Incompressible Euler Equations Research Articles

Linear feedback stabilization of point-vortex equilibria near a Kasper wing

This paper concerns feedback stabilization of point-vortex equilibria above an inclined thin plate and a three-plate configuration known as the Kasper wing in the presence of an oncoming uniform flow. The flow is assumed to be potential and is modelled by the two-dimensional incompressible Euler equations. Actuation has the form of blowing and suction localized on the main plate and is represented in terms of a sink–source singularity, whereas measurement of pressure across the plate serves as system output. We focus on point-vortex equilibria forming a one-parameter family with locus approaching the trailing edge of the main plate and show that these equilibria are either unstable or neutrally stable. Using methods of linear control theory we find that the system dynamics linearized around these equilibria is both controllable and observable for almost all actuator and sensor locations. The design of the feedback control is based on the linear–quadratic–Gaussian (LQG) compensator. Computational results demonstrate the effectiveness of this control and the key finding of this study is that Kasper wing configurations are in general not only more controllable than their single-plate counterparts, but also exhibit larger basins of attraction under LQG feedback control. The feedback control is then applied to systems with additional perturbations added to the flow in the form of random fluctuations of the angle of attack and a vorticity shedding mechanism. Another important observation is that, in the presence of these additional perturbations, the control remains robust, provided the system does not deviate too far from its original state. Furthermore, except in a few isolated cases, introducing a vorticity-shedding mechanism enhanced the effectiveness of the control. Physical interpretation is provided for the results of the controllability and observability analysis as well as the response of the feedback control to different perturbations.

Convergence of a non-isentropic Euler–Poisson system for all time

We consider periodic smooth solutions for a non-isentropic Euler–Poisson system with small parameters, in which the momentum and energy equations are partially dissipative. When initial data are close to constant equilibrium states, we prove the global-in-time convergence of the system as parameters go to zero. The limit systems are incompressible Euler equations with dissipation, the drift-diffusion system and the energy-transport system. The proof of the results is based on compactness arguments and uniform energy estimates with respect to the time and the parameters.

Inviscid incompressible limits for rotating fluids

We consider the inviscid incompressible limits of the rotating compressible Navier–Stokes system for a barotropic fluid. We show that the limit system is represented by the rotating incompressible Euler equation on the whole space.

A variational $\boldsymbol{H}({\rm div})$ finite-element discretization approach for perfect incompressible fluids

We propose a finite element discretisation approach for the incompressible Euler equations which mimics their geometric structure and their variational derivation. In particular, we derive a finite element method that arises from a nonholonomic variational principle and an appropriately defined Lagrangian, where finite element H(div) vector fields are identified with advection operators; this is the first successful extension of the structure-preserving discretisation of Pavlov et al. (2009) to the finite element setting. The resulting algorithm coincides with the energy-conserving scheme presented in Guzm\'an et al. (2016). Through the variational derivation, we discover that it also satisfies a discrete analogous of Kelvin's circulation theorem. Further, we propose an upwind-stabilised version of the scheme which dissipates enstrophy whilst preserving energy conservation and the discrete Kelvin's theorem. We prove error estimates for this version of the scheme, and we study its behaviour through numerical tests.

Gevrey regularity with weight for incompressible euler equation in the half plane

In this work we prove the weighted Gevrey regularity of solutions to the incompressible Euler equation with initial data decaying polynomially at infinity. This is motivated by the well-posedness problem of vertical boundary layer equation for fast rotating fluid. The method presented here is based on the basic weighted L2-estimate, and the main difficulty arises from the estimate on the pressure term due to the appearance of weight function.

A Lagrangian Scheme à la Brenier for the Incompressible Euler Equations

We approximate the regular solutions of the incompressible Euler equations by the solution of ODEs on finite-dimensional spaces. Our approach combines Arnold’s interpretation of the solution of the Euler equations for incompressible and inviscid fluids as geodesics in the space of measure-preserving diffeomorphisms, and an extrinsic approximation of the equations of geodesics due to Brenier. Using recently developed semi-discrete optimal transport solvers, this approach yields a numerical scheme which is able to handle problems of realistic size in 2D. Our purpose in this article is to establish the convergence of this scheme towards regular solutions of the incompressible Euler equations, and to provide numerical experiments on a few simple test cases in 2D.

An h-Adaptive RKDG Method for the Two-Dimensional Incompressible Euler Equations and the Guiding Center Vlasov Model

In this paper, we generalize an h-adaptive Runge–Kutta discontinuous Galerkin scheme developed earlier in Zhu et al. (J Sci Comput 69:1346–1365, 2016) for the 1D Vlasov–Poisson system to the guiding center Vlasov model and the 2D time dependent incompressible Euler equations in the vorticity-stream function formulation. The main difficulty of this generalization lies in solving the 2D Poisson equation due to the irregular adaptive mesh with hanging nodes. We adopt a local discontinuous Galerkin method to solve the Poisson equation. The full adaptive algorithm and the related numerical implementation details are included. Extensive numerical tests have been performed to showcase the effectiveness of the adaptive scheme and its advantage over the fixed-mesh scheme in saving computational cost and improving solution quality.

On the Finite‐Time Blowup of a One‐Dimensional Model for the Three‐Dimensional Axisymmetric Euler Equations

AbstractIn connection with the recent proposal for possible singularity formation at the boundary for solutions of three‐dimensional axisymmetric incompressible Euler's equations (Luo and Hou, Proc. Natl. Acad. Sci. USA (2014)), we study models for the dynamics at the boundary and show that they exhibit a finite‐time blowup from smooth data. © 2017 Wiley Periodicals, Inc.

Spatial asymptotic expansions in the incompressible Euler equation

In this paper we prove that the Euler equation describing the motion of an ideal fluid in $\R^d$ is well-posed in a class of functions allowing spatial asymptotic expansions as $|x|\to\infty$ of any a priori given order. These asymptotic expansions can involve log terms and lead to a family of conservation laws. Typically, the solutions of the Euler equation with initial data in the Schwartz class develop non-trivial spatial asymptotic expansions of the type considered here.

A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization

We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler–Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler–Voigt equations also require less resolution than simulations of the 3D Euler equations for fixed values of the regularization parameter $$\alpha >0$$ . Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly, namely by simulating the better-behaved 3D Euler–Voigt equations. The new criteria are only known to be sufficient criterion for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well known to occur.

On 2d Incompressible Euler Equations with Partial Damping

We consider various questions about the 2d incompressible Navier–Stokes and Euler equations on a torus when dissipation is removed from or added to some of the Fourier modes.

Scale‐selective dissipation in energy‐conserving finite‐element schemes for two‐dimensional turbulence

We analyze the multiscale properties of energy‐conserving upwind‐stabilized finite‐element discretizations of the two‐dimensional incompressible Euler equations. We focus our attention on two particular methods: the Lie derivative discretization introduced by Natale and Cotter and the Streamline Upwind/Petrov–Galerkin (SUPG) discretization of the vorticity advection equation. Such discretizations provide control on enstrophy by modelling different types of scale interactions. We quantify the performance of the schemes in reproducing the non‐local energy backscatter that characterizes two‐dimensional turbulent flows.

Global Existence and Asymptotic Behavior of Affine Motion of 3D Ideal Fluids Surrounded by Vacuum

The 3D compressible and incompressible Euler equations with a physical vacuum free boundary condition and affine initial conditions reduce to a globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in $\rm{GL}^+(3,\mathbb R)$. The evolution of the fluid domain is described by a family ellipsoids whose diameter grows at a rate proportional to time. Upon rescaling to a fixed diameter, the asymptotic limit of the fluid ellipsoid is determined by a positive semi-definite quadratic form of rank $r=1$, 2, or 3, corresponding to the asymptotic degeneration of the ellipsoid along $3-r$ of its principal axes. In the compressible case, the asymptotic limit has rank $r=3$, and asymptotic completeness holds, when the adiabatic index $\gamma$ satisfies $4/3<\gamma<2$. The number of possible degeneracies, $3-r$, increases with the value of the adiabatic index $\gamma$. In the incompressible case, affine motion reduces to geodesic flow in $\rm{SL}(3,\mathbb R)$ with the Euclidean metric. For incompressible affine swirling flow, there is a structural instability. Generically, when the vorticity is nonzero, the domains degenerate along only one axis, but the physical vacuum boundary condition fails over a finite time interval. The rescaled fluid domains of irrotational motion can collapse along two axes.

Ill-posedness for the Incompressible Euler Equations in Critical Sobolev Spaces

For the 2D Euler equation in vorticity formulation, we construct localized smooth solutions whose critical Sobolev norms become large in a short period of time, and solutions which initially belong to $$L^\infty \cap H^1$$ but escapes $$H^1$$ immediately for $$t>0$$ . Our main observation is that a localized chunk of vorticity bounded in $$L^\infty \cap H^1$$ with odd-odd symmetry is able to generate a hyperbolic flow with large velocity gradient at least for a short period of time, which stretches the vorticity gradient.

Global quasi-neutral limit of Euler–Maxwell systems with velocity dissipation

We consider smooth solutions to the Cauchy problem for an isentropic Euler–Maxwell system with velocity dissipation and small physical parameters. For initial data uniformly close to constant equilibrium states, we prove the global-in-time convergence of the system as the parameters go to zero. The limit systems are the e-MHD system and the incompressible Euler equations, both with velocity dissipation. The proof of the results relies on a single uniform energy estimate with respect to the time and the parameters, together with compactness arguments. For this purpose, the classical energy estimates for the symmetrizable hyperbolic system are not sufficient. We construct a Lyapunov type energy by controlling the divergence and the curl of the velocity, the electric and magnetic fields.

Numerical schemes for the coupling of compressible and incompressible fluids in several space dimensions

We present a numerical scheme for immiscible two-phase flows with one compressible phase and one incompressible phase. Special emphasis lies in the discussion of the coupling strategy for compressible and incompressible Euler equations to simulate inviscid liquid–vapor flows. To reduce the computational effort further, we also introduce two approximate coupling strategies. The resulting schemes are compared numerically to a fully compressible scheme and show good agreement with these standard algorithm at lower numerical costs.

Global existence of weak solutions to the three-dimensional Euler equations with helical symmetry

In this paper, we mainly investigate the weak solutions of the three-dimensional incompressible Euler equations with helical symmetry in the whole space when the helical swirl vanishes. Specifically, we establish the global existence of weak solutions when the initial vorticity lies in L1∩Lp with p>1. Our result extends the previous work [2], where the initial vorticity is compactly supported and belongs to Lp with p>4/3. The key ingredient in this paper involves the explicit analysis of Biot–Savart law with helical symmetry in domain R2×[−π,π] via the theories of singular integral operators and second order elliptic equations.

Non-uniqueness and h-Principle for Hölder-Continuous Weak Solutions of the Euler Equations

In this paper we address the Cauchy problem for the incompressible Euler equations in the periodic setting. Based on estimates developed in [Buckmaster-De Lellis-Isett-Sz\'ekelyhidi], we prove that the set of H\"older $1\slash 5-\eps$ wild initial data is dense in $L^2$, where we call an initial datum wild if it admits infinitely many admissible H\"older $1\slash 5-\eps$ weak solutions. We also introduce a new set of stationary flows which we use as a perturbation profile instead of Beltrami flows to recover arbitrary Reynolds stresses.

Asymptotic solution for high-vorticity regions in incompressible three-dimensional Euler equations

Incompressible three-dimensional Euler equations develop high vorticity in very thin pancake-like regions from generic large-scale initial conditions. In this work, we propose an exact solution of the Euler equations for the asymptotic pancake evolution. This solution combines a shear flow aligned with an asymmetric straining flow, and is characterized by a single asymmetry parameter and an arbitrary transversal vorticity profile. The analysis is based on detailed comparison with numerical simulations performed using a pseudospectral method in anisotropic grids of up to $972\times 2048\times 4096$.

Regularity via duality in calculus of variations and degenerate elliptic PDEs

A technique based on duality to obtain H1 or other Sobolev regularity results for solutions of convex variational problems is presented. This technique, first developed in order to study the regularity of the pressure in the variational formulation of the Incompressible Euler equation, has been recently re-employed in Mean Field Games. Here, it is shown how to apply it to classical problems in relation with degenerate elliptic PDEs of p-Laplace type. This allows to recover many classical results via a different point of view, and to have inspiration for new ones. The applications include, among others, variational models for traffic congestion and more general minimization problems under divergence constraints, but the most interesting results are obtained in dynamical problems such as Mean Field Games with density constraints or density penalizations.