In this paper, for compressible Euler equations in multiple space dimensions, we prove the break-down of classical solutions with a large class of initial data by tracking the propagation of radially symmetric expanding wave including compression. The singularity formation is corresponding to the finite time shock formation. We also provide some new global sup-norm estimates on velocity and density functions for classical solutions. The results in this paper have no restriction on the size of solutions, hence are large data results.
We establish a variational inequality formulation that captures the transonic shock for a steady compressible potential flow. Its critical point satisfies the transonic equation; moreover the associated jump conditions across its free boundary match the usual Rankine-Hugoniot jump conditions for a shock. By means of example we validate our formulation, and establish the necessary and sufficient condition for the existence of a transonic shock. Numerical results are also discussed.
We prove the existence of a large class of global-in-time expanding solutions to vacuum free boundary compressible Euler flows without relying on the existence of an underlying finite-dimensional family of special affine solutions of the flow.
We prove nonlinear stability of compactly supported expanding star-solutions of the mass-critical gravitational Euler-Poisson system. These special solutions were discovered by Goldreich and Weber in 1980. The expanding rate of such solutions can be either self-similar or non-self-similar (linear), and we treat both types. An important outcome of our stability results is the existence of a new class of global-in-time radially symmetric solutions, which are not homogeneous and therefore not encompassed by the existing works. Using Lagrangian coordinates we reformulate the associated free-bounda…
In this paper we investigate the question of the existence of global weak solution for the compressible Navier Stokes equations provided that the initial momentum $ρ_0 u_0$ belongs to $\mbox{bmo}^{-1}(\mathbb{R}^N)$ with $N= 2,3$ and is radially symmetric. More precisely we deal with the so called viscous shallow water system when the viscosity coefficients verify $μ(ρ)=μρ$, $λ(ρ)=0$ with $μ>0$. We prove then a equivalent of the so called Koch-Tataru theorem for the compressible Navier-Stokes equations. In addition we assume that the initial density $ρ_0$ is only bounded in $L^\infty(\mathbb{R…
We concern with the global existence and large time behavior of compressible fluids (including the inviscid gases, viscid gases, and Boltzmann gases) in an infinitely expanding ball. Such a problem is one of the interesting models in studying the theory of global smooth solutions to multidimensional compressible gases with time dependent boundaries and vacuum states at infinite time. Due to the conservation of mass, the fluid in the expanding ball becomes rarefied and eventually tends to a vacuum state meanwhile there are no appearances of vacuum domains in any part of the expansive ball, whic…
The article is devoted to the asymptotic limit of the compressible Navier-Stokes system with a pressure obeying a hard--sphere equation of state on a domain expanding to the whole physical space $R^3$. Under the assumptions that acoustic waves generated in the case of ill-prepared data do not reach the boundary of the expanding domain in the given time interval and a certain relation between the Reynolds and Mach numbers and the radius of the expanding domain, we prove that the target system is the incompressible Euler system on $R^3$. We also provide an estimate of the rate of convergence exp…
This paper is a continuation of the works in \cite{Euler} and \cite{NS}, where the authors have established the global existence of smooth compressible flows in infinitely expanding balls for inviscid gases and viscid gases, respectively. In this paper, we are concerned with the global existence and large time behavior of compressible Boltzmann gases in an infinitely expanding ball. Such a problem is one of the interesting models in studying the theory of global smooth solutions to multidimensional compressible gases with time dependent boundaries and vacuum states at infinite time. Due to the…
Expanding solutions to the compressible Navier-Stokes equations with degenerate viscosities in spherical symmetry: global existence and inviscid limit
The Institute of Mathematical Sciences, The Chinese University of Hong Kong（香港中文大学数学科学研究所）；School of Mathematical Sciences, South China Normal University（华南师范大学数学科学学院）
It is shown that the one-dimensional or two-dimensional radially symmetric isothermal compressible Navier-Stokes system has no non-trivial global smooth solutions if the initial density is compactly supported. This result is a generalization of Xin's work \cite{Xin98} to the isothermal case.
In this article we derive C^1-a priori estimates on the Riemann invariants of the Euler compressible equations in the case of cylindrical or spherical symmetry. These estimates allow then to construct shock waves with a time of existence proportional to the distance to the origin at the initial time.
In this paper, we investigate steady inviscid compressible flows with radial symmetry in an annulus. The major concerns are transonic flows with or without shocks. One of the main motivations is to elucidate the role played by the angular velocity in the structure of steady inviscid compressible flows. We give a complete classification of flow patterns in terms of boundary conditions at the inner and outer circle. Due to the nonzero angular velocity, many new flow patterns will appear. There exists accelerating or decelerating smooth transonic flows in an annulus satisfying one side boundary c…
In a recent work Sideris constructed a finite-parameter family of compactly supported affine solutions to the three-dimensional isentropic compressible Euler equations satisfying the physical vacuum condition. The support of these solutions expands at a linear rate in time. We show that if the adiabatic exponent $γ$ belongs to the interval $(1,\frac53]$, then these affine motions are nonlinearly stable. Small perturbations lead to globally-in-time defined solutions that remain in the vicinity of the manifold of affine motions, they remain smooth in the interior of their support, and no shocks …
We are concerned with spherically symmetric solutions of the Euler equations for multidimensional compressible fluids, which are motivated by many important physical situations. Various evidences indicate that spherically symmetric solutions of the compressible Euler equations may blow up near the origin at certain time under some circumstance. The central feature is the strengthening of waves as they move radially inward. A longstanding open, fundamental question is whether concentration could form at the origin. In this paper, we develop a method of vanishing viscosity and related estimate t…
One of the most influential fundamental tools in harmonic analysis is Riesz transform. It maps $L^p$ functions to $L^p$ functions for any $p\in (1,\infty)$ which plays an important role in singular operators. As an application in fluid dynamics, the norm equivalence between $\|\nabla u\|_{L^p}$ and $\|\mbox{div} u\|_{L^p}+\|\mbox{curl} u\|_{L^p}$ is well established for $p\in (1,\infty)$. However, since Riesz operators sent bounded functions only to BMO functions, there is no hope to bound $\|\nabla u\|_{L^\infty}$ in terms of $\|\mbox{div} u\|_{L^\infty}+\|\mbox{curl} u\|_{L^\infty}$. As poin…
In this paper, we obtain a result on the existence and uniqueness of global spherically symmetric classical solutions to the compressible isentropic Navier-Stokes equations with vacuum in a bounded domain or exterior domain Ω of Rn(n >= 2). Here, the initial data could be large. Besides, the regularities of the solutions are better than those obtained in [H.J. Choe and H. Kim, Math. Methods Appl. Sci., 28 (2005), pp. 1-28; Y. Cho and H. Kim, Manuscripta Math., 120 (2006), pp. 91-129; S.J. Ding, H.Y.Wen, and C.J. Zhu, J. Differential Equations, 251 (2011), pp. 1696-1725]. The analysis is based …
For the physical vacuum free boundary problem with the sound speed being $C^{{1}/{2}}$-H$\ddot{\rm o}$lder continuous near vacuum boundaries of the three-dimensional compressible Euler equations with damping, the global existence of spherically symmetric smooth solutions is proved, which are shown to converge to Barenblatt self-similar solutions of the porous media equation with the same total masses when initial data are small perturbations of Barenblatt solutions. The pointwise convergence with a rate of density, the convergence rate of velocity in supreme norm and the precise expanding rate…
On global classical and weak solutions with arbitrary large initial data to the multi-dimensional viscous Saint-Venant system and compressible Navier-Stokes equations subject to the BD entropy condition under spherical symmetry
For the physical vacuum free boundary problem with the sound speed being $C^{{1}/{2}}$-H$\ddot{\rm o}$lder continuous near vacuum boundaries of the one-dimensional compressible Euler equations with damping, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self-similar solution for the the porous media equation with the same total mass when the initial data is a small perturbation of the Barenblatt solution. The pointwise convergence with a rate of density, the convergence rate of velocity in supereme norm and the precise expanding rate of the …
Using recent developments in the theory of globally defined expanding compressible gases, we construct a class of global-in-time solutions to the compressible 3-D Euler-Poisson system without any symmetry assumptions in both the gravitational and the plasma case. Our allowed range of adiabatic indices includes, but is not limited to all $γ$ of the form $γ=1+\frac1n$, $n\in\mathbb N\setminus\{1\}$. The constructed solutions have initially small densities and a compact support. As $t\to\infty$ the density scatters to zero and the support grows at a linear rate in $t$.
The global existence of smooth solutions to the vacuum free boundary problem with physical singularity of compressible Euler equations with damping and gravity is proved in space dimensions $n=1, 2, 3$, for the initial data being small perturbations of the stationary solution. Moreover, the exponential decay of the velocity is obtained for $n=1, 2, 3$. The exponentially fast convergence of the density and vacuum boundary to those of the stationary solution is shown for $n=1$, and it is proved for $n=2, 3$ that they stay close to those of the stationary solution if they do so initially. The pro…
Nonlinearly Exponential Stability for Lions-Feireisl's Weak Solutions to the Barotropic Compressible Navier-Stokes Equations with Large Potential External Forces
We establish global existence of solutions to the compressible Euler equations, in the case that a finite volume of ideal gas expands into vacuum. Vacuum states can occur with either smooth or singular sound speed, the latter corresponding to the so-called physical vacuum singularity when the enthalpy vanishes on the vacuum wave front like the distance function. In this instance, the Euler equations lose hyperbolicity and form a degenerate system of conservation laws, for which a local existence theory has only recently been developed. Sideris found a class of expanding finite degree-of-freedo…
In this paper, the smooth solution of the physical vacuum problem for the one dimensional compressible Euler equations with time-dependent damping is considered. Near the vacuum boundary, the sound speed is $C^{1/2}$-Hölder continuous. The coefficient of the damping depends on time, given by this form $\fracμ{(1+t)^λ}$, $λ$, $μ>0$, which decays by order $-λ$ in time. Under the assumption that $0<λ<1$, $0<μ$ or $λ=1$, $2<μ$, we will prove the global existence of smooth solutions and convergence to the modified Barenblatt solution of the related porous media equation with time-dependent dissipat…
The uniqueness of entropy solution for the compressible Euler equations is a fundamental and challenging problem. In this paper, the uniqueness of a composite wave of shock and rarefaction of 1-d compressible Euler equations is proved in the inviscid limit of compressible Navier-Stokes equations. Moreover, the relative entropy around the original Riemann solution consisting of shock and rarefaction under the large perturbation is shown to be uniformly bounded by the framework developed in \cite{Kang-Vasseur-2021-Invent}. The proof contains two new ingredients: 1) a cut-off technique and the ex…
Radial similarity flow offers a rare instance where concrete inviscid, multi-dimensional, compressible flows can be studied in detail. In particular, there are flows of this type that exhibit imploding shocks and cavities. In such flows the primary flow variables (density, velocity, pressure, temperature) become unbounded at time of collapse. In both cases the solution can be propagated beyond collapse by having an expanding shock wave reflect off the center of motion. These types of flows are of relevance in bomb-making and inertial confinement fusion, and also as benchmarks for computational…
Based on Makino's solutions with radially symmetry, we extend the corresponding ones with elliptic symmetry for the compressible Euler and Navier-Stokes equations in R^{N} (N\geq2). By the separation method, we reduce the Euler and Navier-Stokes equations into 1+N differential functional equations. In detail, the velocity is constructed by the novel Emden dynamical system: {<K1.1/>| <K1.1 ilk="MATRIX" > a_{i}(t)=(ξ/(a_{i}(t)(Πa_{k}(t))^{γ-1})), for i=1,2,....,N a_{i}(0)=a_{i0}>0, a_{i}(0)=a_{i1} </K1.1> with arbitrary constants ξ, a_{i0} and a_{i1}. Some blowup phenomena or global existences o…
We study the construction of analytical non-radially solutions for the 1-dimensional compressible adiabatic Euler equations in this article. We could design the perturbational method to construct a new class of analytical solutions. In details, we perturb the linear velocity:% \begin{equation} u=c(t)x+b(t) \end{equation} and substitute it into the compressible Euler equations. By comparing the coefficients of the polynomial, we could deduce the corresponding functional differential system of $(c(t),b(t),ρ^{γ-1}(0,t)).$ Then by skillfully applying the Hubble's transformation: \begin{equation} c…
In this paper we investigate the problem of multiple expanding Newtonian stars that interact via their gravitational effect on each other. It is clear physically that if two stars at rest are separated initially, and start expanding as well as moving according to the laws of Newtonian gravity, they may eventually collide. Thus, one can ask whether each star can be given an initial position and velocity such that they can keep expanding without touching. We show that even with gravitational interaction between the bodies, a large class of initial positions and velocities give global-in-time sol…
For the three-dimensional vacuum free boundary problem with physical singularity that the sound speed is $C^{ {1}/{2}}$-H$\ddot{\rm o}$lder continuous across the vacuum boundary of the compressible Euler equations with damping, without any symmetry assumptions, we prove the almost global existence of smooth solutions when the initial data are small perturbations of the Barenblatt self-similar solutions to the corresponding porous media equations simplified via Darcy's law. It is proved that if the initial perturbation is of the size of $ε$, then the existing time for smooth solutions is at lea…
This paper considers the immediate blowup of entropy-bounded classical solutions to the vacuum free boundary problem of non-isentropic compressible Navier-Stokes equations. The viscosities and the heat conductivity could be constants, or more physically, the degenerate, temperature-dependent functions which vanish on the vacuum boundary (i.e., $μ=\barμ θ^α, ~ λ=\barλ θ^α,~ κ=\barκ θ^α$, for constants $0\leq α\leq 1/(γ-1)$, $\barμ>0,~2\barμ+n\barλ\geq0,~\barκ\geq 0$, and adiabatic exponent $γ>1$). With prescribed decaying rate of the initial density across the vacuum boundary, we prove that: (1…
Guderley's 1942 work on radial shock waves provides cases of self-similar Euler flows exhibiting blowup of primary (undifferentiated) flow variables: a converging shock wave invades a quiescent region, and the velocity and pressure in its immediate wake become unbounded at time of collapse. However, these solutions are of border-line physicality: the pressure vanishes within the quiescent region due to vanishing temperature there. It is reasonable that the lack of upstream counter-pressure is conducive to large speeds, with concomitant large amplitudes. Based on Guderley's original solutions i…
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 th…
Space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell WENO finite-volume limiting for simulation of non-stationary compressible multicomponent reactive flows
