Entropy solutions of the compressible Euler equations

We consider the three-dimensional Euler equations of gas dynamics on a bounded periodic domain and a bounded time interval. We prove that Lax–Friedrichs scheme can be used to produce a sequence of solutions with ever finer resolution for any appropriately bounded (but not necessarily small) initial data. Furthermore, with some technical assumptions, e.g. that the density remains strictly positive in the sequence of solutions at hand, a subsequence converges to an entropy solution. We provide numerical evidence for these results by computing a sensitive Kelvin–Helmholtz problem.     

Blowup phenomena of the compressible Euler equations

The blowup phenomena of solutions of the compressible Euler equations is investigated. The approach is to construct the special solutions and use phase plane analysis. In particular, the special explicit solutions with velocity of the form c(t)x are constructed to show the blowup and expanding properties.     

Averaging stochastic systems under weakly dependent perturbations

An averaging procedure for stochastic systems with dependence on the whole past subject to an action described by a random process satisfying the strong mixing condition is investigated. For the probability of deviation beyond the level of normalized fluctuations of the solution of the initial stochastic equation relative to a solution of the average equation which turns out to be deterministic, exponential bounds of the type of the well-known S. N. Bernstein inequalities for the sum of independent random variables are constructed. These bounds can be used for construction of tolerance bands for a solution of the initial equation whose boundaries are determined by the deterministic solution of the averaged equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 593–600, May, 1990.     

Liouville type theorems for the Euler and the Navier–Stokes equations

We prove Liouville type theorems for weak solutions of the Navier–Stokes and the Euler equations. In particular, if the pressure satisfies p∈L¹(0,T;L¹(RN)) with , then the corresponding velocity should be trivial, namely v=0 on RN×(0,T). In particular, this is the case when p∈L¹(0,T;Hq(RN)), where Hq(RN), q∈(0,1], the Hardy space. On the other hand, we have equipartition of energy over each component, if p∈L¹(0,T;L¹(RN)) with . Similar results hold also for the magnetohydrodynamic equations.     

Large time behavior and pointwise estimates for compressible Euler equations with damping

The Cauchy problem of the compressible Euler equations with damping in multi-dimensions is considered when the initial perturbation in H3-norm is small. First, by using two new energy functionals together with the Green's function and iteration method, we improve the L2-decay rate in Tan and Wang(2013)and Tan and Wu(2012)when(ρ0-ˉρ,m)˙B-s1,∞×˙B-s+11,∞with s∈[0,2]is bounded.In particular,it holds that the density converges to its equilibrium state at the rate(1+t)-34-s2 in L2-norm and the momentum decays at the rate(1+t)-54-s2 in L2-norm.Moreover,under a weaker and more general condition on the initial data,we show that the density and the momentum have different pointwise estimates in dimension d with d 3on both space variable x and time variable t as|Dαx(ρ-ˉρ)|C(1+t)-d2-|α|2(1+|x|21+t)-rwith rd2and|Dαxm|C(1+t)-d2-|α|+12(1+|x|21+t)-d2 by a more elaborate analysis on the Green’s function.These results improve those in Wang and Yang(2001),where the density and the velocity(the momentum)have the same pointwise estimates.     

On a regularization of the compressible Euler equations for an isothermal gas

We consider a Leray-type regularization of the compressible Euler equations for an isothermal gas. The regularized system depends on a small parameter α>0. Using Riemann invariants, we prove the existence of smooth solutions for the regularized system for every α>0. The regularization mechanism is a non-linear bending of characteristics that prevents their finite-time crossing. We prove that, in the α→0 limit, the regularized solutions converge strongly. However, based on our analysis and numerical simulations, the limit is not the unique entropy solution of the Euler equations. The numerical method used to support this claim is derived from the Riemann invariants for the regularized system. This method is guaranteed to preserve the monotonicity of characteristics.     

Blowup phenomena of solutions to the Euler equations for compressible fluid flow

The blowup phenomena of solutions is investigated for the Euler equations of compressible fluid flow. The approach is to construct special explicit solutions with spherical symmetry to study certain blowup behavior of multi-dimensional solutions. In particular, the special solutions with velocity of the form c(t)x are constructed to show the expanding and blowup properties. The solution with velocity of the form for γ?1 and for any space dimensions is obtained as a corollary. Another conclusion is that there is only trivial solution with velocity of the form c(t)|x|^α-1x for α≠1 and multi-space dimensions.     

Vortical and self-similar flows of 2D compressible Euler equations

This paper presents the vortical and self-similar solutions for 2D compressible Euler equations using the separation method. These solutions complement Makino’s solutions in radial symmetry without rotation. The rotational solutions provide new information that furthers our understanding of ocean vortices and reference examples for numerical methods. In addition, the corresponding blowup, time-periodic or global existence conditions are classified through an analysis of the new Emden equation. A conjecture regarding rotational solutions in 3D is also made.     

Leonard Euler: Addition theorems and superintegrable systems

We consider the Euler approach to constructing to investigating of the superintegrable systems related to the addition theorems. As an example we reconstruct Drach systems and get some new two-dimensional superintegrable Stäckel systems.     

On the Riemann problem for 2D compressible Euler equations in three pieces

The Riemann problem for two-dimensional isentropic Euler equations is considered. The initial data are three constants in three fan domains forming different angles. Under the assumption that only a rarefaction wave, shock wave or contact discontinuity connects two neighboring constant initial states, it is proved that the cases involving three shock or rarefaction waves are impossible. For the cases involving one rarefaction (shock) wave and two shock (rarefaction) waves, only the combinations when the three elementary waves have the same sign are possible (impossible).     

Averaged systems of equations of the theory of elasticity in a medium with weakly compressible inclusions

Translated from Matematicheskie Zametki, Vol. 51, No. 1, pp. 126–133, January, 1992.     

On the local existence for the Euler equations with free boundary for compressible and incompressible fluids

We consider the free boundary compressible and incompressible Euler equations with surface tension. In both cases, we provide a priori estimates for the local existence with the initial velocity in $H^3$, with the $H^3$ condition on the density in the compressible case. An additional condition is required on the free boundary. Compared to the existing literature, both results lower the regularity of initial data for the Lagrangian Euler equation with surface tension.     

Global entropy solutions to exothermically reacting, compressible Euler equations

The global existence of entropy solutions is established for the compressible Euler equations for one-dimensional or plane-wave flow of an ideal gas, which undergoes a one-step exothermic chemical reaction under Arrhenius-type kinetics. We assume that the reaction rate is bounded away from zero and the total variation of the initial data is bounded by a parameter that grows arbitrarily large as the equation of state converges to that of an isothermal gas. The heat released by the reaction causes the spatial total variation of the solution to increase. However, the increase in total variation is proved to be bounded in t>0 as a result of the uniform and exponential decay of the reactant to zero as t approaches infinity.