3D Euler Equations Research Articles

Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs

Consider a general linear Hamiltonian system∂tu=JLu\partial _{t}u=JLuin a Hilbert spaceXX. We assume that L:X→X∗\ L:X\rightarrow X^{\ast }induces a bounded and symmetric bi-linear form⟨L⋅,⋅⟩\left \langle L\cdot ,\cdot \right \rangleonXX, which has only finitely many negative dimensionsn−(L)n^{-}(L). There is no restriction on the anti-self-dual operatorJ:X∗⊃D(J)→XJ:X^{\ast }\supset D(J)\rightarrow X. We first obtain a structural decomposition ofXXinto the direct sum of several closed subspaces so thatLLis blockwise diagonalized andJLJLis of upper triangular form, where the blocks are easier to handle. Based on this structure, we first prove the linear exponential trichotomy ofetJLe^{tJL}. In particular,etJLe^{tJL}has at most algebraic growth in the finite co-dimensional center subspace. Next we prove an instability index theorem to relaten−(L)n^{-}\left ( L\right )and the dimensions of generalized eigenspaces of eigenvalues of JL\ JL, some of which may be embedded in the continuous spectrum. This generalizes and refines previous results, where mostlyJJwas assumed to have a bounded inverse. More explicit information for the indexes with pure imaginary eigenvalues are obtained as well. Moreover, when Hamiltonian perturbations are considered, we give a sharp condition for the structural instability regarding the generation of unstable spectrum from the imaginary axis. Finally, we discuss Hamiltonian PDEs including dispersive long wave models (BBM, KDV and good Boussinesq equations), 2D Euler equation for ideal fluids, and 2D nonlinear Schrödinger equations with nonzero conditions at infinity, where our general theory applies to yield stability or instability of some coherent states.

Well-Balanced Adaptive Compact Approximate Taylor Methods for Systems of Balance Laws

Compact Approximate Taylor (CAT) methods for systems of conservation laws were introduced by Carrillo and Pares in 2019. These methods, based on a strategy that allows one to extend high-order Lax-Wendroff methods to nonlinear systems without using the Cauchy-Kovalevskaya procedure, have arbitrary even order of accuracy 2p and use (2p + 1)-point stencils, where p is an arbitrary positive integer. More recently in 2021 Carrillo, Macca, Pares, Russo and Zorio introduced a strategy to get rid of the spurious oscillations close to discontinuities produced by CAT methods. This strategy led to the so-called Adaptive CAT (ACAT) methods, in which the order of accuracy, and thus the width of the stencils, is adapted to the local smoothness of the solution. The goal of this paper is to extend CAT and ACAT methods to systems of balance laws. To do this, the source term is written as the derivative of its indefinite integral that is formally treated as a flux function. The well-balanced property of the methods is discussed and a variant that allows in principle to preserve any stationary solution is presented. The resulting methods are then applied to a number of systems going from a linear scalar conservation law to the 2D Euler equations with gravity, passing by the Burgers equations with source term and the 1D shallow water equations: the order and well-balanced properties are checked in several numerical tests.

Existence and regularity for vortex patch solutions of the 2D Euler equations

Existence and regularity for vortex patch solutions of the 2D Euler equations

An algebraic approach to the spontaneous formation of spherical jets

<p style='text-indent:20px;'>The global structure of the atmosphere and the oceans is a continuous source of intriguing challenges in geophysical fluid dynamics (GFD). Among these, jets are determinant in the air and water circulation around the Earth. In the last fifty years, thanks to the development of more and more precise and extensive observations, it has been possible to study in detail the atmospheric formations of the giant-gas planets in the solar system. For those planets, jets are the dominant large scale structure. Starting from the 70s, various theories combining observations and mathematical models have been proposed in order to describe their formation and stability. In this paper, we propose a purely algebraic approach to describe the spontaneous formation of jets on a spherical domain. Analysing the algebraic properties of the 2D Euler equations, we give a characterization of the different jets' structures. The calculations are performed starting from the discrete Zeitlin model of the Euler equations. For this model, the classification of the jets' structures can be precisely described in terms of reductive Lie algebras decomposition. The discrete framework provides a simple tool for analysing both from a theoretical and and a numerical perspective the jets' formation. Furthermore, it allows to extend the results to the original Euler equations.</p>

The incompressible Euler equations under octahedral symmetry: Singularity formation in a fundamental domain

We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: {(x1,x2,x3):0<x3<x2<x1}. In this domain, we prove local well-posedness for Cα vorticities not necessarily vanishing on the boundary with any 0<α<1, and establish finite-time singularity formation within the same class for smooth and compactly supported initial data. The solutions can be extended to all of R3 via a sequence of reflections, and therefore we obtain finite-time singularity formation for the 3D Euler equations in R3 with bounded and piecewise smooth vorticities.

Lie symmetry analysis and invariant solutions of 3D Euler equations for axisymmetric, incompressible, and inviscid flow in the cylindrical coordinates

Through the Lie symmetry analysis method, the axisymmetric, incompressible, and inviscid fluid is studied. The governing equations that describe the flow are the Euler equations. Under intensive observation, these equations do not have a certain solution localized in all directions (r,t,z) due to the presence of the term frac{1}{r}, which leads to the singularity cases. The researchers avoid this problem by truncating this term or solving the equations in the Cartesian plane. However, the Euler equations have an infinite number of Lie infinitesimals; we utilize the commutative product between these Lie vectors. The specialization process procures a nonlinear system of ODEs. Manual calculations have been done to solve this system. The investigated Lie vectors have been used to generate new solutions for the Euler equations. Some solutions are selected and plotted as two-dimensional plots.

Finite-time singularity formation for $C^{1,\alpha}$ solutions to the incompressible Euler equations on $\mathbb{R}^3$

It has been known since work of Lichtenstein and Gunther in the 1920s that the 3D incompressible Euler equation is locally well-posed in the class of velocity fields with Hölder continuous gradient and suitable decay at infinity. It is shown here that these local solutions can develop singularities in finite time, even for some of the simplest three-dimensional flows.

Boundary layer models of the Hou-Luo scenario

Finite time blow up vs global regularity question for 3D Euler equation of fluid mechanics is a major open problem. Several years ago, Luo and Hou [16] proposed a new finite time blow up scenario based on extensive numerical simulations. The scenario is axi-symmetric and features fast growth of vorticity near a ring of hyperbolic points of the flow located at the boundary of a cylinder containing the fluid. An important role is played by a small boundary layer where intense growth is observed. Several simplified models of the scenario have been considered, all leading to finite time blow up [3] , [2] , [9] , [13] , [11] , [15] . In this paper, we propose two models that are designed specifically to gain insight in the evolution of fluid near the hyperbolic stagnation point of the flow located at the boundary. One model focuses on analysis of the depletion of nonlinearity effect present in the problem. Solutions to this model are shown to be globally regular. The second model can be seen as an attempt to capture the velocity field near the boundary to the next order of accuracy compared with the one-dimensional models such as [3] , [2] . Solutions to this model blow up in finite time.

Stability and Exponential Decay for the 2D Anisotropic Navier–Stokes Equations with Horizontal Dissipation

Solutions to the 2D Navier–Stokes equations with full dissipation in the whole space $${\mathbb {R}}^2$$ always decay to zero in Sobolev spaces. In particular, any perturbation near the trivial solution is always asymptotically stable. In contrast, solutions to the 2D Euler equations for inviscid flows can grow rather rapidly. An intermediate situation is when the dissipation is anisotropic and only one-directional. The stability and large-time behavior problem for the 2D Navier–Stokes equations with only one-directional dissipation is not well-understood. When the spatial domain is the whole space $${\mathbb {R}}^2$$ , this problem is widely open. This paper solves this problem when the domain is $${\mathbb {T}}\times {\mathbb {R}}$$ with $${\mathbb {T}}$$ being a 1D periodic box. The idea here is to decompose the velocity u into its horizontal average $${\bar{u}}$$ and the corresponding oscillation $${\widetilde{u}}$$ . By making use of special properties of $${\widetilde{u}}$$ , we establish a uniform upper bound and the stability of u in the Sobolev space $$H^2$$ , and show that $${\widetilde{u}}$$ in $$H^1$$ decays to zero exponentially in time.

Symmetry in stationary and uniformly rotating solutions of active scalar equations

We study the radial symmetry properties of stationary and uniformly rotating solutions of the 2D Euler and gSQG equations, both in the smooth setting and the patch setting. For the 2D Euler equation, we show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial, without any assumptions on the connectedness of the support or the level sets. For the 2D Euler equation in the patch setting, we show that every uniformly rotating patch D with angular velocity Ω≤0 or Ω≥1 2 must be radial, where both bounds are sharp. For the gSQG equation, we obtain a similar symmetry result for Ω≤0 or Ω≥Ωα (with the bounds being sharp), under the additional assumption that the patch is simply connected. These results settle several open questions posed by Hmidi, de la Hoz, Hassainia, and Mateu on uniformly rotating patches. Along the way, we close a question by Choksi, Neumayer, and Topaloglu on overdetermined problems for the fractional Laplacian, which may be of independent interest. The main new ideas come from a calculus-of-variations point of view.

Loss of Regularity for the 2D Euler Equations

In this note, we construct solutions to the 2D Euler equations which belong to the Yudovich class but lose \(W^{1,p}\) regularity continuously with time.

Simulation of nozzle flow based on Euler equations

The nozzle is a widely used device in daily life, such as water fountains to rocket engines. It is important to find out the influence of the position of the nozzle throat for the application or the design of the nozzle. To that end, the finite difference method was employed to solve the 1D Euler equations to obtain the flow inside the nozzle. To implement the method, an in-house python code was developed. The relationship among the velocity, pressure and density in the convergent-divergent nozzle flow was found. It is observed that: the velocity rose quickly along with the nozzle and reached the top before a rapid decrease; pressure remained constant initially, which eventually began to drop; density dropped steadily and had a turning point. Moreover, the influence of the nozzle throat position is investigated thoroughly. It is observed that the position of the nozzle throat influences the velocities at the nozzle exit. The faster the flow reaches the throat, the higher the velocity or Mach number at the exit boundary.

Sensitivity Analysis for the 2D Navier–Stokes Equations with Applications to Continuous Data Assimilation

We rigorously prove the well-posedness of the formal sensitivity equations with respect to the viscosity corresponding to the 2D incompressible Navier–Stokes equations. Moreover, we do so by showing a sequence of difference quotients converges to the unique solution of the sensitivity equations for both the 2D Navier–Stokes equations and the related data assimilation equations, which utilize the continuous data assimilation algorithm proposed by Azouani, Olson, and Titi. As a result, this method of proof provides uniform bounds on difference quotients, demonstrating parameter recovery algorithms that change parameters as the system evolves will be well behaved. Furthermore, our analysis can be extended to analyze the sensitivity of the 2D Euler equations to a viscous regularization. We also note that this appears to be the first such rigorous proof of global existence and uniqueness to strong or weak solutions to the sensitivity equations for the 2D Navier–Stokes equations (in the natural case of zero initial data), and that they can be obtained as a limit of difference quotients with respect to the viscosity.

Remarks on Type I Blow-Up for the 3D Euler Equations and the 2D Boussinesq Equations

In this paper, we derive kinematic relations for quantities involving the rate of strain tensor and the Hessian of the pressure for solutions of the 3D Euler equations and the 2D Boussinesq equations. Using these kinematic relations, we prove new blow-up criteria and obtain conditions for the absence of type I singularity for these equations. We obtain both global and localized versions of the results. Some of the new blow-up criteria and type I conditions improve previous results of Chae and Constantin (Int Math Res Notices rnab014, 2021).

Finite-time singularity formation for an active scalar equation

We introduce an active scalar equation with a similar structure to the 3D Euler equations. Through studying the behavior of scale-invariant solutions, we show that compactly supported Lipschitz solutions belonging to can become singular in finite time. The interesting feature here is that we can achieve this in the absence of spatial boundaries.

Localized non blow-up criterion of the Beale-Kato-Majda type for the 3D Euler equations

We prove a localized non blow-up theorem of the Beale–Kato–Majda type for the solution of the 3D incompressible Euler equations.

Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with $$C^{1,\alpha }$$ Velocity and Boundary

Inspired by the numerical evidence of a potential 3D Euler singularity by Luo-Hou [30,31] and the recent breakthrough by Elgindi [11] on the singularity formation of the 3D Euler equation without swirl with $C^{1,\alpha}$ initial velocity, we prove the finite time singularity for the 2D Boussinesq and the 3D axisymmetric Euler equations with boundary and $C^{1,\alpha}$ initial data for the velocity (and density in the case of Boussinesq equations). Our finite time blowup solution for the 3D Euler equations and the singular solution considered in [30,31] share many essential features, including the symmetry properties of the solution, the flow structure, and the sign of the solution in each quadrant, except that we use $C^{1,\alpha}$ initial velocity. We use a dynamic rescaling formulation and follow the general framework of analysis developed in [11]. We also use some strategy proposed in our recent joint work with Huang in [7] and adopt several methods of analysis in [11] to establish the linear and nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the 3D Euler or the 2D Boussinesq equations with $C^{1,\alpha}$ initial data will develop a finite time singularity. Moreover, the velocity field has finite energy before the singularity time. In the previous version of this paper, we proved the blowup results for the 3D axisymmetric Euler equations with initial data $(u_0^{\theta})^2, u_0^r, u_0^z \in C^{1,\alpha}$. Though the velocity $u^r, u^z$ in the axisymmetric setting is $C^{1,\alpha}$, our interpretation that the velocity is $C^{1,\alpha}$ is not correct since the velocity in 3D also depends on $u^{\theta}$, which is not $C^{1,\alpha}$. This oversight can be fixed easily with a minor change in the construction of the approximate steady state and a minor modification to localize the approximate steady state.

Fokker–Planck equation for dissipative 2D Euler equations with cylindrical noise

After a short review of recent progresses in 2D Euler equations with random initial conditions and noise, some of the recent results are improved by exploiting a priori estimates on the associated infinite dimensional Fokker-Planck equation. The regularity class of solutions investigated here does not allow energy- or enstrophy-type estimates, but only bounds in probability with respect to suitable distributions of the initial conditions. This is a remarkable application of Fokker-Planck equations in infinite dimensions. Among the example of random initial conditions we consider Gibbsian measures based on renormalized kinetic energy.

Higher-dimensional Euler fluids and Hasimoto transform: counterexamples and generalizations

The binormal (or vortex filament) equation provides the localized induction approximation of the 3D incompressible Euler equation. We present explicit solutions of the binormal equation in higher-dimensions that collapse in finite time. The local nature of this phenomenon suggests a possibility of the singularity appearance in nearby vortex blob solutions of the Euler equation in 5D and higher. Furthermore, the Hasimoto transform takes the binormal equation to the NLS and barotropic fluid equations. We show that in higher dimensions the existence of such a transform would imply the conservation of the Willmore energy in skew-mean-curvature flows and present counterexamples for vortex membranes based on products of spheres. These (counter)examples imply that there is no straightforward generalization to higher dimensions of the 1D Hasimoto transform. We derive its replacement, the evolution equations for the mean curvature and torsion form for membranes, thus generalizing the barotropic fluid and Da Rios equations.

Strong Convergence of the Vorticity for the 2D Euler Equations in the Inviscid Limit

In this paper we prove the uniform-in-time L^p convergence in the inviscid limit of a family omega ^nu of solutions of the 2D Navier–Stokes equations towards a renormalized/Lagrangian solution omega of the Euler equations. We also prove that, in the class of solutions with bounded vorticity, it is possible to obtain a rate for the convergence of omega ^{nu } to omega in L^p. Finally, we show that solutions of the Euler equations with L^p vorticity, obtained in the vanishing viscosity limit, conserve the kinetic energy. The proofs are given by using both a (stochastic) Lagrangian approach and an Eulerian approach.