taylor-Rayleigh instability does not occur. This is in contrast to the free-boundary Euler problem, where the stability condition on the initial pressure needs to be imposed. [ABSTRACT FROM AUTHOR]

In this paper, two newly proposed generalized Hamiltonian Conservative Chaotic Systems (CCSs) based on integrated 4D Euler equations are first discussed. Second, another generalized Hamiltonian CCS is also proposed. Furthermore, a class of generalized Hamiltonian CCSs based on integrated 4D Euler equations is given and studied. Subsequently, numerical analysis for the class of generalized Hamiltonian CCSs is further investigated to show their advantages over most of the existing CCSs, where the corresponding Lyapunov exponents' diagrams, bifurcation diagrams and phase portraits are all given to show their multistability and complex dynamics. Finally, the class of generalized Hamiltonian CCSs is implemented by using FPGA technology, and the results observed in FPGA implementation are consistent with those observed in the numerical analysis. All these results not only show multistability from a physical point of view, but also provide new physical models for chaos applications. [ABSTRACT FROM AUTHOR]

This document explores the impact of wind on the shape of shoaling and overturning waves using a two-phase DNS model, focusing on the wind Reynolds number (Re ∗) and its effects on wave shape. The simulations reveal that wind influences wave faces' steepness and geometrical parameters of overturning waves, with pressure and viscous stresses playing a role at the air-water interface. The findings suggest that wind significantly affects wave shape and overturning dynamics, potentially impacting coastal engineering and morphological evolution. The research compiled in this document provides valuable insights into fluid mechanics, surf zone dynamics, wind wave growth, and numerical modeling of wind effects on breaking waves, published in reputable journals. [Extracted from the article]

Purpose: Rollover of a semi-trailer vehicle is a common type of instability that can occur when turning at high speeds on roads with high adhesion coefficients, or when colliding with other vehicles. In order to design effective early warning and anti-roll control systems, it is important to accurately determine the rollover thresholds. This research aims to determine the dynamic rollover thresholds of a tractor semi-trailer vehicle during turning maneuvers based on a dynamic model. Methods: A full dynamic model of a semi-trailer vehicle with 48 degrees of freedom has been established using the Multibody Dynamic analysis method and Newton Euler equations. The non-linear tire model is used to determine tire-road interaction forces when affected by tire-road deformation, adhesion coefficient of road, road wheel angle, and vehicle velocity. To validate the accuracy of the model, experiments were conducted on a road with a radius of 40 m. The verified model was then used to investigate the vehicle's dynamics during turning maneuvers at velocities ranging from 40 to 80 km/h and the magnitude of steering wheel angles ranging from 12.5 to 300 deg. Results: The results show the combined impact of the magnitude of the steering wheel angle and initial velocity on the rollover condition of the tractor semi-trailer vehicle. The 3D maps and table of maximum values for load transfer ratio and lateral acceleration of the vehicle bodies reveal the regions of roll stability and rollover. The dynamic rollover instability threshold for the tractor ranges from 4.382 to 4.838 m/s2, while that of the semi-trailer ranges from 4.022 to 4.433 m/s2. Conclusions: The article presents a method for determining the dynamic rollover threshold of tractor semi-trailer vehicles based on a full dynamic model. The dynamics rollover thresholds are the boundaries that separate the rollover region from the stability region of the lateral acceleration of the tractor and semi-trailer bodies. The research findings in this article serve as a basis for determining input parameters for early warning and anti-rollover control systems in tractor semi-trailer vehicles. These results will be further discussed in the next articles. [ABSTRACT FROM AUTHOR]

This paper investigates the theoretical implications of applying Darcy's law to premixed flames, a topic of growing interest in research on flame propagation in porous media and confined geometries. A multiple-scale analysis is carried out treating the flame as a hydrodynamic discontinuity in density, viscosity, and permeability. The analysis accounts in particular for the inner structure of the flame. A simple model is derived allowing the original conservation equations to be replaced by Laplace's equation for pressure, applicable on both sides of the flame front, subject to specific conditions across the front. Such model is useful for investigating general problems under confinement including flame instabilities in porous media or Hele-Shaw channels. In this context, two Markstein numbers are identified, for which explicit expressions are provided. In particular, our analysis reveals novel contributions to the local propagation speed arising from discontinuities in the tangential components of velocity and gravitational force, which are permissible in Darcy's flows to leading order, but not in flows obeying Euler or Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

In this paper we consider the incompressible 2D Euler equation in an annular domain with non-penetration boundary condition. In this setting, we prove the existence of a family of non-trivially smooth time-periodic solutions at an arbitrarily small distance from the stationary Taylor-Couette flow in H s , with s < 3 2 , at the vorticity level. [ABSTRACT FROM AUTHOR]

This work is concerned with limit behavior of the Navier–Stokes–Voigt equation with degenerate white noise in a 3D thin domain. Although the individual solutions may be chaotic in fluid dynamics, the stationary statistical solutions are essential to capture complex dynamical behaviors in the view of statistic. We therefore prove that the stationary statistical solution of the system converges weakly to the counterpart of the 2D stochastic Euler equation as the viscosity, the elasticity parameter and the thickness of the thin domain tend to zero. [ABSTRACT FROM AUTHOR]

In this paper, we consider two-dimensional periodic capillary-gravity waves traveling under the influence of a vertical electric field. The full system is a nonlinear, two-layered, free boundary problem. The interface dynamics are derived by coupling Euler equations for the velocity field of the fluid with voltage potential equations governing the electric field. We first introduce the naive flattening technique to transform the free boundary problem into a fixed boundary problem. We then prove the existence of small-amplitude electrohydrodynamic waves with constant vorticity using local bifurcation theory. Moreover, we show that these electrohydrodynamic waves are formally stable in the linearized sense. Furthermore, we obtain a secondary bifurcation curve that emerges from the primary branch, consisting of ripple solutions on the interface. As far as we know, such solutions in electrohydrodynamics are established for the first time. It is worth noting that the electric field E 0 plays a key role in controlling the shapes and types of waves on the interface. [ABSTRACT FROM AUTHOR]

The entropic regularization of the conservative stable discontinuous Galerkin method (DGM) in conservative variables for three-dimensional Euler equations is constructed through the use of a special slope limiter. This limiter ensures the fulfillment of the three-dimensional analogs of the monotonicity conditions and a discrete analog of the entropic inequality. The developed method is tested on a three-dimensional model problem of a Taylor–Green vortex. [ABSTRACT FROM AUTHOR]

We propose Coadjoint Orbit FLIP (CO-FLIP), a high order accurate, structure preserving fluid simulation method in the hybrid Eulerian-Lagrangian framework. We start with a Hamiltonian formulation of the incompressible Euler Equations, and then, using a local, explicit, and high order divergence free interpolation, construct a modified Hamiltonian system that governs our discrete Euler flow. The resulting discretization, when paired with a geometric time integration scheme, is energy and circulation preserving (formally the flow evolves on a coadjoint orbit) and is similar to the Fluid Implicit Particle (FLIP) method. CO-FLIP enjoys multiple additional properties including that the pressure projection is exact in the weak sense, and the particle-to-grid transfer is an exact inverse of the grid-to-particle interpolation. The method is demonstrated numerically with outstanding stability, energy, and Casimir preservation. We show that the method produces benchmarks and turbulent visual effects even at low grid resolutions. [ABSTRACT FROM AUTHOR]

We propose a novel framework for simulating ink as a particle-laden flow using particle flow maps. Our method addresses the limitations of existing flow-map techniques, which struggle with dissipative forces like viscosity and drag, thereby extending the application scope from solving the Euler equations to solving the Navier-Stokes equations with accurate viscosity and laden-particle treatment. Our key contribution lies in a coupling mechanism for two particle systems, coupling physical sediment particles and virtual flow-map particles on a background grid by solving a Poisson system. We implemented a novel path integral formula to incorporate viscosity and drag forces into the particle flow map process. Our approach enables state-of-the-art simulation of various particle-laden flow phenomena, exemplified by the bulging and breakup of suspension drop tails, torus formation, torus disintegration, and the coalescence of sedimenting drops. In particular, our method delivered high-fidelity ink diffusion simulations by accurately capturing vortex bulbs, viscous tails, fractal branching, and hierarchical structures. [ABSTRACT FROM AUTHOR]

This paper presents a WENO-based upwind rotated lattice Boltzmann flux solver (WENO-URLBFS) in the finite difference framework for simulating compressible flows with contact discontinuities and strong shock waves. In the method, the original rotating lattice Boltzmann flux solver is improved by applying the theoretical solution of the Euler equation in the tangential direction of the cell interface to reconstruct the tangential flux so that the numerical dissipation can be reduced. The fluxes at each interface are evaluated using a weighted summation of lattice Boltzmann solutions in two local perpendicular directions decomposed from the direction vector so that the stability performance can be improved. To achieve high-order accuracy, both fifth and seventh-order WENO reconstructions of the flow variables in the characteristic spaces are carried out. The order accuracy of the WENO-URLBFS is evaluated and compared with the traditional Lax–Friedrichs scheme, Roe scheme, and the LBFS by simulating the advection of the density disturbance problem. It is shown that the fifth and seventh-order accuracy can be achieved by all considered flux-evaluation schemes, and the present WENO-URLBFS has the lowest numerical dissipation. The performance of the WENO-URLBFS is further examined by simulating several 1D and 2D examples, including shock tube problems, Shu–Osher problems, blast wave problems, double Mach reflections, 2D Riemann problems, K-H instability problems, and High Mach number astrophysical jets. Good agreements with published data have been achieved quantitatively. Moreover, complex flow structures, including shock waves and contact discontinuities, are successfully captured. The present WENO-URLBFS scheme seems to present an effective numerical tool with high-order accuracy, lower numerical dissipation, and strong robustness for simulating challenging compressible flow problems. [ABSTRACT FROM AUTHOR]

We present a formulation of special relativistic dissipative hydrodynamics (SRDHD) derived from the well-established Müller–Israel–Stewart (MIS) formalism using an expansion in deviations from ideal behaviour. By re-summing the non-ideal terms, our approach extends the Euler equations of motion for an ideal fluid through a series of additional source terms that capture the effects of bulk viscosity, shear viscosity, and heat flux. For efficiency these additional terms are built from purely spatial derivatives of the primitive fluid variables. The series expansion is parametrized by the dissipation strength and time-scale coefficients, and is therefore rapidly convergent near the ideal limit. We show, using numerical simulations, that our model reproduces the dissipative fluid behaviour of other formulations. As our formulation is designed to avoid the numerical stiffness issues that arise in the traditional MIS formalism for fast relaxation time-scales, it is roughly an order of magnitude faster than standard methods near the ideal limit. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem of a two-phase flow system consisting of the compressible isothermal Euler equations and the incompressible Navier-Stokes equations coupled through the drag force, which can be formally derived from the Vlasov-Fokker-Planck/incompressible Navier-Stokes equations. When the initial data is a small perturbation around an equilibrium state, we prove the global well-posedness of the classical solutions to this system and show the solutions tends to the equilibrium state as time goes to infinity. In order to resolve the main difficulty arising from the pressure term of the incompressible Navier-Stokes equations, we properly use the Hodge decomposition, spectral analysis, and energy method to obtain the L 2 time decay rates of the solution when the initial perturbation belongs to L 1 space. Furthermore, we show that the above time decay rates are optimal. [ABSTRACT FROM AUTHOR]

We consider the patch problem for the α$\alpha$‐(surface quasi‐geostrophic) SQG system with the values α=0$\alpha =0$ and α=12$\alpha = \frac{1}{2}$ being the 2D Euler and the SQG equations respectively. It is well‐known that the Euler patches are globally wellposed in non‐endpoint Ck,β$C^{k,\beta }$ Hölder spaces, as well as in W2,p$W^{2,p}$, 1every C2,β$C^{2,\beta }$ Hölder space with β<1$\beta <1$. Moreover, in a suitable range of regularity, the same strong illposedness holds for every W2,p$W^{2,p}$ Sobolev space unless p=2$p=2$. [ABSTRACT FROM AUTHOR]

We present a condition under which the thermal quasi-geostrophic (TQG) model possesses a solution that is holomorphic in time with values in the Gevrey space of complex analytic functions. This can be seen as the complex extension of the work by Levermore and Oliver (1997) for the generalised Euler equation but applied to the TQG model. [ABSTRACT FROM AUTHOR]

In this paper we study the stability of a special magnetic Bénard system near equilibrium, where there exists Laplacian magnetic diffusion and temperature damping but the velocity equation involves no dissipation. Without any velocity dissipation, the fluid velocity is governed by the two-dimensional incompressible Euler equation, whose solution can grow rapidly in time. However, when the fluid is coupled with the magnetic field and temperature through the magnetic Bénard system, we show that the solution is stable. Our results mathematically illustrate that the magnetic field and temperature have the effect of enhancing dissipation and contribute to stabilize the fluid. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial value problem to the higher dimensional Euler equations in the whole space. Based on the local well-posedness result and the lifespan, we prove that the data-to-solution map of this problem is not uniformly continuous in nonhomogeneous Besov spaces. Our obtained result improves considerably the previous results given by Himonas-Misiołek (2010) [8] and Pastrana (2021) [21]. [ABSTRACT FROM AUTHOR]

The article in the Journal of Fluid Mechanics discusses time-periodic generalised solitary waves with a hydraulic fall in open channel flow. The study focuses on unsteady solutions emerging from an unstable steady solution, featuring large-amplitude time-periodic ripples caused by a sudden decrease in water depth known as a hydraulic fall. The research highlights the importance of considering solution stability and the long-term trends of unstable configurations in free-boundary problems, offering insights into fluid dynamics over topography. [Extracted from the article]

The article "Exact planetary waves and jet streams" delves into the study of nonlinear waves on surfaces resembling a rotating sphere for two-dimensional inviscid incompressible flow. It examines exact wave solutions in the Lagrangian reference frame within the β-plane and γ-approximation systems, showcasing non-trivial Lagrangian mean flow akin to polar jet streams. Through numerical experiments, the research investigates the temporal evolution of these waves and their connection to vorticity, shedding light on atmospheric dynamics and climate studies. The text also discusses the derivation of governing equations for two-dimensional flow on a sphere using stereographic coordinates and β and γ approximations, highlighting the stability and symmetry of vortex waves on the sphere. [Extracted from the article]

Most engineering tools dealing with detonation waves in condensed explosives are based on the reactive Euler equations, which involve a thermodynamic closure based on temperature and pressure equilibrium conditions. Indeed, the reactant and the detonation products are assumed to evolve in mechanical and thermal equilibrium. Although the assumption of thermal equilibrium is physically questionable, the reactive Euler equations are used for convenience. This choice is supported by several reasons. When considering thermal equilibrium, the assessment of thermal exchanges between the reactant and the detonation products is simplified, as no exchange coefficient needs to be determined. Furthermore, the reactive Euler equations are in conservative form and the shock relations are well-defined. This model appears to be the simplest option for addressing detonations in condensed explosives. However, this simplicity has limitations and conceals a subtle difficulty that can lead to pathological detonations. The present contribution investigates this difficulty and presents a global exothermic condition to preserve exothermic detonations in the frame of temperature–pressure equilibrium flow models. This condition depends on the equations of state only. It is independent of the kinetics used to model the decomposition of the explosive. By meticulously selecting the thermodynamic parameters of the equations of state for the reactant and detonation products, pathological detonations are prevented, thereby maintaining a globally exothermic reaction consistent with the Zeldovich–von Neumann–Döring theory. [ABSTRACT FROM AUTHOR]

This paper is concerned with the analytical construction of piecewise smooth solutions containing a single shock wave for the radially symmetric relativistic Euler equations with polytropic gases. We derive meticulously the a prioriC1$C^1$‐estimates on the Riemann invariants of the governing system under some assumptions on the piecewise initial data. Based on these estimates, we show that the long time of existence of smooth solutions in the angular region bounded by a characteristic curve and a shock curve. The piecewise smooth initial conditions ensured the existence of smooth solutions in the angular region are discussed. Moreover, it is verified that the existence time is proportional to the initial discontinuous position. [ABSTRACT FROM AUTHOR]

We study the formation of singularities of smooth solutions to the relativistic Euler equations of Chaplygin gases in Schwarzschild spacetime. The system is in the spherically symmetric form, and its coefficients and nonhomogeneous terms contain a parameter reflecting the mass of the black hole, which makes it highly nonlinear and complicated. To overcome the influence of the mass parameter of black hole, we introduce a pair of suitable auxiliary variables related to it and derive their characteristic decompositions to establish the estimates of the smooth solution. We show that, for a kind of initial data, the smooth solution develops singularity in finite time and the mass‐energy density itself approaches infinity at the blowup point. [ABSTRACT FROM AUTHOR]

Euler deconvolution is widely used as a semi-automatic interpretation technique for potential field data. In its original form it uses derivatives of orders 0 and 1, while second order Euler deconvolution uses derivatives of orders 0 and 2. Complete Euler deconvolution is introduced here, and this combines both methods to use derivatives of orders 0, 1, and 2. The use of the additional information in the three orders of derivatives provides increased accuracy compared to the original method, and when combined with other Euler equations it can also determine the structural index of the source. The methods are applied to both synthetic data and to two aeromagnetic data profiles from South Africa. [ABSTRACT FROM AUTHOR]

We propose a simple geometric optics analog of a gravitational lens with a refractive index equal to one at large distances and scaling like n (r) 2 = 1 + C 2 / r 2 , where C is a constant. We obtain the equation for ray trajectories from Fermat's principle of least time and the Euler equation. Our model yields a very simple ray trajectory equation. The optical rays bending, reflecting, and looping around the lens are all described by a single trigonometric function in polar coordinates. Optical rays experiencing fatal attraction are described by a hyperbolic function. We use our model to illustrate the formation of Einstein rings and multiple images. Editor's Note: This article describes a simple theoretical model for gravitational lensing. The authors analyze a graded index of refraction that reproduces the behavior for light passing near the event horizon of a black hole. The mathematical simplicity of the model permits exploration of the effects of gravitational lensing—including bending, reflection, and the formation of Einstein rings—using only integral calculus and Fermat's principle. The authors illustrate many interesting lensing phenomena with 2D and 3D graphics. The model described in this paper could be introduced as a "theoretical toy model" to complement classroom demonstrations of gravitational lensing such as a "logarithmic lens" or the stem of a wine glass, making gravitational lensing and its use in modern astrophysics accessible to introductory physics students. [ABSTRACT FROM AUTHOR]

In this article, we present a formally fourth‐order accurate hybrid‐variable (HV) method for the Euler equations in the context of method of lines. The HV method seeks numerical approximations to both cell averages and nodal solutions and evolves them in time simultaneously; and it is proved in previous work that these methods are supraconvergent, that is, the order of the method is higher than that of the local truncation error. Taking advantage of the supraconvergence, the method is built on a third‐order discrete differential operator, which approximates the first spatial derivative at each grid point using only the information in the two neighboring cells. Analyses of stability, accuracy, and pointwise convergence are conducted in the one‐dimensional case for the linear advection equation; whereas extension to nonlinear systems including the Euler equations is achieved using characteristic decomposition and the incorporation of a residual‐consistent viscosity to capture strong discontinuities. Extensive numerical tests are presented to assess the numerical performance of the method for both 1D and 2D problems. [ABSTRACT FROM AUTHOR]

A novel physical effect has been found to be illuminated of being physically self-consistent to take into account in further calculating the dynamics of magnetically-induced asteroid rotation, surface of which has mostly a conducting structure (whereas such mostly metallic-formed asteroid is assumed to be moving in an external magnetic field of planet but in the meantime to be orbiting preferably outside its sphere of effective attraction). The last condition of asteroid surface's having conducting structure means the existence of the applied external torques stemming from the physical interactions between external magnetic field and iron asteroid itself (including long-term effect due to torques stemming from arising eddy-currents on metallic surface of asteroid). Such eddy-currents should heat the conducting surface of asteroid with further non-uniformly re-directing the heat flow from surface of asteroid into outer space via fluxes of thermal photons which carry momentum (in orthogonal direction to the surface which is in most cases far from the ideal surface of sphere). This leads by taking into account the overall outcome of heat flow (from the non-ideal surface of asteroid) to a kind of long-term magnetic second-grade YORP effect. System of Euler equations for aforementioned dynamics of asteroid magnetic rotation has been investigated in regard to existence of semi-analytical solution. Various perturbations (such as collisions, YORP effect) may destabilize the rotation of asteroid deviating it from the current spin state, whereas the electric(eddy)current-induced dissipation of energy reduces kinetic one of asteroid spin. So, dynamics of the asteroid rotation should result in a spinning about maximal-inertia axis with the proper spin state corresponding to minimal energy with a fixed angular momentum. • New physical effect is found for magnetically-induced metallic asteroid rotation. • Metallic-formed asteroid is assumed to be moving in an external magnetic field. • Conducting structure means torques stemming from an eddy-currents on its surface. • Overall outcome of non-iniform heat flow forms magnetic second-grade YORP effect. • Evolution of spin towards rotation about maximal-inertia axis is approximated. [ABSTRACT FROM AUTHOR]

In this paper, we develop a high-order well-balanced discontinuous Galerkin method for hyperbolic balance laws based on the Gauss-Lobatto quadrature rules. Important applications of the method include preserving the non-hydrostatic equilibria of shallow water equations with non-flat bottom topography and Euler equations in gravitational fields. The well-balanced property is achieved through two essential components. First, the source term is reformulated in a flux-gradient form in the local reference equilibrium state to mimic the true flux gradient in the balance laws. Consequently, the source term integral is discretized using the same approach as the flux integral at Gauss-Lobatto quadrature points, ensuring that the source term is exactly balanced by the flux in equilibrium states. Our method differs from existing well-balanced DG methods for shallow water equations with non-hydrostatic equilibria, particularly in the aspect that it does not require the decomposition of the source term integral. The effectiveness of our method is demonstrated through ample numerical tests. [ABSTRACT FROM AUTHOR]

In this article, we investigate the incompressible 2D Euler equations on a rotating biaxial ellipsoid, which model the dynamics of the atmosphere of a Jovian planet. We study the non-zonal Rossby–Haurwitz solutions of the Euler equations on an ellipsoid, while previous works only considered the case of a sphere. Our main results include: the existence and uniqueness of the stationary Rossby–Haurwitz solutions; the construction of the traveling-wave solutions; and the demonstration of the Lyapunov instability of both the stationary and the traveling-wave solutions. [ABSTRACT FROM AUTHOR]

In this paper, a kind of classic generalized Riemann problem for 2-dimensional isothermal Euler equations for compressible gas dynamics is considered. The problem is the gas (u 0 , v 0 , r 0 ∣ x ∣ β) in the rectangular region expands into the vacuum. We construct the solution of the following form u = u (ξ , η) , v = v (ξ , η) , ρ = t β ϱ (ξ , η) , ξ = x t , η = y t , where ρ and (u, v) denote the density and the velocity fields respectively, and u 0 , v 0 , r 0 > 0 and β ∈ (- 1 , 0) ∪ (0 , + ∞) are constants. The continuity of the self-similar solution depends on the value of β . Under certain conditions, we get a weak solution with shock wave, which is necessarily generated initially and move apart along a plane. Furthermore, by the method of characteristic analysis, we explain the mechanism of the shock wave. [ABSTRACT FROM AUTHOR]

We consider a complexification of the Euler equations introduced by Šverák in [35] which conserves energy. We prove that these complex Euler equations are nonlinearly ill-posed below analytic regularity and, moreover, we exhibit solutions which lose analyticity in finite time. Our examples are complex shear flows and, hence, one-dimensional. This motivates us to consider fully nonlinear systems in one spatial dimension which are non-hyperbolic near a constant equilibrium. We prove nonlinear ill-posedness and finite-time singularity for these models. Our approach is to construct an infinite-dimensional unstable manifold to capture the high frequency instability at the nonlinear level. [ABSTRACT FROM AUTHOR]

It is well-known that the first energy shell, $ \mathcal{S}_1^{c_0}: = \{\alpha \cos(x+\mu)+\beta\cos(y+\lambda): \alpha^2+\beta^2 = c_0\, \, \&\, \, (\mu, \lambda)\in\mathbb{R}^2\} $of solutions to the 2d Euler equation is Lyapunov stable on $ \mathbb{T}^2 $. This is simply a consequence of the conservation of energy and enstrophy. Using the idea of Wirosoetisno and Shepherd [17], which is to take advantage of conservation of a properly chosen Casimir, we give a simple and quantitative proof of the $ L^2 $ stability of single modes up to translation. In other words, each$ \mathcal{S}_1^{\alpha, \beta}: = \{\alpha \cos(x+\mu)+\beta\cos(y+\lambda): (\mu, \lambda)\in\mathbb{R}^2\} $is Lyapunov stable. Interestingly, our estimates indicate that the extremal cases $ \alpha = 0, $ $ \beta = 0 $, and $ \alpha = \pm\beta $ may be markedly less stable than the others. [ABSTRACT FROM AUTHOR]

From an open set of initial data, we construct a family of classical solutions to the 1D nonisentropic compressible Euler equations which form $ C^{0,\nu} $ cusps as a first singularity, for any $ \nu \in [\frac{1}{2}, 1) $. For this range of $ \nu $, this is the first result demonstrating the stable formation of such $ C^{0,\nu} $ cusp-type singularities, also known as pre-shocks. The proof uses a new formulation of the differentiated Euler equations along the fast acoustic characteristic, and relies on a novel set of $ L^p $ energy estimates for all $ 1

In this article, we investigate the Euler-α\alpha equations in a three-dimensional bounded domain. On the one hand, we prove in the Euler setting that the equations are locally well-posed with initial data in Hs(s≥3){H}^{s}\left(s\ge 3). On the other hand, the relationship between the Hs{H}^{s}-norm of the velocity field and the parameter α\alpha is clarified.

A recently initiated variant of density functional theory utilizes a set of spherically symmetric densities instead of the density. The exact functionals are unknown in the new theory akin to the standard density functional theory. In order to test approximate functionals exactly solvable models are introduced. A harmonic molecular ion, the analogue to the hydrogen molecule ion and a harmonic two-electron molecule showing analogy to the hydrogen molecule are proposed. It has been found that the wave function and the density can be given analytically. The exact spherical densities and the effective potentials of the Euler equations also have analytical form. It has been shown that the models can be easily extended to several "nuclei." [ABSTRACT FROM AUTHOR]

The Pauli kinetic energy functional and its functional derivative, termed Pauli potential, play a crucial role in the successful implementation of orbital-free density functional theory for electronic structure calculations. However, the exact forms of these two quantities are not known. Therefore, perforce, one employs the approximate forms for the Pauli functional or Pauli potential for performing orbital-free density functional calculations. In the present study, we developed a feed-forward neural network-based representation for the Pauli potential using a 1-dimensional (1-D) model system. We expanded density in terms of basis functions, and the coefficients of the expansion were used as input to a feed-forward neural network. Using the neural network-based representation of the Pauli potential, we calculated the ground-state densities of the 1-D model system by solving the Euler equation. We calculated the Pauli kinetic energy using the neural network-based Pauli potential employing the exact relation between the Pauli kinetic energy functional and the potential. The sum of the neural network-based Pauli kinetic energy and the von Weizsäcker kinetic energy resulted in an accurate estimation of the total kinetic energy. The approach presented in this paper can be employed for the calculation of Pauli potential and Pauli kinetic energy, obviating the need for a functional derivative. The present study is an important step in the advancement of application of machine learning-based techniques toward the orbital-free density functional theory-based methods. [ABSTRACT FROM AUTHOR]

We introduce new second-order adaptive low-dissipation central-upwind (LDCU) schemes for the one- and two-dimensional hyperbolic systems of conservation laws. The new adaptive LDCU schemes employ the recently proposed LDCU numerical fluxes computed using the point values reconstructed with the help of adaptively selected nonlinear limiters. To this end, we use a smoothness indicator to detect "rough" parts of the computed solution, where the piecewise linear reconstruction is performed using an overcompressive limiter, which leads to extremely sharp resolution of shock and contact waves. In the "smooth" areas, we use a more dissipative limiter to prevent appearance of artificial kinks and staircase-like structures there. In order to avoid oscillations, we perform the reconstruction in the local characteristic variables obtained using the local characteristic decomposition. We use a smoothness indicator from Löhner (1987) [34] and apply the developed schemes to the one- and two-dimensional Euler equations of gas dynamics. The obtained numerical results clearly demonstrate that the new adaptive LDCU schemes outperform the original ones. [ABSTRACT FROM AUTHOR]

This paper concerns the structural stability of subsonic flows with a contact discontinuity in a finitely long axisymmetric cylinder. We establish the existence and uniqueness of axisymmetric subsonic flows with a contact discontinuity by prescribing the horizontal mass flux distribution, the swirl velocity, the entropy and the Bernoulli's quantity at the entrance and the radial velocity at the exit. It can be formulated as a free boundary problem with the contact discontinuity to be determined simultaneously with the flows. Compared with the two-dimensional case, a new difficulty arises due to the singularity near the axis. An invertible modified Lagrangian transformation is introduced to overcome this difficulty and straighten the contact discontinuity. The key elements in our analysis are to utilize the deformation-curl decomposition introduced in [S. Weng and Z. Xin, A deformation-curl decomposition for three dimensional steady Euler equations, Sci. Sin. Math.49 (2019) 307–320 (in Chinese): doi:10.1360/N012018-00125] to effectively decouple the hyperbolic and elliptic modes in the steady axisymmetric Euler system and to use the implicit function theorem to locate the contact discontinuity. [ABSTRACT FROM AUTHOR]