Compressible Euler System Research Articles

Limit of a Consistent Approximation to the Complete Compressible Euler System

The goal of the present paper is to prove that if a weak limit of a consistent approximation scheme of the compressible complete Euler system in full space mathbb {R}^d,; d=2,3 is a weak solution of the system, then the approximate solutions eventually converge strongly in suitable norms locally under a minimal assumption on the initial data of the approximate solutions. The class of consistent approximate solutions is quite general and includes the vanishing viscosity and heat conductivity limit. In particular, they may not satisfy the minimal principle for entropy.

An extended seventh-order compact nonlinear scheme with positivity-preserving property

A seventh-order accurate high-resolution compact nonlinear scheme is developed based on the fifth-order scheme, TCNS-II (Zhang et al., 2019), such that the stencil used for the interpolation is extended to include more endpoints in the smooth flowfield, thus maximizing the order of accuracy, while in the vicinity of discontinuities, the scheme can fall back to the original five-point scheme, thus maintaining strong stability. In the proposed extension strategy, the evaluation procedure of the smoothness measurement is simplified by following the method of the typical five-point scheme, i.e., no more complex local or global smoothness indicators are involved. The extension strategy is further optimized through the reuse of the available smoothness information, which significantly reduces the computational effort. A positivity-preserving flux limiting technique is incorporated into the flux differencing of the proposed high-order scheme for the compressible Euler system of equations in both one and two dimensions. Positive values of density and pressure are strictly enforced even for flow problems involving severe discontinuities. A variety of benchmark cases is used to demonstrate the improved performance of the present scheme particularly in shock-capturing and multi-scale wave-resolving capabilities.

Numerical study of non-uniqueness for 2D compressible isentropic Euler equations

In this paper, we numerically study a class of solutions with spiraling singularities in vorticity for two-dimensional, inviscid, compressible Euler systems, where the initial data have an algebraic singularity in vorticity at the origin. These are different from the multi-dimensional Riemann problems widely studied in the literature. Our computations provide numerical evidence of the existence of initial value problems with multiple solutions, thus revealing a fundamental obstruction toward the well-posedness of the governing equations. The compressible Euler equations are solved using the positivity-preserving discontinuous Galerkin method.

Entropy stable, robust and high-order DGSEM for the compressible multicomponent Euler equations

This work concerns the numerical approximation of a multicomponent compressible Euler system for a fluid mixture in multiple space dimensions on unstructured meshes with a high-order discontinuous Galerkin spectral element method (DGSEM). We first derive an entropy stable (ES) and robust (i.e., that preserves the positivity of the partial densities and internal energy) three-point finite volume scheme using relaxation-based approximate Riemann solvers from Bouchut (2004) [9] and Coquel and Perthame (1998) [22]. Then, we consider the DGSEM based on collocation of quadrature and interpolation points which relies on the framework introduced by Fisher and Carpenter (2013) [28] and Gassner (2013) [31]. We replace the physical fluxes in the integrals over discretization elements by entropy conservative numerical fluxes [Tadmor (1987) [71]], while ES numerical fluxes are used at element interfaces. We thus derive a two-point numerical flux satisfying the Tadmor's entropy conservation condition and use the numerical flux from the three-point scheme as ES flux. Time discretization is performed with a strong-stability preserving Runge-Kutta scheme. We then derive conditions on the numerical parameters to guaranty a semi-discrete entropy inequality as well as positivity of the cell average of the partial densities and internal energy of the fully discrete DGSEM at any approximation order. The later results allow to use existing limiters in order to restore positivity of nodal values within elements. The scheme also resolves exactly stationary material interfaces. Numerical experiments in one and two space dimensions on flows with discontinuous solutions support the conclusions of our analysis and highlight stability, robustness and high resolution of the scheme.

Simple waves of the two dimensional compressible Euler system for a class of pressure laws

Simple waves of the two dimensional compressible Euler system for a class of pressure laws

Uniqueness of a Planar Contact Discontinuity for 3D Compressible Euler System in a Class of Zero Dissipation Limits from Navier–Stokes–Fourier System

We prove the stability of contact discontinuities without shear, a family of special discontinuous solutions for the three-dimensional full Euler systems, in the class of vanishing dissipation limits of the corresponding Navier-Stokes-Fourier system. We also show that solutions of the Navier-Stokes-Fourier system converge to the contact discontinuity when the initial datum converges to the contact discontinuity itself. This implies the uniqueness of the contact discontinuity in the class that we are considering. Our results give an answer to the open question, whether the contact discontinuity is unique for the multi-D compressible Euler system. Our proof is based on the relative entropy method, together with the theory of $a$-contraction up to a shift.

A note on the long-time behavior of dissipative solutions to the Euler system

We show that the Reynolds defect measure for a dissipative weak solution of the compressible Euler system vanishes for large time. This may be seen as a piece of evidence that the dissipative solutions are asymptotically close to weak solutions in the turbulent regime, whence suitable for describing compressible fluid flows in the long run.

Small Knudsen Rate of Convergence to Rarefaction Wave for the Landau Equation

In this paper, we are concerned with the hydrodynamic limit to rarefaction waves of the compressible Euler system for the Landau equation with Coulomb potentials as the Knudsen number $\epsilon>0$ is vanishing. Precisely, whenever $\epsilon>0$ is small, for the Cauchy problem on the Landau equation with suitable initial data involving a scaling parameter $a\in [\frac{2}{3},1]$, we construct the unique global-in-time uniform-in-$\epsilon$ solution around a local Maxwellian whose fluid quantities are the rarefaction wave of the corresponding Euler system. In the meantime, we establish the convergence of solutions to the Riemann rarefaction wave uniformly away from $t=0$ at a rate $\epsilon^{\frac{3}{5}-\frac{2}{5}a}|\ln \epsilon|$ as $\epsilon\to 0$. The proof is based on the refined energy approach combining [19] and [32] under the scaling transformation $(t,x)\to (\epsilon^{-a}t,\epsilon^{-a}x)$.

Isentropic Approximation and Gevrey Regularity for the Full Compressible Euler Equations in $${\mathbb {R}}^{N}$$

The article is devoted to the study of isentropic approximation and Gevrey regularity for the full compressible Euler system in $${\mathbb {R}}^{N}$$ (or $${\mathbb {T}}^{N}$$ ) with any dimension $$N\ge 1$$ . We first establish the existence and uniqueness of solution in Gevrey function spaces $$G_{\sigma ,s}^{r}({\mathbb {R}}^{N})$$ , then with the definition modulus of continuity, we show that the solution of Euler system is continuously dependent of the initial data $$v_{0}$$ in $$G_{\sigma ,s}^{r}({\mathbb {R}}^{N})$$ . Finally, the isentropic approximation is investigated in Banach spaces $${\mathcal {B}}_{T}^\nu ({\mathbb {R}}^{N})$$ , provided the initial entropy $$S_{0}(x)$$ changes closing a constant $${\bar{S}}$$ in Gevrey function spaces $$G_{\sigma ,s}^{r}({\mathbb {R}}^{N})$$ .

On Strong Continuity of Weak Solutions to the Compressible Euler System

Let \({\mathcal {S}} = \{ \tau _n \}_{n=1}^\infty \subset (0,T)\) be an arbitrary countable (dense) set. We show that for any given initial density and momentum, the compressible Euler system admits (infinitely many) admissible weak solutions that are not strongly continuous at each \(\tau _n\), \(n=1,2,\dots \). The proof is based on a refined version of the oscillatory lemma of De Lellis and Székelyhidi with coefficients that may be discontinuous on a set of zero Lebesgue measure.

Computing oscillatory solutions of the Euler system via 𝒦-convergence

We develop a method to compute effectively the Young measures associated to sequences of numerical solutions of the compressible Euler system. Our approach is based on the concept of [Formula: see text]-convergence adapted to sequences of parameterized measures. The convergence is strong in space and time (a.e. pointwise or in certain [Formula: see text] spaces) whereas the measures converge narrowly or in the Wasserstein distance to the corresponding limit.

The global existence issue for the compressible Euler system with Poisson or Helmholtz couplings

We consider the Cauchy problem for the barotropic Euler system coupled to Helmholtz or Poisson equations, in the whole space. We assume that the initial density is small enough, and that the initial velocity is close to some reference vector field [Formula: see text] such that the spectrum of [Formula: see text] is positive and bounded away from zero. We prove the existence of a global unique solution with (fractional) Sobolev regularity, and algebraic time decay estimates. Our work extends Grassin and Serre’s papers [Existence de solutions globales et régulières aux équations d’Euler pour un gaz parfait isentropique, C. R. Acad. Sci. Paris Sér. I 325 (1997) 721–726, 1997; Global smooth solutions to Euler equations for a perfect gas, Indiana Univ. Math. J. 47 (1998) 1397–1432; Solutions classiques globales des équations d’Euler pour un fluide parfait compressible, Ann. Inst. Fourier Grenoble 47 (1997) 139–159] dedicated to the compressible Euler system without coupling and with integer regularity exponents.

The 3D Vlasov–Poisson–Landau System Near 1D Local Maxwellians

We consider the Cauchy problem on the Vlasov–Poisson–Landau system with the Coulomb interaction in the three dimensional space domain $${\mathbb {R}}\times {\mathbb {T}^2}$$ . Although there have been extensive studies on global existence and large time behavior of solutions near global Maxwellians either in $$\mathbb {T}^3$$ or $$\mathbb {R}^3$$ , it is unknown whether solutions can be constructed around some non-trivial asymptotic profiles. In this paper, we obtain the global solutions near a spatially one-dimensional local Maxwellian connecting two distinct global Maxwellians at $$x_1=\pm \infty $$ , where the fluid components of the local Maxwellian are the smooth approximate rarefaction wave of the corresponding full compressible Euler system in $$x_1\in \mathbb {R}$$ . We also prove the large time asymptotics of global solutions towards such planar rarefaction waves, and establish the propagation of the high-order moments and regularity of solutions with large amplitude. As a byproduct, all these results can be carried over to the pure Landau equation in the same setting.

On the global classical solution to compressible Euler system with singular velocity alignment

We consider a compressible Euler system with singular velocity alignment, known as the Euler-alignment system, describing the flocking behaviors of large animal groups. We establish a local well-posedness theory for the system, as well as a global well-posedness theory for small initial data. We also show the asymptotic flocking behavior, where solutions converge to a constant steady state exponentially in time.

一类广义压力下的二维可压缩欧拉方程的特征分解

一类广义压力下的二维可压缩欧拉方程的特征分解

Incompressible limit of Euler equations with damping

<abstract><p>The Cauchy problem for the compressible Euler system with damping is considered in this paper. Based on previous global existence results, we further study the low Mach number limit of the system. By constructing the uniform estimates of the solutions in the well-prepared initial data case, we are able to prove the global convergence of the solutions in the framework of small solutions.</p></abstract>

Hydrodynamic Limit of the Boltzmann Equation to the Planar Rarefaction Wave in Three Dimensional Space

In this paper, we establish the global-in-time hydrodynamic limit of the Boltzmann equation to the planar rarefaction wave of a compressible Euler system in three dimensional space $x\in\mathbb{R}^3$ for general collision kernels. Our approach is based on the Hilbert expansion and a recent $L^2$-$L^\infty$ framework. In particular, we improve the $L^2$-estimate to be a localized version because the planar rarefaction wave is indeed a one dimensional wave which makes the source terms not integrable in the $L^2$ energy estimate of the three dimensional problem. We also point out that the wave strength of rarefaction may be large.

On Admissible Locations of Transonic Shock Fronts for Steady Euler Flows in an Almost Flat Finite Nozzle with Prescribed Receiver Pressure

AbstractThis paper concerns the existence of transonic shock solutions to the 2‐D steady compressible Euler system in an almost flat finite nozzle (in the sense that it is a generic small perturbation of a flat one), under physical boundary conditions proposed by Courant‐Friedrichs in [14], in which the receiver pressure is prescribed at the exit of the nozzle. In the resulting free boundary problem, the location of the shock front is some of the most desirable information one would like to determine. However, the location of the normal shock front in a flat nozzle can be anywhere in the nozzle so that it provides little information on the possible location of the shock front when the nozzle's boundary is perturbed. So one of the key difficulties in looking for transonic shock solutions is to determine the shock front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution will be taken as an initial approximation for the transonic shock solution. Once an initial approximation is obtained, a further nonlinear iteration can be constructed and proved to lead to a transonic shock solution. In this paper, a sufficient condition in terms of the geometry of the nozzle and the given exit pressure is derived that yields the existence of the solutions to the proposed free boundary problem. By this condition, it will be shown that, for the proposed linear free boundary problem, if the nozzle is either strictly expanding or contracting, there exists only one solution as long as the given pressure at the exit lies within a certain interval; while if the nozzle is generic with both expanding and contracting portions, there may exist more than one solution for the same given receiver pressure, which implies the existence of more than one initial approximating shock front. This nonuniqueness of the initial approximations will lead to the nonuniqueness of the transonic shock solutions for generic nozzles with the same given receiver pressure, which shows the instability of the unperturbed normal shock solution under general perturbations of the nozzle boundary and how this instability may behave. © 2020 Wiley Periodicals LLC

Non-relativistic limits of contact discontinuities to 1-D piston problem for the relativistic full Euler system

This paper is contributed to the 1-D piston problem for the compressible relativistic full Euler system. Given some small BV perturbations of both the piston velocity and the initial state, we establish the structural stability of a contact discontinuity when the piston moves relatively still to the gas. We also consider the non-relativistic limits of such entropy solutions to piston problem as the light speed c→+∞. Moreover, we study the long time behavior of these solutions as t→+∞.

On the global existence for the compressible Euler–Poisson system, and the instability of static solutions

We consider the Cauchy problem for the barotropic Euler system coupled to Poisson equation, in the whole space. Our main aim is to exhibit a simple functional framework that allows to handle solutions with density going to zero at infinity, but that need not be compactly supported. We have in mind in particular the 3D static solution, when the polytropic index $$\gamma $$ of the gas is equal to 6/5. Our first result is the local existence of classical solutions in a simple functional framework that does not require the velocity to tend to 0 at infinity and the density to be compactly supported. Next, following the work by Grassin and Serre dedicated to the compressible Euler system Grassin and Serre (C R Acad Sci Paris Ser I 325:721–726, 1997, Grassin (Indiana Univ Math J, 47:1397–1432, 1998), we show that if the initial density is small enough, and the initial velocity is close to some reference vector field $$u_0$$ such that the spectrum of $$Du_0$$ is positive and bounded away from zero, then the corresponding classical solution is global, and satisfies algebraic time decay estimates. Compared to our recent paper (Blanc et al. in The global existence issue for the compressible Euler system with Poisson or Helmholtz coupling), we are able to handle the 3D static solution that was mentioned above, and to show its instability, within our functional framework.