Multidimensional Compressible Euler Equations Research Articles

Global smooth solutions of multidimensional compressible Euler equations with critical time depending damping and vorticity

The authors (Hou and Yin, 2017 [9]; Hou et al., 2018 [10]) proved that the 2D and 3D irrotational compressible Euler equations with critical time depending damping admit global classical solutions. In this paper, we will study this problem with non-zero vorticity and show the global existence of smooth solutions to the 2D and 3D compressible Euler equations with critical time depending damping. They used the Lorentz vector field and their method fails in the presence of the vorticity. The main ingredients in this paper are the auxiliary energies which will make up for the loss of the Lorentz vector field and the improved decay estimates of the energy which can be achieved by a new multiplier.

Optimal Decay Rates of the Compressible Euler Equations with Time-Dependent Damping in \({\mathbb {R}}^n\): (II) Overdamping Case

This paper is concerned with the large time behavior of the multidimensional compressible Euler equations with time-dependent overdamping of the form in , where , , and . This continues our previous work dealing with the underdamping case for . We show the optimal decay estimates of the solutions such that for and , and , which indicates that a stronger damping gives rise to solutions decaying optimally slower. For the critical case of , we prove the optimal logarithmical decay of the perturbation of density for the damped Euler equations such that and for . The overdamping effect reduces the decay rates of the solutions to be slow, which causes us some technical difficulty in obtaining the optimal decay rates by the Fourier analysis method and the Green function method. Here, we propose a new idea to overcome such a difficulty by artfully combining the Green function method and the time-weighted energy method.

Optimal Decay Rates of the Compressible Euler Equations with Time-Dependent Damping in $${\mathbb {R}}^n$$: (I) Under-Damping Case

This paper is concerned with the multi-dimensional compressible Euler equations with time-dependent damping of the form \(-\frac{\mu }{(1+t)^\lambda }\rho {\varvec{u}}\) in \({\mathbb {R}}^n\), where \(n\ge 2\), \(\mu >0\), and \(\lambda \in [0,1)\). When \(\lambda >0\) is bigger, the damping effect time-asymptotically gets weaker, which is called under-damping. We show the optimal decay estimates of the solutions such that \(\Vert \partial _x^\alpha (\rho -1)\Vert _{L^2(\mathbb R^n)}\approx (1+t)^{-\frac{1+\lambda }{2}\left( \frac{n}{2}+|\alpha |\right) }\), and \(\Vert \partial _x^\alpha {\varvec{u}}\Vert _{L^2({\mathbb {R}}^n)}\approx (1+t)^{-\frac{1+\lambda }{2}\left( \frac{n}{2}+|\alpha |\right) -\frac{1-\lambda }{2}}\), and see how the under-damping effect influences the structure of the Euler system. Different from the traditional view that the stronger damping usually makes the solutions decaying faster, here we recognize that the weaker damping with \(0\le \lambda <1\) enhances the faster decay for the solutions. The adopted approach is the technical Fourier analysis and the Green function method. The main difficulties caused by the time-dependent damping lie in twofold: non-commutativity of the Fourier transform of the linearized operator precludes explicit expression of the fundamental solution; time-dependent evolution implies that the Green matrix G(t, s) is not translation invariant, i.e., \(G(t,s)\ne G(t-s,0)\). We formulate the exact decay behavior of the Green matrices G(t, s) with respect to t and s for both linear wave equations and linear hyperbolic system, and finally derive the optimal decay rates for the nonlinear Euler system.

Surviving time estimates of local classical solutions to compressible Euler equations with logarithmic equation of state

In this paper, two sets of initial conditions that guarantee a positive lower bound and a finite upper bound of existence time of local classical solutions to multi-dimensional compressible Euler equations with Logarithmic equation of state are established.

Global Solutions of the Compressible Euler Equations with Large Initial Data of Spherical Symmetry and Positive Far-Field Density

We are concerned with the global existence theory for spherically symmetric solutions of the multidimensional compressible Euler equations with large initial data of positive far-field density. The central feature of the solutions is the strengthening of waves as they move radially inward toward the origin. Various examples have shown that the spherically symmetric solutions of the Euler equations blow up near the origin at certain time. A fundamental unsolved problem is whether the density of the global solution would form concentration to become a measure near the origin for the case when the total initial-energy is unbounded and the wave propagation is not at a finite speed starting initially. In this paper, we establish a global existence theory for spherically symmetric solutions of the compressible Euler equations with large initial data of positive far-field density and relative finite-energy. This is achieved by developing a new approach via adapting a class of degenerate density-dependent viscosity terms, so that a rigorous proof of the vanishing viscosity limit of global weak solutions of the Navier-Stokes equations with the density-dependent viscosity terms to the corresponding global solution of the Euler equations with large initial data of spherical symmetry and positive far-field density can be obtained. One of our main observations is that the adapted class of degenerate density-dependent viscosity terms not only includes the viscosity terms for the Navier-Stokes equations for shallow water (Saint Venant) flows but also, more importantly, is suitable to achieve our key objective of this paper. These results indicate that concentration is not formed in the vanishing viscosity limit for the Navier-Stokes approximations constructed in this paper even when the total initial-energy is unbounded.

Finite-time singularity formation for the original multidimensional compressible Euler equations for generalized Chaplygin gas

It is shown that singularity will be developed in finite time for the original (without symmetry) multidimensional compressible Euler equations for generalized Chaplygin gas if the initial momentum is sufficiently strong and an initial averaged quantity is nonnegative.

Finite-time singularity formation for C 1 solutions to the compressible Euler equations with time-dependent damping

In this paper, the initial-boundary value problem of the multidimensional compressible Euler equations with time-dependent damping in radial symmetry is considered. It is shown that finite-time singularity will be developed for the solutions of the compressible Euler equations with time-dependent damping coefficients if the initial value of a newly introduced functional, with a time-dependent parameter is sufficiently large. The blowup conditions imply that the initial kinetic energy of the fluid must not be less than a given constant.

High-order residual distribution scheme for the time-dependent Euler equations of fluid dynamics

In the present work, a high order finite element type residual distribution scheme is designed in the framework of multidimensional compressible Euler equations of gas dynamics. The strengths of the proposed approximation rely on the generic spatial discretization of the model equations using a continuous finite element type approximation technique, while avoiding the solution of a large linear system with a sparse mass matrix which would come along with any standard ODE solver in a classical finite element approach to advance the solution in time. In this work, we propose a new Residual Distribution (RD) scheme, which provides an arbitrary explicit high order approximation of the smooth solutions of the Euler equations both in space and time. The design of the scheme via the coupling of the RD formulation (Ricchiuto and Abgrall, 2010; Abgrall, 2006) with a Deferred Correction (DeC) type method (Liu et al., 2008; Minion, 2003), allows to have the matrix associated to the update in time, which needs to be inverted, to be diagonal. The use of Bernstein polynomials as shape functions, guarantees that this diagonal matrix is invertible and ensures strict positivity of the resulting diagonal matrix coefficients. This work is the extension of Abgrall et al. (2016) and Abgrall (2017) to multidimensional systems. We have assessed our method on several challenging benchmark problems for one- and two-dimensional Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions.

On the global existence of smooth solutions to the multi-dimensional compressible euler equations with time-depending damping in half space

This paper is a continue work of [4,5]. In the previous two papers, we studied the Cauchy problem of the multi-dimensional compressible Euler equations with time-depending damping term -μ(1+t)λρu, where λ≥0 and μ > 0 are constants. We have showed that, for all λ≥0 and μ>0, the smooth solution to the Cauchy problem exists globally or blows up in finite time. In the present paper, instead of the Cauchy problem we consider the initial-boundary value problem in the half space ℝd+ with space dimension d = 2,3. With the help of the special structure of the equations and the fluid vorticity, we overcome the difficulty arisen from the boundary effect. We prove that there exists a global smooth solution for 0 ≤ λ <1 when the initial data is close to its equilibrium state. In addition, exponential decay of the fluid vorticity will also be established.

On the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending damping

In this paper, we are concerned with the global existence and blowup of smooth solutions to the multi-dimensional compressible Euler equations with time-depending dampingwhere (d = 2, 3), the frictional coefficient is with and , is a constant, , , , and is sufficiently small. One can totally divide the range of and into the following four cases:Case 1: , for d = 2, 3;Case 2: , for d = 2, 3;Case 3: , for d = 2;Case 4: , for d = 2, 3.We show that there exists a global smooth solution in Case 1, and Case 2 with , while in Case 3 and Case 4, for some classes of , the solution will blow up in finite time. Therefore, and appear to be the critical power and critical value, respectively, for the global existence of small amplitude smooth solution in d−dimensional compressible Euler equations with time-depending damping.

Weak continuity and compactness for nonlinear partial differential equations

We present several examples of fundamental problems involving weak continuity and compactness for nonlinear partial differential equations, in which compensated compactness and related ideas have played a significant role. We first focus on the compactness and convergence of vanishing viscosity solutions for nonlinear hyperbolic conservation laws, including the inviscid limit from the Navier-Stokes equations to the Euler equations for homentropy flow, the vanishing viscosity method to construct the global spherically symmetric solutions to the multidimensional compressible Euler equations, and the sonic-subsonic limit of solutions of the full Euler equations for multidimensional steady compressible fluids. We then analyze the weak continuity and rigidity of the Gauss-Codazzi-Ricci system and corresponding isometric embeddings in differential geometry. Further references are also provided for some recent developments on the weak continuity and compactness for nonlinear partial differential equations.

On the formation of shocks to the compressible Euler equations

We present an explicit upper-bound for the life-span of C3-smooth solutions to the multi-dimensional compressible Euler equations with a certain class of initial data containing compression vacuum states. We also show that the divergence of a fluid velocity will blow up along the particle trajectories issued from the compression vacuum states, which represent a shock formation at the vacuum states.

Existence and asymptotic behavior of C1 solutions to the multi-dimensional compressible Euler equations with damping

In this paper, the existence and asymptotic behavior of C 1 solutions to the multi-dimensional compressible Euler equations with damping on the framework of Besov space are considered. Comparing with the well-posedness results of Sideris–Thomases–Wang [T. Sideris, B. Thomases, D.H. Wang, Long time behavior of solutions to the three-dimensional compressible Euler with damping, Comm. Partial Differential Equations 28 (2003) 953–978], we weaken the regularity assumptions on the initial data. The global existence lies on a crucial a-priori estimate which is obtained by the spectral localization method. The main analytic tools are the Littlewood–Paley decomposition and Bony’s paraproduct formula.

Explicit and Implicit Multidimensional Compact High-Resolution Shock-Capturing Methods:Formulation

Two families of explicit and implicit compact high-resolution shock-capturing methods for the multidimensional compressible Euler equations for fluid dynamics are constructed. Some of these schemes can be fourth- and sixth-order accurate away from discontinuities. For the semi-discrete case their shock-capturing properties are of the total variation diminishing (TVD), total variation bounded (TVB), total variation diminishing in the mean (TVDM), essentially nonoscillatory (ENO), or positive type of scheme for 1D scalar hyperbolic conservation laws and are positive schemes in more than one dimension. These higher-order compact schemes require the same grid stencil per spatial direction as their second-order noncompact cousins. The added terms over the second-order noncompact cousins involve extra vector additions but no added flux evaluations. Due to the construction, these schemes can be viewed as approximations to genuinely multidimensional schemes in the sense that they might produce less distortion in spherical type shocks and are more accurate in vortex type flows than schemes based purely on 1D extensions. The extension of these families of compact schemes to coupled nonlinear systems can be accomplished using the Roe approximate Riemann solver, the generalized Steger and Warming flux-vector splitting, or the van Leer type flux-vector splitting. Modification to existing high-resolution second- or third-order noncompact shock-capturing computer codes is minimal. High-resolution shock-capturing properties can also be achieved via a variant of the second-order Lax–Friedrichs numerical flux without the use of Riemann solvers for coupled nonlinear systems with comparable operations count to their classical shock-capturing counterparts. An efficient and compatible high-resolution shock-capturing filter for spatially fourth- and sixth-order classical compact and noncompact schemes is discussed. The simplest extension to viscous flows can be achieved by using the standard fourth-order compact or noncompact formula for the viscous terms.