Three-dimensional Compressible Euler Equations Research Articles

Rough solutions of the relativistic Euler equations

We prove that the time of classical existence of smooth solutions to the relativistic Euler equations can be bounded from below in terms of norms that measure the “(sound) wave-part” of the data in Sobolev space and “transport-part” in higher regularity Sobolev space and Hölder spaces. The solutions are allowed to have nontrivial vorticity and entropy. We use the geometric framework from [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270], where the relativistic Euler flow is decomposed into a “wave-part”, that is, geometric wave equations for the velocity components, density and enthalpy, and a “transport-part”, that is, transport-div-curl systems for the vorticity and entropy gradient. Our main result is that the Sobolev norm [Formula: see text] of the variables in the “wave-part” and the Hölder norm [Formula: see text] of the variables in the “transport-part” can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption [Formula: see text] is the optimal result for the variables in the “wave-part”. Compared to low-regularity results for quasilinear wave equations and the three-dimensional (3D) non-relativistic compressible Euler equations, the main new challenge of the paper is that when controlling the acoustic geometry and bounding the wave equation energies, we must deal with the difficulty that the vorticity and entropy gradient are four-dimensional space-time vectors satisfying a space-time div-curl-transport system, where the space-time div-curl part is not elliptic. Due to lack of ellipticity, one cannot immediately rely on the approach taken in [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270] to control these terms. To overcome this difficulty, we show that the space-time div-curl systems imply elliptic div-curl-transport systems on constant-time hypersurfaces plus error terms that involve favorable differentiations and contractions with respect to the four-velocity. By using these structures, we are able to adequately control the vorticity and entropy gradient with the help of energy estimates for transport equations, elliptic estimates, Schauder estimates and Littlewood–Paley theory.

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.

High-order accurate entropy stable adaptive moving mesh finite difference schemes for (multi-component) compressible Euler equations with the stiffened equation of state

This paper extends the high-order entropy stable (ES) adaptive moving mesh finite difference schemes developed in Duan and Tang (2022) to the two- and three-dimensional (multi-component) compressible Euler equations with the stiffened equation of state (EOS). The two-point entropy conservative (EC) flux is first constructed in the curvilinear coordinates, which is nontrivial in the case of the stiffened EOS. With the aid of the high-order discretization of the geometric conservation laws and the linear combinations of the two-point EC fluxes, the high-order semi-discrete EC schemes are derived. The high-order semi-discrete ES schemes are constructed by adding suitable high-order dissipation term to the EC schemes such that the semi-discrete entropy inequality is satisfied and unphysical oscillations are suppressed. The high-order dissipation term is built on the multi-resolution weighted essentially non-oscillatory (WENO) reconstruction and the newly derived scaled eigenvector matrices. The explicit strong-stability-preserving Runge–Kutta methods are used for the time discretization and the mesh points are adaptively redistributed by iteratively solving the mesh redistribution equations with an appropriate monitor function, which is adapted to the multi-component flow and encodes more physical characteristics of the solutions. Several 2D and 3D numerical tests are conducted on the parallel computer system with the MPI programming to validate the accuracy and the ability to effectively capture the localized structures of the proposed schemes.

Optimal L p – L q -Type Decay Rates of Solutions to the Three-Dimensional Nonisentropic Compressible Euler Equations with Relaxation

In this paper, we consider the three-dimensional Cauchy problem of the nonisentropic compressible Euler equations with relaxation. Following the method of Wu et al. (2021, Adv. Math. Phys. Art. ID 5512285, pp. 1–13), we show the existence and uniqueness of the global small H k k ⩾ 3 solution only under the condition of smallness of the H 3 norm of the initial data. Moreover, we use a pure energy method with a time-weighted argument to prove the optimal L p – L q 1 ⩽ p ⩽ 2 , 2 ⩽ q ⩽ ∞ -type decay rates of the solution and its higher-order derivatives.

Global Existence and Long-Time Behavior of Solutions to the Full Compressible Euler Equations with Damping and Heat Conduction in ℝ 3

We study the Cauchy problem of the three-dimensional full compressible Euler equations with damping and heat conduction. We prove the existence and uniqueness of the global small H N N ≥ 3 solution; in particular, we only require that the H 4 norms of the initial data be small when N ≥ 5 . Moreover, we use a pure energy method to show that the global solution converges to the constant equilibrium state with an optimal algebraic decay rate as time goes to infinity.

Simulations of the Early Postbounce Phase of Core-collapse Supernovae in Three-dimensional Space with Full Boltzmann Neutrino Transport

Abstract We report on the core-collapse supernova simulation we conducted for a 11.2M ⊙ progenitor model in three-dimensional space up to 20 ms after bounce, using a radiation-hydrodynamics code with full Boltzmann neutrino transport. We solve the six-dimensional Boltzmann equations for three neutrino species and the three-dimensional compressible Euler equations with Furusawa and Togashi's nuclear equation of state. We focus on the prompt convection at ∼10 ms after bounce and investigate how neutrinos are transported in the convective matter. We apply a new analysis based on the eigenvalues and eigenvectors of the Eddington tensor and make a comparison between the Boltzmann transport results and the M1 closure approximation in the transition regime between the optically thick and thin limits. We visualize the eigenvalues and eigenvectors using an ellipsoid, in which each principal axis is parallel to one of the eigenvectors and has a length proportional to the corresponding eigenvalue. This approach enables us to understand the difference between the Eddington tensor derived directly from the Boltzmann simulation and the one given by the M1 prescription from a new perspective. We find that the longest principal axis of the ellipsoid is almost always nearly parallel to the energy flux in the M1 closure approximation, whereas in the Boltzmann simulation it becomes perpendicular in some transition regions, where the mean free path is ∼0.1 times the radius. In three spatial dimensions, the convective motions make it difficult to predict where this happens and to possibly improve the closure relation there.

The Lifespan of Classical Solutions to the (Damped) Compressible Euler Equations

In this paper, the initial-boundary value problem of the original three-dimensional compressible Euler equations with (or without) time-dependent damping is considered. By considering a functional $$F(t,\alpha ,f)$$ weighted by a general time-dependent parameter function $$\alpha $$ and a general radius-dependent parameter function f, we show that if the initial value $$F|_{t=0}$$ is sufficiently large, then the lifespan of the system is finite. Here, f can be any $$C^1$$ strictly increasing function such that the sum of initial values of f and $$\alpha $$ is non-negative. It follows that a class of conditions for non-existence of global classical solutions is established. Moreover, the conditions imply that a strong $$\alpha $$ will lead to a more unrestrained necessary condition for classical solutions of the system to exist globally in time.

Subsonic flows passing a duct for three-dimensional steady compressible Euler systems with friction

This paper studies steady motion of gas in a rectilinear duct with square cross-sections,governed by the three-dimensional (3-d) non-isentropic compressible Euler equations with a friction term.Such flows are called Fanno flows in engineering. We construct respectively special subsonic flows, supersonic flows and transonic shocks in the duct.Since the 3-d steady compressible Euler equations are of quasi-linear hyperbolic-elliptic composite type for subsonic flows, and there is no generaltheory up to now, we formulate a boundary value problem arising from studies of transonic shocks, and prove the well-posednessof this problem by showing that the special subsonic flows constructed above are stable under small multi-dimensional perturbations.The proof depends on separation of the elliptic and hyperbolic parts in the Euler equations, and designation of a suitable nonlinear iteration scheme.Particularly, there are strong interactions between the elliptic part and the hyperbolic part due to the appearance of friction, and we deducea linear mixed boundary value problem of a second-order elliptic equation with an integral-type nonlocal term. Its well-posedness is establishedby applying methods of Fourier analysis and regularity theory of second-order elliptic equations.

New weighted functional for non-existence of global solutions to the non-isentropic compressible Euler equations

In this paper, we consider the Cauchy problem of the original three-dimensional non-isentropic compressible Euler equations. Among the two main results, a class of finite-time blowup conditions without assumptions on γ>1 and η(0)≔∫R3ρeS(0,x)∕γ(0,x)−ρ̄eS̄∕γdx is established. It is shown that together with the requirement on the sign of initial momentum, sufficiently strong F(0) will develop finite-time singularity and the C1 solutions cannot exist globally. Here, F(t)=∫R3[α(t)+f(r)]x⋅ρudx is a newly introduced functional weighted by a sum of a time-dependent parameter function α(t) and a radius-dependent parameter function f(t) satisfying some mild conditions. As one of the applications, it is analysed that stronger α implies that the necessary conditions for solutions of the original three-dimensional non-isentropic compressible Euler equations to exist on or before a given finite time is looser.

Expanding large global solutions of the equations of compressible fluid mechanics

Without any symmetry assumptions on the initial data we construct global-in-time unique solutions to the vacuum free boundary three-dimensional isentropic compressible Euler equations when the adiabatic exponent $$\gamma $$ lies in the interval $$(1,\frac{5}{3}]$$ . Our initial data lie sufficiently close to the expanding compactly supported affine motions recently constructed by Sideris and they satisfy the physical vacuum boundary condition.

A Study of Preconditioned Jacobian-Free Newton-Krylov Discontinuous Galerkin Method for Compressible Flows on 3D Hexahedral Grids

Storage requirement and computational efficiency have always been challenges for the efficient implementation of discontinuous Galerkin (DG)methods for real life applications. In this paper, a fully implicit Jacobian-Free Newton-Krylov (JFNK) method is developed in the context of DG discretizations for the three-dimensional compressible Euler and Navier-Stokes equations. Compared with the Jacobian-based methods, the Jacobian-Free approach saves the storage for the Jacobian matrix which can be of great importance for DG methods. Three types of preconditioners are investigated in which the block diagonal preconditioner requires the least storage, while the block LU-SGS and ILU0 preconditioners require more storage but are more computationally efficient. An implicit time-stepping strategy is adopted for the stability of the current solver,which is based upon a hexahedral spatialmesh and the nonlinear solver package Kinsol is used to improve the computational efficiency and robustness. Numerical results demonstrate that the preconditioned JFNK-DG solver can substantially reduce the storage requirement compared with the Jacobian based method without significantly compromising accuracy or efficiency. Furthermore, as a good compromise between efficiency and storage requirement, the ILU0 preconditioner shows the best choice of the preconditioners presented.

Time periodic solution to the compressible Euler equations with damping in a periodic domain

In this paper, we consider the existence and uniqueness of a time periodic solution to the three-dimensional compressible damped Euler equations in a periodic domain. By adapting a regularized approximation scheme and applying the topological degree theory, we establish the existence of the time periodic solution under some smallness and structure assumptions imposed on a time periodic force. And based on energy estimates, the uniqueness of the periodic solution is proved.

Compatible finite element spaces for geophysical fluid dynamics

Compatible finite elements provide a framework for preserving important structures in equations of geophysical fluid dynamics, and are becoming important in their use for building atmosphere and ocean models. We survey the application of compatible finite element spaces to geophysical fluid dynamics, including the application to the nonlinear rotating shallow water equations, and the three-dimensional compressible Euler equations. We summarise analytic results about dispersion relations and conservation properties, and present new results on approximation properties in three dimensions on the sphere, and on hydrostatic balance properties.

Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities

In this paper, we establish a priori estimates for three-dimensional compressible Euler equations with the moving physical vacuum boundary, the $\gamma$-gas law equation of state for $\gamma=2$ and the general initial density $\rho_0 \in H^5$. Because of the degeneracy of the initial density, we investigate the estimates of the horizontal spatial and time derivatives and then obtain the estimates of the normal or full derivatives through the elliptic-type estimates. We derive a mixed space-time interpolation inequality which plays a vital role in our energy estimates and obtain some extra estimates for the space-time derivatives of the velocity in $L^3$.

Well-Posedness for the Motion of Physical Vacuum of the Three-dimensional Compressible Euler Equations with or without Self-Gravitation

This paper concerns the well-posedness theory of the motion of physical vacuum for the compressible Euler equations with or without self-gravitation. First, a general uniqueness theorem of classical solutions is proved for the three dimensional general motion. Second, for the spherically symmetric motions, without imposing the compatibility condition of the first derivative being zero at the center of symmetry, a new local-in-time existence theory is established in a functional space involving less derivatives than those constructed for three-dimensional motions in \cite{10',7,16'} by constructing suitable weights and cutoff functions featuring the behavior of solutions near both the center of the symmetry and the moving vacuum boundary.

Weak stability of transonic contact discontinuities in three-dimensional steady non-isentropic compressible Euler flows

In this paper, we study the stability of contact discontinuities that separate a C1 supersonic flow from a static gas, governed by the three-dimensional steady non-isentropic compressible Euler equations. The linear stability problem of this transonic contact discontinuity is formulated as a one-phase free boundary problem for a hyperbolic system with the boundary being characteristics. By calculating the Kreiss–Lopatinskii determinant for this boundary value problem, we conclude that this transonic contact discontinuity is always stable, but only in a weak sense because the Kreiss–Lopatinskii condition fails exactly at the poles of the symbols associated with the linearized hyperbolic operators. Both of the planar and nonplanar contact discontinuities are studied. We establish the energy estimates of solutions to the linearized problem at a contact discontinuity, by constructing the Kreiss symmetrizers microlocally away from the poles of the symbols, and studying the equations directly at each pole. The nonplanar case is studied by using the calculus of para-differential operators. The failure of the uniform Kreiss–Lopatinskii condition leads to a loss of derivatives of solutions in estimates.

Well-Posedness of the Free-Boundary Compressible 3-D Euler Equations with Surface Tension and the Zero Surface Tension Limit

We prove that the three-dimensional compressible Euler equations with surface tension along the moving free-boundary are well-posed; we then establish the limit as surface tension tends to zero. Specifically, we consider isentropic dynamics and consider an equation of state, modeling a liquid, given by Courant and Friedrichs [Supersonic Flow and Shock Waves, Appl. Math. Sci. 21, Springer-Verlag, New York, 1976] as $p(\rho) = \alpha \rho^ \gamma - \beta$ for consants $\gamma >1$ and $ \alpha , \beta > 0$. The analysis is made difficult by two competing nonlinearities associated with the potential energy: compression in the bulk and surface area dynamics on the free-boundary. Unlike the analysis of the incompressible Euler equations, wherein boundary regularity controls regularity in the interior, the compressible Euler equation requires the additional analysis of nonlinear wave equations generating sound waves. An existence theory is developed by a specially chosen parabolic regularization together with th...

Well-Posedness in Smooth Function Spaces for the Moving-Boundary Three-Dimensional Compressible Euler Equations in Physical Vacuum

We prove well-posedness for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) = C γ ρ γ for γ > 1. The physical vacuum singularity requires the sound speed c to go to zero as the square-root of the distance to the moving boundary, and thus creates a degenerate and characteristic hyperbolic free-boundary system wherein the density vanishes on the free-boundary, the uniform Kreiss–Lopatinskii condition is violated, and manifest derivative loss ensues. Nevertheless, we are able to establish the existence of unique solutions to this system on a short time-interval, which are smooth (in Sobolev spaces) all the way to the moving boundary, and our estimates have no derivative loss with respect to initial data. Our proof is founded on an approximation of the Euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term, chosen to preserve as much of the geometric structure of the Euler equations as possible. We first construct solutions to this degenerate parabolic regularization using a higher-order version of Hardy’s inequality; we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter. Solutions to the compressible Euler equations are found in the limit as the artificial viscosity tends to zero. Our regular solutions can be viewed as degenerate viscosity solutions. Our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws.

A three-dimensional compressible flow model for rotating waves in vaneless diffusers with unparallel walls

A three-dimensional compressible flow model is presented to study the occurrence of weak rotating waves in unparallel wall vaneless diffusers in centrifugal compressors. The model extends the three-dimensional compressible flow model for parallel wall diffusers recently developed by present authors. Linearised three-dimensional compressible Euler equations casted on a rotating frame of reference travelling at the same speeds as the waves are employed and the viscous effects are ignored. Complex functions of the solutions to the linearised Euler equations are then obtained by a second-order finite difference method and the singular value decomposition technique. Undisturbed flow is assumed potential and first solved by numerical method of strongly implicit procedure. Critical inlet flow rate and rotating wave speed of diffusers of three different shroud wall shapes, namely, convergent, convergent then divergent and constant area tapered, are studied for three different diffuser outlet-to-inlet radius ratios and for different inlet Mach numbers, and results compared with those from diffusers with parallel walls. The results show suppression effects on rotating stall by the contracting walls and the suppression effects vary with wall contraction rate, wall shape, inlet Mach number and the diffuser radius ratio. Further, the effects of diffuser inlet contraction are studied and prediction of the model is compared with experimental result.

A Priori Estimates for the Free-Boundary 3D Compressible Euler Equations in Physical Vacuum

We prove a priori estimates for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) = C γ ρ γ for γ > 1. The vacuum condition necessitates the vanishing of the pressure, and hence density, on the dynamic boundary, which creates a degenerate and characteristic hyperbolic free-boundary system to which standard methods of symmetrizable hyperbolic equations cannot be applied.