Euler Case Research Articles

The α$\alpha$‐SQG patch problem is illposed in C2,β$C^{2,\beta }$ and W2,p$W^{2,p}$

AbstractWe consider the patch problem for the ‐(surface quasi‐geostrophic) SQG system with the values and being the 2D Euler and the SQG equations respectively. It is well‐known that the Euler patches are globally wellposed in non‐endpoint Hölder spaces, as well as in , spaces. In stark contrast to the Euler case, we prove that for , the ‐SQG patch problem is strongly illposed in every Hölder space with . Moreover, in a suitable range of regularity, the same strong illposedness holds for every Sobolev space unless .

Top on a smooth plane.

We investigate the dynamics of a sliding top that is a rigid body with an ideal sharp tip moving in a perfectly smooth horizontal plane, so no friction forces act on the body. We prove that this system is integrable only in two cases analogous to the Euler and Lagrange cases of the classical top problem. The cases with the constant gravity field with acceleration g≠0 and without external field g=0 are considered. The non-integrability proof for g≠0 is based on the fact that the equations of motion for the sliding top are a perturbation of the classical top equations of motion. We show that the integrability of the classical top is a necessary condition for the integrability of the sliding top. Among four integrable classical top cases, the corresponding two cases for the sliding top are also integrable, and for the two remaining cases, we prove their non-integrability by analyzing the differential Galois group of variational equations along a certain particular solution. In the absence of a constant gravitational field g=0, the integrability is much more difficult. First, we proved that if the sliding top problem is integrable, then the body is symmetric. In the proof, we applied the Ziglin theorem concerning the splitting of separatrices phenomenon. Then, we prove the non-integrability of the symmetric sliding top using the differential Galois group of variational equations except two the same as for g≠0 cases. The integrability of these cases is also preserved when we add to equations of motion a gyrostatic term.

A Note on the Laguerre-Type Appell and Hypergeometric Polynomials

The Laguerre derivative and its iterations have been used to define new sets of special functions, showing the possibility of generating a kind of parallel universe for mathematical entities of this kind. In this paper, we introduce the Laguerre-type Appell polynomials, in particular, the Bernoulli and Euler case, and we examine a set of hypergeometric Laguerre–Bernoulli polynomials. We show their main properties and indicate their possible extensions.

Realizing integrable Hamiltonian systems by means of billiard books

Fomenko’s conjecture that the topology of the Liouville foliations associated with integrable smooth or analytic Hamiltonian systems can be realized by means of integrable billiard systems is discussed. An algorithm of Vedyushkina and Kharcheva’s realizing 3-atoms by billiard books, which has been simplified significantly by formulating it in terms of f f -graphs, is presented. Note that, using another algorithm, Vedyushkina and Kharcheva have also realized an arbitrary type of the base of the Liouville foliation on the whole 3-dimensional isoenergy surface. This algorithm is illustrated graphically by an example where the invariant of the well-known Joukowsky system (the Euler case with a gyrostat) is realized for a certain energy range. It turns out that the entire Liouville foliation, rather than just the class of its base, is realized there; that is, the billiard and mechanical systems turn out to be Liouville equivalent. Results due to Vedyushkina and Kibkalo on constructing billiards with arbitrary values of numerical invariants are also presented. For billiard books without potential that possess a certain property, the existence of a Fomenko–Zieschang invariant is shown; it is also proved that they belong to the class of topologically stable systems. Finally, an example is presented when the addition of a Hooke potential to a planar billiard produces a splitting nondegenerate 4-singularity of rank 1.

On Distributional Solutions of a Singular Differential Equation of 2-order in the Space K’

The main purpose of this work is to describe all the zero-centered solutions of the second order linear singular differential equation with Dirac delta function (or it derivatives of some order) in the second right hand side in the space<i> K’. </i>All the coefficients and the exponents of the polynomials under the unknown function and it derivatives up to second order respectively, are real and natural numbers in the considered equation. We conduct investigations for both the euler case and left euler case situations of this equation, when it is fulfilled some particular conditions in the relationships between the parameters <i>A, B, C, m, n and r</i>. In each of these cases, we look for the zero-centered solutions and substitute the form of the particular solution into the equation. We then after, determinate the unknown coefficients and formulate the related theorems to describe all the solutions depending of the cases to be investigated.

Evolutionary force billiards

A new class of integrable billiards has been introduced: evolutionary force billiards. They depend on a parameter and change their topology as energy (time) increases. It has been proved that they realize some important integrable systems with two degrees of freedom on the entire symplectic four-dimensional phase manifold at a time, rather than on only individual isoenergy 3-surfaces. For instance, this occurs in the Euler and Lagrange cases. It has also been proved that these two well-known systems are "billiard-equivalent", despite the fact that the former one is square integrable, and the latter one allows a linear integral.

Plasma transport simulations of Rayleigh–Taylor instability in near-ICF deceleration regimes

Rayleigh–Taylor (R–T) instability between plasma species is examined in a kinetic test and near-inertial confinement fusion (ICF) regimes. A transport approximation to the plasma species kinetics is used to represent viscosity and species mass transport within a hydrodynamic fluid code (xRage). R–T simulation results are compared in a kinetic test regime with a fully kinetic particle-in-cell approach [vectorized particle-in-cell (VPIC)] and with an analytic model for the growth rate of R–T instability. Single-mode growth rates from both codes and the analytic model are in reasonable agreement over a range of initial wavelengths including the wavenumber of maximum growth rate. Both codes exhibit similar diffusive mixing fronts. Small code-to-code differences arise from the kinetics, while simulation-analytic model differences arise from several sources dominated by the choice of gradients establishing the hydrostatic equilibrium initial conditions. After demonstrating code agreement in the kinetic test regime, which is practically accessible to the VPIC code, then the xRage code, with the fluid plasma transport approximation, is applied to single mode R–T instability under deceleration conditions closer to an ICF implosion, approximated with a carbon (C) shell imploding on a deuterium (D) fuel. The analytic wavelength of maximum instability is limited by the kinetics, primarily in the viscosity, and is found to be ≈10 μm for an ion temperature near 1 keV at this C–D interface, with the most unstable wavelength increasing as temperature increases. The analytic viscous model agrees with simulation results over a range of initial perturbation wavelengths, provided the simulation results are analyzed over a sufficiently short duration (⪅0.2 ns in this case). Details of the fluid structure evolution during this R–T deceleration are compared between the inviscid Euler equations and cases, which include plasma transport over a range in initial wavelengths and initial perturbation amplitudes. The inviscid Euler solutions show a grid-dependent cascade to smaller scale structures often seen in the R–T instability, while simulations with plasma transport in this deceleration regime develop a single vortex roll-up, as the plasma transport smoothes all hydrodynamic fluid structures smaller than several micrometers. This leads to a grid-converged transient solution for the R–T instability when kinetic effects are included in the simulations, and thus represents a direct numerical simulation of the thermal ions during R–T unstable mixing in ICF relevant conditions.

Canonical structure of minimal varying Λ theories

Minimal varying Λ theories are defined by an action built from the Einstein–Cartan–Holst first order action for gravity with the cosmological constant Λ as an independent scalar field, and supplemented by the Euler and Pontryagin densities multiplied by 1/Λ. We identify the canonical structure of these theories which turn out to represent an example of irregular systems. We find five degrees of freedom on generic backgrounds and for generic values of parameters, whereas if the parameters satisfy a certain condition (which includes the most commonly considered Euler case) only three degrees of freedom remain. On de Sitter-like backgrounds the canonical structure changes, and due to an emergent conformal symmetry one degree of freedom drops from the spectrum. We also analyze the self-dual case with an holomorphic action depending only on the self-dual part of the connection. In this case we find two (complex) degrees of freedom, and further discuss the Kodama state, the restriction to de Sitter background and the effect of reality conditions.

Force Evolutionary Billiards and Billiard Equivalence of the Euler and Lagrange Cases

A class of force evolutionary billiards is discovered that realizes important integrable Hamiltonian systems on all regular isoenergy 3-surfaces simultaneously, i.e., on the phase 4-space. It is proved that the well-known Euler and Lagrange integrable systems are billiard equivalent, although the degrees of their integrals are different (two and one).

Discrete Geodesic Flows on Stiefel Manifolds

We study integrable discretizations of geodesic flows of Euclidean metrics on the cotangent bundles of the Stiefel manifolds $$V_{n,r}$$ . In particular, for $$n=3$$ and $$r=2$$ , after the identification $$V_{3,2}\cong\mathrm{SO}(3)$$ , we obtain a discrete analog of the Euler case of the rigid body motion corresponding to the inertia operator $$I=(1,1,2)$$ . In addition, billiard-type mappings are considered; one of them turns out to be the “square root” of the discrete Neumann system on $$V_{n,r}$$ .

Separatrix splitting and nonintegrability in the nonholonomic rolling of a generalized Chaplygin sphere

We consider a nonholonomic system that describes the rolling without slipping of a spherical shell inside which a frame rotates with constant angular velocity (this system is one of the possible generalizations of the problem of the rolling of a Chaplygin sphere). After a suitable scale transformation of the radius of the shell or the mass of the system the equations of motion can be represented as a perturbation of the integrable Euler case in rigid body dynamics. Using this representation, we explicitly calculate a Melnikov integral, which contains an isolated zero under some restrictions on the system parameters. Thereby we prove the absence of an additional integral in this system and the existence of chaotic trajectories. We conclude by presenting numerical experiments that illustrate the system dynamics depending on the behavior of the Melnikov function.

NERONE: An Open-Source Based Tool for Aerodynamic Transonic Optimization of Nonplanar Wings

Transonic shape optimization of nonplanar wings is a relevant topic in current aeronautical world. Although widely studied, an appropriate level of robustness and automation is still lacking for the inclusion of aerodynamic shape optimization in a Multi-Disciplinary Optimization (MDO) process. In this paper, a new framework, opeNsource-mEsh-geneRation-fOr-aerodyNamic-Evaluations (NERONE), is presented that allows for an automatic transonic shape optimization process within an MDO loop design using open-source libraries, namely OpenCasCade (OCC) for the geometric description, GMSH for grid generation and SU2 multi-physics for aerodynamic analysis and optimization. Wing geometry is defined in the CPACS standard, converted to a mathematical continuous description (NURBS) and passed to the mesher. Grid generation process is driven by user-defined mesh dimensions defined on wing regions and on support domain boundary surfaces. The aerodynamic analysis and optimization are then launched on the obtained grid. SU2 software has been augmented to overcome robustness issues regarding point-inversion algorithm and to control geometric quality during optimization of nonplanar wings. Capability of NERONE is first demonstrated for 2D Euler and RANS simulations (on the RAE2822 airfoil) and on a 3D Euler case (ONERA M6). With very few user-specified parameters, high-quality grids are obtained providing results that correlate well with the literature data. Finally, NERONE is applied to the local shape optimization of a wing–winglet configuration in transonic flight conditions.

A holistic approach for local buckling of composite laminated beams under compressive load

The present paper deals with a new holistic closed-form analytical model for the local buckling load of thin-walled composite beams with I-, Z-, C-, L- and T-cross sections under axial compressive load. The beam is simply supported at both ends (Euler case II), and the plate behaviour of web and flanges is described by the Classical Laminated Plate Theory. Furthermore, symmetric and orthotropic laminates are considered. In previous investigations on composite beams under compression, the web and flange plates are considered as separate composite plates. The present analysis is performed using the Ritz method in which an approach for the entire cross section is realized. The individual webs and flanges of the beam are assembled by suitable continuity conditions into one system. In order to achieve that, new displacement shape functions for web and flange that fulfil all boundary conditions have been developed. The present closed-form analytical method enables the explicit representation of the buckling load for the entire composite beam under axial compression. The comparison between the present approach and comparative finite element simulations shows a very satisfactory agreement. The present method is ideal for pre-designing such structures, highly efficient in terms of computational effort and very suitable for practical engineering work.

Second-order implicit-explicit total variation diminishing schemes for the Euler system in the low Mach regime

In this work, we consider the development of implicit-explicit total variation diminishing (TVD) methods (also termed SSP: strong stability preserving) for the compressible isentropic Euler system in the low Mach number regime. The proposed scheme is asymptotically stable with a CFL condition independent of the Mach number. In addition, it degenerates, in the low Mach number regime, to a consistent discretization of the incompressible system. Since it has been proved that implicit schemes of order higher than one cannot be TVD (SSP) [30], we construct a new paradigm of implicit time integrators by coupling first-order in time schemes with second-order ones in the same spirit as highly accurate shock-capturing TVD methods in space. For this particular class of schemes, the TVD property is first proved on a linear model advection equation and then extended to the isentropic Euler case. The result is a method which interpolates from the first- to the second-order both in space and time. It preserves the monotonicity of the solution, and is highly accurate for all choices of the Mach number. Moreover, the time step is only restricted by the non-stiff part of the system. We finally show, thanks to one- and two-dimensional test cases, that the method indeed possesses the claimed properties.

All Mach Number Second Order Semi-implicit Scheme for the Euler Equations of Gas Dynamics

This paper presents an asymptotic preserving (AP) all Mach number finite volume shock capturing method for the numerical solution of compressible Euler equations of gas dynamics. Both isentropic and full Euler equations are considered. The equations are discretized on a staggered grid. This simplifies flux computation and guarantees a natural central discretization in the low Mach limit, thus dramatically reducing the excessive numerical diffusion of upwind discretizations. Furthermore, second order accuracy in space is automatically guaranteed. For the time discretization we adopt an Semi-IMplicit/EXplicit (S-IMEX) discretization getting an elliptic equation for the pressure in the isentropic case and for the energy in the full Euler case. Such equations can be solved linearly so that we do not need any iterative solver thus reducing computational cost. Second order in time is obtained by a suitable S-IMEX strategy taken from Boscarino et al. (J Sci Comput 68:975–1001, 2016). Moreover, the CFL stability condition is independent of the Mach number and depends essentially on the fluid velocity. Numerical tests are displayed in one and two dimensions to demonstrate performance of our scheme in both compressible and incompressible regimes.

Finite time singularity for the modified SQG patch equation

It is well known that the incompressible Euler equations in two dimensions have globally regular solutions. The inviscid surface quasi-geostrophic (SQG) equation has a Biot-Savart law which is one derivative less regular than in the Euler case, and the question of global regularity for its solutions is still open. We study here the patch dynamics in the half-plane for a family of active scalars which interpolates between these two equations, via a parameter α ∈ [0, 12 ] appearing in the kernels of their Biot-Savart laws. The values α = 0 and α = 12 correspond to the 2D Euler and SQG cases, respectively. We prove global in time regularity for the 2D Euler patch model, even if the patches initially touch the boundary of the half-plane. On the other hand, for any sufficiently small α > 0, we exhibit initial data which lead to a singularity in finite time. Thus, these results show a phase transition in the behavior of solutions to these equations, and provide a rigorous foundation for classifying the 2D Euler equations as critical. ∗Department of Mathematics, Rice University, 6100 S. Main St., Houston, TX 77005-1892; kiselev@rice.edu †Department of Mathematics, Stanford University, Stanford, CA 94305; ryzhik@math.stanford.edu ‡School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160; yaoyao@math.gatech.edu §Department of Mathematics, University of Wisconsin, 480 Lincoln Dr. Madison, WI 53706-1325; zlatos@math.wisc.edu

Uniqueness Results for Weak Solutions of Two-Dimensional Fluid–Solid Systems

In this paper, we consider two systems modelling the evolution of a rigid body in an incompressible fluid in a bounded domain of the plane. The first system corresponds to an inviscid fluid driven by the Euler equation whereas the other one corresponds to a viscous fluid driven by the Navier–Stokes system. In both cases we investigate the uniqueness of weak solutions, à la Yudovich for the Euler case, à la Leray for the Navier–Stokes case, as long as no collision occurs.

Contact of ropes and orthotropic rough surfaces

AbstractContact between arbitrary curved ropes and arbitrary curved rough orthotropic surfaces has been revised from the geometrical point of view. Variational equations for the equilibrium of ropes on orthotropic rough surfaces are derived, first, using the consistent variational inclusion of frictional contact constraints via Karush‐Kuhn‐Tucker conditions expressed in Darboux basis. Then, the systems of differential equations are derived for both statics and dynamics of ropes on a rough surface depending on the sticking‐sliding condition for orthotropic Coulomb's friction. Three criteria are found to be fulfilled during the static equilibrium of a rope on a rough surface: “no separation”, condition for dragging coefficient of friction and inequality for tangential forces at the end of the rope. The limit tangential loads still preserve the famous “Euler view” T = T0eωs for the curves and surfaces of constant curvature. It is shown that the curve of the maximum tension of a rough orthotropic surface is geodesic. Equations of motion are derived in the case if the sliding criteria is fulfilled and there is “no separation”. Various cases possessing analytical solutions of the derived system, including Euler case and a spiral rope on a cylinder are shown as examples of application of the derived theory. (© 2014 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations

In this paper, we propose implicit and semi-implicit in time finite volume schemes for the barotropic Euler equations (hence, as a particular case, for the shallow water equations) and for the full Euler equations, based on staggered discretizations. For structured meshes, we use the MAC finite volume scheme, and, for general mixed quadrangular/hexahedral and simplicial meshes, we use the discrete unknowns of the Rannacher−Turek or Crouzeix−Raviart finite elements. We first show that a solution to each of these schemes satisfies a discrete kinetic energy equation. In the barotropic case, a solution also satisfies a discrete elastic potential balance; integrating these equations over the domain readily yields discrete counterparts of the stability estimates which are known for the continuous problem. In the case of the full Euler equations, the scheme relies on the discretization of the internal energy balance equation, which offers two main advantages: first, we avoid the space discretization of the total energy, which involves cell-centered and face-centered variables; second, we obtain an algorithm which boils down to a usual pressure correction scheme in the incompressible limit. Consistency (in a weak sense) with the original total energy conservative equation is obtained thanks to corrective terms in the internal energy balance, designed to compensate numerical dissipation terms appearing in the discrete kinetic energy inequality. It is then shown in the 1D case, that, supposing the convergence of a sequence of solutions, the limit is an entropy weak solution of the continuous problem in the barotropic case, and a weak solution in the full Euler case. Finally, we present numerical results which confirm this theory.

Thermodynamics of a photon gas in nonlinear electrodynamics

In this paper we analyze the thermodynamic properties of a photon gas under the influence of a background electromagnetic field in the context of any nonlinear electrodynamics. Neglecting the self-interaction of photons, we obtain a general expression for the grand canonical potential. Particularizing for the case when the background field is uniform, we determine the pressure and the energy density for the photon gas. Although the pressure and the energy density change when compared with the standard case, the relationship between them remains unaltered, namely ρ=3p. Finally, we apply the developed formulation to the cases of Heisenberg–Euler and Born–Infeld nonlinear electrodynamics. For the Heisenberg–Euler case, we show that our formalism recovers the results obtained with the 2-loop thermal effective action approach.