Incompressible Euler Equations Research Articles

Derivation of Euler’s Equations of Perfect Fluids from von Neumann’s Equation with Magnetic Field

We give a rigorous derivation of the incompressible 2D Euler equation from the von Neumann equation with an external magnetic field. The convergence is with respect to the modulated energy functional, and implies weak convergence in the sense of measures. This is the semi-classical counterpart of theorem 1.5 in (Han-Kwan and Iacobelli in Proc Am Math Soc 149(7):3045–3061, 2021). Our proof is based on a Gronwall estimate for the modulated energy functional, which in turn heavily relies on a recent functional inequality due to (Serfaty in Duke Math J 169:2887–2935, 2020).

Energy conservation of weak solutions for the incompressible Euler equations via vorticity

Motivated by the works of Cheskidov, Lopes Filho, Nussenzveig Lopes and Shvydkoy in [8, Commun. Math. Phys. 348: 129-143, 2016] and Chen and Yu in [5, J. Math. Pures Appl. 131: 1-16, 2019], we address how the Lp control of vorticity could influence the energy conservation for the incompressible homogeneous and nonhomogeneous Euler equations in this paper. For the homogeneous flow in the periodic domain or whole space, we provide a self-contained proof for the criterion ω=curlv∈L3(0,T;L3nn+2(Ω))(n=2,3), that generalizes the corresponding result in [8] and can be viewed as in Onsager critical spatio-temporal spaces. Regarding the nonhomogeneous flow, it is shown that the energy is conserved as long as the vorticity lies in the same space as before and ∇ρ belongs to L∞(0,T;Ln(Tn))(n=2,3), which gives an affirmative answer to a problem proposed by Chen and Yu in [5].

A Liouville-type theorem for the 3D primitive equations

The 3D primitive equations are used in most geophysical fluid models to approximate the large scale oceanic and atmospheric dynamics. We prove that there do not exist smooth stationary solutions to the 3D primitive equations with compact support, independently of the presence of the Coriolis rotation term or the viscosity. This result is in strong contrast with the recently established existence of compactly supported smooth solutions to the incompressible 3D Euler equations.

Maximum-principle-preserving high-order discontinuous Galerkin methods for incompressible Euler equations on overlapping meshes

In this paper, we construct a new local discontinuous Galerkin (LDG) algorithm to solve the incompressible Euler equation in two space dimensions on overlapping meshes. This method solves the vorticity, velocity field and potential function on different meshes. Different from the traditional LDG method, the overlapping meshes used in this paper make the velocity to be continuous along the interfaces of the primitive meshes. Therefore, the upwind fluxes can be applied. We derive two sufficient conditions to obtain the maximum principle of vorticity. The first one is the divergence-free numerical approximation of the velocity field. This condition further grants that the scheme of the vorticity equation keeps constant solutions. The second one is to preserve the positivity of the numerical vorticity. We select suitable time step sizes to construct positive numerical cell averages of the vorticity provided the vorticity in the previous time step is positive. Then a slope limiter can be applied to enforce the positivity of the numerical approximation of the vorticity. Thanks to the above two conditions, we can arbitrarily add constants to the vorticity function and construct high-order MPP LDG methods on overlapping meshes for the two-dimensional incompressible Euler equation in the vorticity stream function formulation. Numerical tests will be given to demonstrate the good performance of the proposed method.

Adaptive nonlinear optimization of district heating networks based on model and discretization catalogs

We propose an adaptive optimization algorithm for operating district heating networks in a stationary regime. The behavior of hot water flow in the pipe network is modeled using the incompressible Euler equations and a suitably chosen energy equation. By applying different simplifications to these equations, we derive a catalog of models. Our algorithm is based on this catalog and adaptively controls where in the network which model is used. Moreover, the granularity of the applied discretization is controlled in a similar adaptive manner. By doing so, we are able to obtain optimal solutions at low computational costs that satisfy a prescribed tolerance w.r.t. the most accurate modeling level. To adaptively control the switching between different levels and the adaptation of the discretization grids, we derive error measure formulas and a posteriori error measure estimators. Under reasonable assumptions we prove that the adaptive algorithm terminates after finitely many iterations. Our numerical results show that the algorithm is able to produce solutions for problem instances that have not been solvable before.

Singularity and mesh divergence of inviscid adjoint solutions at solid walls

• Comprehensive review of the mesh dependence anomaly observed in two and three dimensional numerical adjoint solutions for subsonic and transonic inviscid flows past wings, airfoils and blunt bodies. • Review of the analytic adjoint solution obtained for the 2D incompressible Euler equations. • Identification of the origin of the mesh dependence anomaly as a singularity of the analytic solution. • Cross-comparison of analytic and numerical adjoint solutions for lift and drag.

Probabilistic Descriptions of Fluid Flow: A Survey

Fluids can behave in a highly irregular, turbulent way. It has long been realised that, therefore, some weak notion of solution is required when studying the fundamental partial differential equations of fluid dynamics, such as the compressible or incompressible Navier–Stokes or Euler equations. The standard concept of weak solution (in the sense of distributions) is still a deterministic one, as it gives exact values for the state variables (like velocity or density) for almost every point in time and space. However, observations and mathematical theory alike suggest that this deterministic viewpoint has certain limitations. Thus, there has been an increased recent interest in the mathematical fluids community in probabilistic concepts of solution. Due to the considerable number of such concepts, it has become challenging to navigate the corresponding literature, both classical and recent. We aim here to give a reasonably concise yet fairly detailed overview of probabilistic formulations of fluid equations, which can roughly be split into measure-valued and statistical frameworks. We discuss both approaches and their relationship, as well as the interrelations between various statistical formulations, focusing on the compressible and incompressible Euler equations.

Weak-strong uniqueness and vanishing viscosity for incompressible Euler equations in exponential spaces

In the class of admissible weak solutions, we prove a weak-strong uniqueness result for the incompressible Euler equations assuming that the symmetric part of the gradient belongs to Lloc1([0,+∞);Lexp(Rd;Rd×d)), where Lexp denotes the Orlicz space of exponentially integrable functions. Moreover, under the same assumptions on the limit solution to the Euler system, we obtain the convergence of vanishing-viscosity Leray–Hopf weak solutions of the Navier–Stokes equations.

Onsager’s energy conservation of solutions for density-dependent Euler equations in [formula omitted

This article is devoted to the study of energy conservation of weak solutions for the incompressible inhomogeneous Euler equations in Td, d>1. We give two sufficient conditions to prove the energy conservation, which only include a regularity of space (and time) on the velocity u. This is different from these results in Chen (2020), Chen and Yu (2019) and Nguyen et al. (2020), which contain the regularity of ∂tρ and the pressure p. In particular, our assumption in theorems is weaker than the assumption of Theorem 1.2 in Chen (2020) and the assumption of regularity of density of Theorem 1.1 in Chen and Yu (2019).

Passage from the Boltzmann equation with diffuse boundary to the incompressible Euler equation with heat convection

We derive the incompressible Euler equations with heat convection with the no-penetration boundary condition from the Boltzmann equation with the diffuse boundary in the hydrodynamic limit for the scale of large Reynold number. Inspired by the recent framework in [30], we consider the Navier-Stokes-Fourier system with no-slip boundary conditions as an intermediary approximation and develop a Hilbert-type expansion of the Boltzmann equation around the global Maxwellian that allows the nontrivial heat transfer by convection in the limit. To justify our expansion and the limit, a new direct estimate of the heat flux and its derivatives in the Navier-Stokes-Fourier system is established adopting a recent Green's function approach in the study of the inviscid limit.

Non-linear singularity formation for circular vortex sheets

We study the evolution of vortex sheets according to the Birkhoff-Rott equation, which describe the motion of sharp shear interfaces governed by the incompressible Euler equation in two dimensions. In a recent work, the authors demonstrated within this context a marginal linear stability of circular vortex sheets, standing in sharp contrast with classical instability of the flat vortex sheet, which is known as the Kelvin-Helmholtz instability. This article continues that analysis by investigating how non-linear effects induce singularity formation near the circular vortex sheet. In high-frequency regimes, the singularity formation is primarily driven by a complex-valued, conjugated Burgers equation, which we study by modifying a classical argument from hyperbolic conservation laws. This provides a deeper understanding of the mechanisms driving the breakdown of circular vortex sheets, which are observed both numerically and experimentally.

Interaction of mode-one internal solitary waves of opposite polarity in double-pycnocline stratifications

Numerical simulations of the interaction of internal solitary waves (ISWs) of opposite polarity are conducted by solving the incompressible Euler equations under the Boussinesq approximation. A double-pycnocline stratification is used. A method to determine when ISWs of both polarities exist is also presented. The simulations confirm previous work that the interaction of waves of the same polarity are soliton-like; however, here it is shown that when a fast ISW with the same polarity as a Korteweg–de Vries (KdV) solitary wave catches up and interacts with a slower ISW of opposite polarity, the interaction can be far from soliton-like. The energy in the fast KdV-polarity wave can increase by more than a factor of 5 while the energy in the slower negative-KdV-polarity wave can decrease by 50 %. Large trailing wave trains may be generated and in some cases multiple ISWs with KdV polarity may be formed by the interaction.

Asymptotic limits of the Euler–Poisson equations for dissipative measure-valued solutions

We prove that the dissipative measure-valued solutions of the 3D compressible Euler–Poisson equations converge to the solution of the incompressible Euler equations in an asymptotic regime, where the Mach number and Debye length tend to zero simultaneously. Furthermore, the convergence rates are also obtained.

Helical symmetry vortices for 3D incompressible Euler equations

In this paper, we study the desingularization of vortices for the 3D incompressible Euler equations in an infinite pipe. We construct a family of traveling-rotating helical vortices for the Euler equations with a general vorticity function, which tends asymptotically to singular helical vortex filament evolved by the binormal curvature flow. The results are obtained by using an improved vorticity method. Some asymptotic properties of this family of solutions have also been studied.

Remarks on orbital stability of steady vortex rings

In this paper, we study nonlinear orbital stability of steady vortex rings without swirl, which are special global solutions of the three-dimensional incompressible Euler equations. We prove the existence of orbitally stable steady vortex rings. The proof is based on the classical variational method.

Structure of Green’s function of elliptic equations and helical vortex patches for 3D incompressible Euler equations

We develop a new structure of the Green’s function of a second-order elliptic operator in divergence form in a 2D bounded domain. Based on this structure and the theory of rearrangement of functions, we construct concentrated traveling-rotating helical vortex patches to 3D incompressible Euler equations in an infinite pipe. By solving an equation for vorticity $$\begin{aligned} w=\frac{1}{\varepsilon ^2}f_\varepsilon \left( \mathcal {G}_{K_H}w-\frac{\alpha }{2}|x|^2|\ln \varepsilon |\right) \ \ \text {in}\ \Omega \end{aligned}$$ for small $$ \varepsilon >0 $$ and considering a certain maximization problem for the vorticity, where $$ \mathcal {G}_{K_H} $$ is the inverse of an elliptic operator $$ \mathcal {L}_{K_H} $$ in divergence form, we get the existence of a family of concentrated helical vortex patches, which tend asymptotically to a singular helical vortex filament evolved by the binormal curvature flow. We also get nonlinear orbital stability of the maximizers in the variational problem under $$ L^p $$ perturbation when $$ p\ge 2. $$

An intermittent Onsager theorem

For any regularity exponent $$\beta <\frac{1}{2}$$ , we construct non-conservative weak solutions to the 3D incompressible Euler equations in the class $$C^0_t (H^{\beta } \cap L^{\frac{1}{(1-2\beta )}})$$ . By interpolation, such solutions belong to $$C^0_t B^{s}_{3,\infty }$$ for s approaching $$\frac{1}{3}$$ as $$\beta $$ approaches $$\frac{1}{2}$$ . Hence this result provides a new proof of the flexible side of the $$L^3$$ -based Onsager conjecture. Of equal importance is that the intermittent nature of our solutions matches that of turbulent flows, which are observed to possess an $$L^2$$ -based regularity index exceeding $$\frac{1}{3}$$ . Thus our result does not imply, and is not implied by, the work of Isett (Ann Math 188(3):871, 2018), who gave a proof of the Hölder-based Onsager conjecture. Our proof builds on the authors’ previous joint work with Buckmaster et al. (Intermittent convex integration for the 3D Euler equations: (AMS-217), Princeton University Press, 2023.), in which an intermittent convex integration scheme is developed for the 3D incompressible Euler equations. We employ a scheme with higher-order Reynolds stresses, which are corrected via a combinatorial placement of intermittent pipe flows of optimal relative intermittency.

Potential Singularity Formation of Incompressible Axisymmetric Euler Equations with Degenerate Viscosity Coefficients

In this paper, we present strong numerical evidence that the incompressible axisymmetric Euler equations with degenerate viscosity coefficients and smooth initial data of finite energy develop a potential finite-time locally self-similar singularity at the origin. An important feature of this potential singularity is that the solution develops a two-scale traveling wave that travels toward the origin. The two-scale feature is characterized by the scaling property that the center of the traveling wave is located at a ring of radius surrounding the symmetry axis while the thickness of the ring collapses at a rate . The driving mechanism for this potential singularity is due to an antisymmetric vortex dipole that generates a strong shearing layer in both the radial and axial velocity fields. Without the viscous regularization, the three-dimensional Euler equations develop a sharp front and some shearing instability in the far field. On the other hand, the Navier–Stokes equations with a constant viscosity coefficient regularize the two-scale solution structure and do not develop a finite-time singularity for the same initial data.

On the Grad–Rubin boundary value problem for the two-dimensional magneto-hydrostatic equations

In this work, we study the solvability of a boundary value problem for the magneto-hydrostatic equations originally proposed by Grad and Rubin (Proceedings of the 2nd UN conference on the peaceful uses of atomic energy. IAEA, Geneva, 1958). The proof relies on a fixed point argument which combines the so-called current transport method together with Hölder estimates for a class of non-convolution singular integral operators. The same method allows to solve an analogous boundary value problem for the steady incompressible Euler equations.

Rotation-dominant three-scale limit of the Cauchy problem to the inviscid rotating stratified Boussinesq equations

In this paper, we investigate the rotation-dominant three-scale singular limit of the initial-value problem to the inviscid rotating stratified Boussinesq equations. We first prove a dispersive decay estimate for the linear propagator of that limit regime, which requires novel techniques involving an elaborate frequency cut-off since the phase function is degenerate on both horizontal and vertical directions in some areas. Using the dispersive decay estimate, we then investigate rigorously the rotation-dominant limit of local strong solutions to Boussinesq equations with ill-prepared initial data. The limiting system is shown to be the two-dimensional incompressible Euler equations coupled with a two-dimensional transport equation but in three spatial dimensions.