This paper is concerned with the Riemann problem of one-dimensional Euler equations with a singular source.
The exact solution of this Riemann problem contains a stationary discontinuity induced by the singular source, which is different from all the simple waves in the Riemann solution of classical Euler equations.
We propose an eigenvalue-based monotonicity criterion to select the physical curve of this stationary discontinuity.
By including this stationary discontinuity as an elementary wave, the structure of Riemann solution becomes diverse, e.g. the number of waves is not fixed and interactions between two waves become possible.
Under the double CRP framework, we prove all possible structures of the Riemann solution.
The convergence of the Boltzmann equaiton to the compressible Euler equations when the Knudsen number tends to zero has been a long standing open problem in the kinetic theory. In the setting of Riemann solution that contains the generic superposition of shock, rarefaction wave and contact discontinuity to the Euler equations, we succeed in justifying this limit by introducing hyperbolic waves with different solution backgrounds to capture the extra masses carried by the hyperbolic approximation of the rarefaction wave and the diffusion approximation of contact discontinuity.
In this paper, we analyze various types of critical phenomena in one-dimensional gas flows described by Euler equations. We give a geometrical interpretation of thermodynamics with a special emphasis on phase transitions. We use ideas from the geometrical theory of PDEs, in particular, symmetries and differential constraints to find solutions to the Euler system. Solutions obtained are multivalued, have singularities of projection to the plane of independent variables. We analyze the propagation of the shock wave front along with phase transitions.
We propose a two-dimensional generalization of Constantin-Lax-Majda model [2]. Some results about singular solutions are given. This model might be the first step toward the singular solutions of the Euler equations. Along the same line (vorticity formulation), we present some further model equations. They possibly models various aspects of difficulties related with the singular solutions of the Euler and Navier-Stokes equations. We also make some discussions on the possible connection between turbulence and the singular solutions of the Navier-Stokes equations.
In this paper, the Riemann problem for the pressureless Euler equations with a discontinuous source term is considered. The delta shock wave solution is obtained by combining the generalized Rankine-Hugoniot conditions together with the method of characteristics for different situations, which reflects the impact of the source term on the delta shock front. Moreover, during the construction process of the Riemann solution, some interesting phenomena are also observed, such as the disappearance of the delta shock wave and the occurrence of the vacuum state, etc.
We study the Riemann problem for the multidimensional compressible isentropic Euler equations. Using the framework developed by Chiodaroli, De Lellis, Kreml and based on the techniques of De Lellis and Székelyhidi, we extend our previous results and prove that whenever the initial Riemann data give rise to a self-similar solution consisting of one admissible shock and one rarefaction wave and are not too far from lying on a simple shock wave, the problem admits also infinitely many admissible weak solutions.
In this paper, a system of one-dimensional gas dynamics equations is considered. This system is a particular case of Jacobi type systems and has a natural representation in terms of 2-forms on 0-jet space. We use this observation to find a new class of multivalued solutions for an arbitrary thermodynamic state model and discuss singularities of their projections to the space of independent variables for the case of an ideal gas. Caustics and discontinuity lines are found.
The behaviour of the solutions to the Riemann problem for the isentropic Euler equations when the pressure vanishes is analysed. It is shown that any solution composed of a 1-shock wave and a 2-rarefaction wave tends to a two-shock wave when the pressure gets smaller than a fixed value determined by the Riemann data; by contrast, any solution composed of a 1-rarefaction wave and a 2-shock wave tends to a two-rarefaction wave. The two situations are illustrated with numerical tests.
In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem), for which the smoothness of the interface breaks down in finite time into a splash singularity or a splat singularity.
We consider the complete Euler system describing the time evolution of an inviscid non-isothermal gas. We show that the rarefaction wave solutions of the 1D Riemann problem are stable, in particular unique, in the class of all bounded weak solutions to the associated multi-D problem. This may be seen as a counterpart of the non-uniqueness results of physically admissible solutions emanating from 1D shock waves constructed recently by the method of convex integration.
The question of well- and ill-posedness of entropy admissible solutions to the multi-dimensional systems of conservation laws has been studied recently in the case of isentropic Euler equations. In this context special initial data were considered, namely the 1D Riemann problem which is extended trivially to a second space dimension. It was shown that there exist infinitely many bounded entropy admissible weak solutions to such a 2D Riemann problem for isentropic Euler equations, if the initial data give rise to a 1D self-similar solution containing a shock. In this work we study such a 2D Rie…
We study an Eulerian droplet model which can be seen as the pressureless gas system with a source term, a subsystem of this model and the inviscid Burgers equation with source term. The condition for loss of regularity of a solution to Burgers equation with source term is established. The same condition applies to the Eulerian droplet model and its subsystem. The Riemann problem for the Eulerian droplet model is constructively solved by going through the solution of the Riemann problems for the inviscid Burgers equation with a source term and the subsystem, respectively. Under suitable general…
In this paper, we study the limits of Riemann solutions to the inhomogeneous Euler equations of one-dimensional compressible fluid flow as the adiabatic exponent $γ$ tends to one. Different from the homogeneous equations, the Riemann solutions of the inhomogeneous system are non self-similar. It is rigorously shown that, as $γ$ tends to one, any two-shock Riemann solution tends to a delta shock solution of the pressureless Euler system with a source term, and the intermediate density between the two shocks tends to a weighted $δ$-mesaure which forms the delta shock; while any two-rarefaction-w…
We solve the Riemann problems for isentropic compressible Euler equations of polytropic gases in the class of Radon measures, and the solutions admit the concentration of mass. It is found that, under the requirement of satisfying the over-compressing entropy condition: (i) there is a unique delta shock solution, corresponding to the case that has two strong classical Lax shocks; (ii) for the initial data that the classical Riemann solution contains a shock wave and a rarefaction wave, or two shocks with one being weak, there are infinitely many solutions, each consists of a delta shock and a …
Fluid dynamic limit to compressible Euler equations from compressible Navier-Stokes equations and Boltzmann equation has been an active topic with limited success so far. In this paper, we consider the case when the solution of the Euler equations is a Riemann solution consisting two rarefaction waves and a contact discontinuity and prove this limit for both Navier-Stokes equations and the Boltzmann equation when the viscosity, heat conductivity coefficients and the Knudsen number tend to zero respectively. In addition, the uniform convergence rates in terms of the above physical parameters ar…
Some classical and recent results on the Euler equations governing perfect (incompressible and inviscid) fluid motion are collected and reviewed, with some small novelties scattered throughout. The perspective and emphasis will be given through the lens of infinite-dimensional dynamical systems, and various open problems are listed and discussed.
We study the stability of Riemann solutions to pressureless Euler equations with Coulomb-like friction under the nonlinear approximation of flux functions with one parameter. The approximated system can be seen as the generalized Chaplygin pressure Aw-Rascle model with Coulomb-like friction, which is also equivalent to the nonsymmetric system of Keyfitz-Kranzer type with generalized Chaplygin pressure and Coulomb-like friction. Compared with the original system, The approximated system is strictly hyperbolic, which has one eigenvalue genuinely nonlinear and the other linearly degenerate. Hence…
We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: $\{ (x_1,x_2,x_3): 0<x_3<x_2<x_1 \}.$ In this domain, we prove local well-posedness for $C^α$ vorticities not necessarily vanishing on the boundary with any $0<α<1$, and establish finite-time singularity formation within the same class for smooth and compactly supported initial data. The solutions can be extended to all of $\mathbb{R}^3$ via a sequence of reflections, and therefore we obtain finite-time singularity formation for the 3D Euler equations in $\…
The Riemann solutions to Chaplygin Euler equations with a scaled pressure are considered. When the pressure vanishes, there are three cases. The Riemann solution containing two shock waves converges to the delta shock wave solution of the transport equations. During this process, both the strength and propagation speed of the delta shock are investigated in detail. We find that there is something different from that for polytropic or isothermal gas. The Riemann solution containing two rarefaction waves tends to the two contact discontinuity solution to the transport equations as the pressure g…
Homotopy probability theory is a version of probability theory in which the vector space of random variables is replaced with a chain complex. A natural example extends ordinary probability theory on a finite volume Riemannian manifold M. In this example, initial conditions for fluid flow on M are identified with collections of homotopy random variables and solutions to the Euler equation are identified with homotopies between collections of homotopy random variables. Several ideas about using homotopy probability theory to study fluid flow are introduced.
The question of (non-)uniqueness of one-dimensional self-similar solutions to the Riemann problem for hyperbolic systems of gas dynamics in sets of multi-dimensional admissible weak solutions was addressed in recent years in several papers culminating in [17] with the proof that the Riemann problem for the isentropic Euler system with a power law pressure is ill-posed if the one-dimensional self-similar solution contains a shock. Natural question then arises whether the same holds also for a more involved system of equations, the full Euler system. After the first step in this direction was ma…
We prove that bounded weak solutions of the compressible Euler equations will conserve thermodynamic entropy unless the solution fields have sufficiently low space-time Besov regularity. A quantity measuring kinetic energy cascade will also vanish for such Euler solutions, unless the same singularity conditions are satisfied. It is shown furthermore that strong limits of solutions of compressible Navier-Stokes equations that are bounded and exhibit anomalous dissipation are weak Euler solutions. These inviscid limit solutions have non-negative anomalous entropy production and kinetic energy di…
In this short note we partially extend the recent nonuniqueness results on admissible weak solutions to the Riemann problem for the 2D compressible isentropic Euler equations. We prove nonuniqueness of admissible weak solutions that start from the Riemann initial data allowing a contact discontinuity to emerge.
In this paper we discuss the existence of stationary incompressible fluids with splash singularities. Specifically, we show that there are stationary solutions to the Euler equations with two fluids whose interfaces are arbitrarily close to a splash, and that there are stationary water waves with splash singularities.
In this paper, vacuum and singularity formation are considered for compressible Euler equations with time-dependent damping. For $1<γ\leq 3$, by constructing some new control functions ingeniously, we obtain the lower bounds estimates on density for arbitrary classical solutions. Basing on these lower estimates, we succeed in proving the singular formation theorem for all $λ$, which was open in [1] for some cases.Moreover, the singularity formation of the compressible Euler equations when $γ=3$ is investigated, too.
We give an extremely short proof that the free-surface incompressible, irrotational Euler equations with regular initial condition can form a finite time singularity in 2D or 3D. Thus, we provide a simple view of the problem studied by Castro, Cordoba, Fefferman, Gancedo, Lopez-Fernadez, Gomez-Serrano and Coutand, Shkoller.
Singular vorticty solutions of the incompressible 3D-Euler equation are constructed which satisfy the BKM criterion (cf. [2]). The construction is done by inviscid limits of vorticity solutions of transformed incompressible Navier Stokes type equations with a damping potential term, where the latter equations admit a global regular solution for positive viscosity. The inviscid limit vorticity solution of the incompressible Euler vorticity equation becomes singular at a point of the boundary of a finite domain.
We consider the Riemann problem composed of two shocks for the 1D Euler system. We show that the Riemann solution with two shocks is stable and unique in the class of weak inviscid limits of solutions to the Navier-Stokes equations with initial data with bounded energy. This work extends to the case of two shocks a previous result of the authors in the case of a single shock. It is based on the method of weighted relative entropy with shifts known as $a$-contraction theory. A major difficulty due to the method is that very little control is available on the shifts. A modification of the constr…
This paper deals with the derivation and analysis of a compressible Euler-type equation with singular commutator, which is derived from a hyperbolic limit of the kinetic description to the Cucker-Smale model of interacting individuals
An example of a solution branch of the three dimensional Euler equation Cauchy problem is constructed which develops a singular velocity component and a singular vorticity component after finite time for some data which have Hoelder continuous first order spatial derivatives. Such a solution branch can be extended beyond a time section at some positive finite time where a two dimensional set of singularities is located.
We study the formation of singularities for the Euler-Alignment system with influence function $ψ=\frac{k_α}{|x|^α}$ in 1D. As in [20] the problem is reduced to the analysis of a nonlocal 1D equation. We show the existence of singularities in finite time for any $α$ in the range $0<α<2$ in both the real line and the periodic case.
In this paper, we develop an abstract framework to establish ill-posedness in the sense of Hadamard for some nonlocal PDEs displaying unbounded unstable spectra. We apply it to prove the ill-posedness for the hydrostatic Euler equations as well as for the kinetic incompressible Euler equations and the Vlasov-Dirac-Benney system.
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.
By introducing a new averaged quantity with a fast decay weight to perform Sideris's argument (Commun Math Phys, 1985) developed for the Euler Equations, we extend the formation of singularities of classical solution to the 3D Euler Equations established in Sideris (1985) and Makino et al. (Jpn J Appl Math, 1986) for the initial data with compactly supported disturbances to the spherically symmetric solution with general initial data in Sobolev space. Moreover, we also prove the formation of singularities of the spherically symmetric solutions to the 3D Euler-Poisson Equations, but remove the …
In this paper, the singularity formation of classical solutions for the compressible Euler equations with general pressure law is considered. The gradient blow-up of classical solutions is shown without any smallness assumption by the delicate analysis on the decoupled Riccati type equations. The proof also relies on a new estimate for the upper bound of density.
We investigate the self-similar singularity of a 1D model for the 3D axisymmetric Euler equations, which is motivated by a particular singularity formation scenario observed in numerical computation. We prove the existence of a discrete family of self-similar profiles for this model and analyze their far-field properties. The self-similar profiles we find agree with direct simulation of the model and seem to have some stability.
This paper is devoted to a rigorous derivation of the isentropic Euler-alignment system with singular communication weights $ϕ_α(x) = |x|^{-α}$ for some $α> 0$. We consider a kinetic BGK-alignment model consisting of a kinetic BGK-type equation with a singular Cucker-Smale alignment force. By taking into account a small relaxation parameter, which corresponds to the asymptotic regime of a strong effect from BGK operator, we quantitatively derive the isentropic Euler-alignment system with pressure $p(ρ) = ρ^γ$, $γ= 1 + \frac2d$ from that kinetic equation.
We consider the Euler alignment system with mildly singular interaction kernels. When the local repulsion term is of the fractional type, global in time existence of smooth solutions was proved in\cite{do2018global,shvydkoy2017eulerian1,shvydkoy2017eulerian2,shvydkoy2017eulerian3}. Here, we consider a class of less singular interaction kernels and establish the global regularity of solutions as long as the interaction kernels are not integrable. The proof relies on modulus of continuity estimates for a class of parabolic integro-differential equations with a drift and mildly singular kernels.
We define compressive and rarefactive waves and give the differential equations describing smooth wave steepening for the compressible Euler equations with a varying entropy profile and general pressure laws. Using these differential equations, we directly generalize P. Lax's singularity (shock) formation results for hyperbolic systems with two variables to the compressible Euler equations for a polytropic ideal gas. Our results are valid globally without restriction on the size of the variation of initial data.
We study the pressureless Euler equations with nonlocal alignment interactions, which arises as a macroscopic representation of complex biological systems modeling animal flocks. For such Euler-Alignment system with bounded interactions, a critical threshold phenomenon is proved by Tadmor-Tan in 2014, where global regularity depends on initial data. With strongly singular interactions, global regularity is obtained by Do-Kiselev-Ryzhik-Tan in 2018, for all initial data. We consider the remaining case when the interaction is weakly singular. We show a critical threshold, similar to the system w…
A class of singular 3D-velocity vector fields is constructed which satisfy the incompressible 3D-Euler equation. It is shown that such a solution scheme does not exist in dimension 2. The solutions constructed are bounded and smooth up to finite time where they become singular. Although the solution is smooth and bounded there seems to be no bound in L2 of the velocity field.
We show that weak solutions of degenerate Navier-Stokes equations converge to the strong solutions of the pressureless Euler system with linear drag term, Newtonian repulsion and quadratic confinement. The proof is based on the relative entropy method using the artificial velocity formulation for the one-dimensional Navier-Stokes system.
We construct a special $C^{1,α}$ blow up solution to the three dimensional system modeling electro-hydrodynamics, which is strongly coupled with incompressible Euler equation and Nernst-Planck-Poisson equation. Our construction lies on the framework established in Elgindi [11] and relies on a special solution to variant spherical Laplacian.
The work addresses a singular limit for a rotating compressible Euler system in the low Mach number and low Rossby number regime. Based on the concept of dissipative measure-valued solution, the quasi-geostrophic system is identified as the limit problem in case of ill-prepared initial data. The ill-prepared initial data will cause rapidly oscillating acoustic waves. Using dispersive estimates of Strichartz type, the effect of the acoustic waves in the asymptotic limit is eliminated.
A sufficient condition for a finite-time $ L_2 $ singularity of the 3d Euler Equation
通过对压力Hessian特征值的适当假设，得到了三维不可压Euler方程有限时间$L_2$奇异的充分条件.在此条件下，$\lim_{t \to T_*} \sup | \frac{D ø} {Dt} |_{L_2(\vO)} = \infty$，其中$~ \vO \subset \R3$随流体移动。特别地，$|ø|$、$|§_{ij}| , and $| ¶_{ij}| $ all become unbounded at one point $（x_1，T_1）$, $ T_1 $ being the first blow-up time in $ L_2$.
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.
It has been known since work of Lichtenstein [42] and Gunther [29] in the 1920's that the $3D$ incompressible Euler equation is locally well-posed in the class of velocity fields with Hölder continuous gradient and suitable decay at infinity. It is shown here that these local solutions can develop singularities in finite time, even for some of the simplest three-dimensional flows.
On compact Riemannian manifolds, we prove a decomposition theorem for arbitrarily bounded energy sequence of solutions of a singular elliptic equation.
This letter studies the Euler-alignment system with weakly singular influence functions by introducing a novel technique to bound the density. Instead of resorting to a nonlinear maximum principle used in [C. Tan, Nonlinearity, 33: 1907--1924, 2020] to bound the interaction term $ψ\astρ$ by $ρ^s$ with $s\in (0, 1)$ for $ψ$ with an algebraic singularity at origin, we bound $ψ*ρ$ by a relaxed constant for any $ψ\in L^1$. We thus establish the global-in-time existence results with weaker assumptions on $ψ$ and refined solution bounds, characterized by the structure of $ψ$.
