Compressible Euler System Research Articles

Structural stability of transonic shocks for the full compressible Euler system in a two-dimensional general nozzle with an external force

In this paper, we investigate the transonic shock solutions to the full compressible Euler system in a general two-dimensional nozzle with an external force. In their book, Courant and Friedrichs [Supersonic Flow and Shock Waves (Interscience Publishers, Inc., New York, 1948)] described the transonic shock phenomena in a de Laval nozzle: given the appropriate receiver pressure, if the upcoming flow is still supersonic after passing through the throat of the nozzle, then a shock front intervenes at some place in the diverging part of the nozzle, and the gas is compressed and slowed down to subsonic speed. We first establish the existence and uniqueness of one-dimensional transonic shock solutions to the steady full Euler system with an external force by prescribing suitable pressure at the exit of the nozzle when the upstream flow is a uniform supersonic flow. With the help of external forces, both existence and uniqueness are established by solving a nonlinear free boundary value problem in a weighted Hölder space.

Well-posedness and decay structure of a quantum hydrodynamics system with Bohm potential and linear viscosity

In this paper, a compressible viscous-dispersive Euler system in one space dimension in the context of quantum hydrodynamics is considered. The purpose of this study is twofold. First, it is shown that the system is locally well-posed. For that purpose, the existence of classical solutions which are perturbation of constant states is established. Second, it is proved that in the particular case of subsonic equilibrium states, sufficiently small perturbations decay globally in time. In order to prove this stability property, the linearized system around the subsonic state is examined. Using an appropriately constructed compensating matrix symbol in the Fourier space, it is proved that solutions to the linear system decay globally in time, underlying a dissipative mechanism of regularity gain type. These linear decay estimates, together with the local existence result, imply the global existence and the decay of perturbations to constant subsonic equilibrium states as solutions to the full nonlinear system.

Compressible fluid limit for smooth solutions to the Landau equation

Although the compressible fluid limit of the Boltzmann equation with cutoff has been extensively investigated in Caflisch [Comm. Pure Appl. Math. 33 (1980), 651–666] and Guo, Jang, and Jiang [Comm. Pure Appl. Math. 63 (2010), 337–361], obtaining analogous results in the case of the angular non-cutoff or even in the grazing limit which gives the Landau equation, still remains largely open, essentially due to the velocity diffusion effect of the collision operator such that L^{\infty} estimates are hard to obtain without using Sobolev embeddings. In this paper, we are concerned with the compressible Euler and acoustic limits of the Landau equation for Coulomb potentials in the whole space. Specifically, over any finite time interval where the full compressible Euler system admits a smooth solution around constant states, we construct a unique solution in a high-order weighted Sobolev space for the Landau equation with suitable initial data and also show the uniform estimates independent of the small Knudsen number \varepsilon>0 , yielding the O(\varepsilon) convergence of the Landau solution to the local Maxwellian whose fluid quantities are the given Euler solution. Moreover, the acoustic limit for smooth solutions to the Landau equation in an optimal scaling is also established. For the proof, by using the macro-micro decomposition around local Maxwellians, together with techniques for viscous compressible fluid and properties of Burnett functions, we design an \varepsilon -dependent energy functional to capture the dissipation in the compressible fluid limit with the feature that only the highest-order derivatives are most singular.

Global well-posedness and asymptotic behavior for the Euler-alignment system with pressure

We study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We establish a global well-posedness theory for the system with small smooth initial data. Additionally, we derive asymptotic emergent behaviors for the system, providing time decay estimates with optimal decay rates. Notably, the optimal decay rate we obtain does not align with the corresponding fractional heat equation within our considered range, where the parameter α∈(0,1). This highlights the distinct feature of the alignment operator.

Asymptotic stability of rarefaction wave for compressible Euler system with velocity alignment

In this paper, we study the asymptotic stability of the rarefaction wave for the one-dimensional compressible Euler system with nonlocal velocity alignment. Namely, for the initial data approaching to rarefaction wave, we prove the corresponding solution converges toward the rarefaction wave. Moreover, we obtain this system has weak alignment behavior. We develop some promoted estimates for the smooth approximate rarefaction wave and new a priori estimates by Fourier analysis tools. Moreover, we introduce the weighted energy method and Besov spaces to obtain the key high-order derivative estimates, in which we overcome the difficulties caused by the nonlocal velocity alignment. It is worth mentioning that this is the first stability result of rarefaction wave for compressible Euler system with velocity alignment.

On uniqueness of steady 1-D shock solutions in a finite nozzle via asymptotic analysis for physical parameters

In this paper, we study the uniqueness of the steady 1-D shock solutions for the inviscid compressible Euler system in a finite nozzle via asymptotic analysis for physical parameters. The parameters for the heat conductivity and the temperature-depending viscosity are investigated for both barotropic gases and polytropic gases. It finally turns out that the hypotheses on the physical effects have significant influences on the asymptotic behaviors as the parameters vanish. In particular, the positions of the shock front for the limit shock solution (if exists) are different for different hypotheses. Hence, it seems impossible to figure out a criterion selecting the unique shock solution within the framework of the inviscid Euler flows.

Inviscid limit for the compressible Navier-Stokes equations with density dependent viscosity

We consider the compressible Navier-Stokes system describing the motion of a barotropic fluid with density dependent viscosity confined in a three-dimensional bounded domain Ω. We show the convergence of the weak solution to the compressible Navier-Stokes system to the strong solution to the compressible Euler system when the viscosity and the damping coefficients tend to zero.

Hilbert expansion of Boltzmann equations for gas mixture

The aim of this paper is to study the hydrodynamic limit of Boltzmann equations for a gas mixture containing two kinds of particles, where one is assumed to be near a vacuum and the other is near a Maxwellian equilibrium state. By the method of Hilbert expansion, we could derive the compressible Euler systems together with the terms of Boltzmann equations according to the different orders of the Knudsen number. Next, we truncate the expansion and establish the uniform estimates of the remainder term by using L 2 − L ∞ estimate.

Optimal Boundary Control of the Isothermal Semilinear Euler Equation for Gas Dynamics on a Network

The analysis and boundary optimal control of the nonlinear transport of gas on a network of pipelines is considered. The evolution of the gas distribution on a given pipe is modeled by an isothermal semilinear compressible Euler system in one space dimension. On the network, solutions satisfying (at nodes) the Kirchhoff flux continuity conditions are shown to exist in a neighborhood of an equilibrium state. The associated nonlinear optimization problem then aims at steering such dynamics to a given target distribution by means of suitable (network) boundary controls while keeping the distribution within given (state) constraints. The existence of local optimal controls is established and a corresponding Karush–Kuhn–Tucker (KKT) stationarity system with an almost surely non-singular Lagrange multiplier is derived.

Global non-relativistic quasi-neutral limit for a two-fluid Euler-Maxwell system

This paper concerns a two-fluid Euler-Maxwell system for plasmas with two small parameters. We study the non-relativistic quasi-neutral limit for smooth solutions of the system in whole space R3. When the initial data are smooth and sufficiently close to constant equilibrium states, we give a rigorous justification of the limit for all time. The limit system is governed by a compressible Euler system. The proof is based on uniform energy estimates and various dissipative estimates of smooth solutions with respect to the parameters and time. A key step in these estimates is to control the quasi-neutrality of the velocities by using a projection operator.

Global well-posedness and asymptotic behavior in critical spaces for the compressible Euler system with velocity alignment

In this paper, we study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We prove the existence and uniqueness of global solutions in critical Besov spaces to the considered system with small initial data. The local-in-time solvability is also addressed. Moreover, we show the large-time asymptotic behaviour and optimal decay estimates of the solutions as .

A semi-implicit finite volume scheme for dissipative measure-valued solutions to the barotropic Euler system

A semi-implicit in time, entropy stable finite volume scheme for the compressible barotropic Euler system is designed and analyzed and its weak convergence to a dissipative measure-valued (DMV) solution [Feireisl et al., Calc. Var. Part. Differ. Equ. 55 (2016) 141] of the Euler system is shown. The entropy stability is achieved by introducing a shifted velocity in the convective fluxes of the mass and momentum balances, provided some CFL-like condition is satisfied to ensure stability. A consistency analysis is performed in the spirit of the Lax’s equivalence theorem under some physically reasonable boundedness assumptions. The concept of Ƙ-convergence [Feireisl et al., IMA J. Numer. Anal. 40 (2020) 2227–2255] is used in order to obtain some strong convergence results, which are then illustrated via rigorous numerical case studies. The convergence of the scheme to a DMV solution, a weak solution and a strong solution of the Euler system using the weak–strong uniqueness principle and relative entropy are presented.

Finite-time blow-up for the compressible Euler system in the exterior domain

Finite-time blow-up for the compressible Euler system in the exterior domain

Asymptotic behaviors for the compressible Euler system with nonlinear velocity alignment

We consider the compressible Euler system with a family of nonlinear velocity alignments. The system is a nonlinear extension of the Euler-alignment system in collective dynamics. We show the asymptotic emergent phenomena of the system: alignment and flocking. Different types of nonlinearity and nonlocal communication protocols are investigated, resulting in a variety of different asymptotic behaviors.

A convergent finite volume scheme for the stochastic barotropic compressible Euler equations

In this paper, we analyze a semi-discrete finite volume scheme for the three-dimensional barotropic compressible Euler equations driven by a multiplicative Brownian noise. We derive necessary a priori estimates for numerical approximations, and show that the Young measure generated by the numerical approximations converge to a dissipative measure-valued martingale solution to the stochastic compressible Euler system. These solutions are probabilistically weak in the sense that the driving noise and associated filtration are integral part of the solution. Moreover, we demonstrate strong convergence of numerical solutions to the regular solution of the limit systems at least on the lifespan of the latter, thanks to the weak (measure-valued)–strong uniqueness principle for the underlying system. To the best of our knowledge, this is the first attempt to prove the convergence of numerical approximations for the underlying system.

A remark on the multi-dimensional compressible Euler system with damping in the 𝐿^{𝑝} critical Besov spaces

We study the global-in-time well-posedness and relaxation limit of the compressible Euler system with damping in L p L^p -type critical Besov spaces. In comparison with the results obtained by Crin-Barat and Danchin [Pure Appl. Anal. 4 (2022), pp. 85–125; Math. Ann. 386 (2023), pp. 2159–2206], the more general pressure law satisfying P ′ ( ρ ¯ ) > 0 P’(\bar {\rho })>0 is allowed. To achieve it, a new composition estimate is established in the L 2 L^2 - L p L^p hybrid Besov spaces with explicit dependence on the threshold between high frequencies and low frequencies.

Small Knudsen rate of convergence to contact wave for the Landau equation

In this paper, we study the hydrodynamic limit to contact discontinuity of the compressible Euler system for the Landau equation with the physical Coulomb interaction as the Knudsen number ε>0 is vanishing. Our results are twofold. First, we construct the unique global-in-time solution to the Landau equation with suitably small initial data as ε>0 is sufficiently small. Second, we prove that the solution of the Landau equation converges to a local Maxwellian whose fluid quantities are the contact discontinuity of the corresponding Euler system uniformly away from the discontinuity as ε→0. Moreover, the uniform convergence rate is also obtained. The main idea is to introduce a new time-velocity weigh function to induce an extra quartic dissipation term for treating the large-velocity growth in the nonlinear estimates due to degeneration of the linearized Landau operator in the Coulomb case.

Decay rate to contact discontinuities for the one-dimensional compressible Navier–Stokes equations with a reacting mixture

In this paper, we investigate the nonlinear stability of contact waves for the Cauchy problem to the compressible Navier–Stokes equations for a reacting mixture in one dimension. If the corresponding Riemann problem for the compressible Euler system admits a contact discontinuity solution, it is shown that the contact wave is nonlinearly stable, while the strength of the contact discontinuity and the initial perturbation are suitably small. Especially, we obtain the convergence rate by using anti-derivative methods and elaborated energy estimates.

Classification of nonlocal symmetries and exact solutions for 3 × 3 Chaplygin gas equation with conservation laws

In this paper, we study the one-dimensional isentropic compressible Euler system for the Chaplygin gas through Lie symmetry analysis. The one-dimensional optimal subalgebras are classified using the adjoint transformation and the invariant functions. We derived several new exact solutions from the optimal subalgebras and investigated the physical behavior of some solutions graphically. Next, a tree of nonlocally related partial differential equations (PDEs) is presented and we classify the nonlocal symmetry of the given system. Futher, some nontrivial exact solutions for the given model are constructed using nonlocal symmetries. Furthermore, using the traveling wave transformation, which is invariant under the symmetry group, we obtain solutions of the nature of peakon-type and kink-type solitons. Then, conservation laws are constructed through the direct multipliers method. Finally, the evolutionary behavior of a C1-wave is investigated using one of the developed solutions.

Second Order Two-Species Systems with Nonlocal Interactions: Existence and Large Damping Limits

We study the mathematical theory of second order systems with two species, arising in the dynamics of interacting particles subject to linear damping, to nonlocal forces and to external ones, and resulting into a nonlocal version of the compressible Euler system with linear damping. Our results are limited to the 1 space dimensional case but allow for initial data taken in a Wasserstein space of probability measures. We first consider the case of smooth nonlocal interaction potentials, not subject to any symmetry condition, and prove existence and uniqueness. The concept of solutions relies on a stickiness condition in case of collisions, in the spirit of previous works in the literature. The result uses concepts from classical Hilbert space theory of gradient flows (cf. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, 1973) and a trick used in Brenier et al. (J. Math. Pures Appl. 99(5):577–617, 2013). We then consider a large-time and large-damping scaled version of our system and prove convergence to solutions to the corresponding first order system. Finally, we consider the case of Newtonian potentials - subject to symmetry of the cross-interaction potentials - and external convex potentials. After showing existence in the sticky particles framework in the spirit of Brenier et al. (J. Math. Pures Appl. 99(5):577–617, 2013), we prove convergence for large times towards Dirac delta solutions for the two densities. All the results share a common technical framework in that solutions are considered in a Lagrangian framework, which allows to estimate the behavior of solutions via L^{2} estimates of the pseudo-inverse variables corresponding to the two densities. In particular, due to this technique, the large-damping result holds under a rather weak condition on the initial data, which does not require well-prepared initial velocities. We complement the results with numerical simulations.