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).     

A two-dimensional inverse problem for the viscoelasticity equation

For the integrodifferential equation that corresponds to the two-dimensional viscoelasticity problem, we study the problem of determining the density, the elasticity coefficient, and the spaceintegral term in the equation. We assume that the sought functions differ from the given constants only inside the unit disk D = {x ∈ ?² | |x| < 1}. As information for solving this inverse problem, we consider the one-parameter family of solutions to the integrodifferential equation corresponding to impulse sources localized on straight lines and, on the boundary of D, there are defined the traces of the solutions for some finite time interval. It is shown that the use of a comparatively small part of the given information about the kinematics and the elements of dynamics of the propagating waves makes it possible to reduce the problem under consideration to three consecutively and uniquely solvable inverse problems that together give a solution to the initial inverse problem.     

Well-posedness of transonic characteristic discontinuities in two-dimensional steady compressible Euler flows

In our previous work, we have established the existence of transonic characteristic discontinuities separating supersonic flows from a static gas in two-dimensional steady compressible Euler flows under a perturbation with small total variation of the incoming supersonic flow over a solid right wedge. It is a free boundary problem in Eulerian coordinates and, across the free boundary (characteristic discontinuity), the Euler equations are of elliptic–hyperbolic composite-mixed type. In this paper, we further prove that such a transonic characteristic discontinuity solution is unique and L ¹–stable with respect to the small perturbation of the incoming supersonic flow in Lagrangian coordinates.     

Exterior Problem for the Three-dimensional Euler Equations

A priori estimates for the exterior initial boundary value problems of the Euler equations are considered. The existence and uniqueness of a local solution is proved.     

Stability of contact discontinuities to 1-D piston problem for the compressible Euler equations

We consider 1-D piston problem for the compressible Euler equations when the piston is static relatively to the gas in the tube. By a modified wave front tracking method, we prove that a contact discontinuity is structurally stable under the assumptions that the total variation of the initial data and the perturbation of the piston velocity are both sufficiently small. Meanwhile, we study the asymptotic behavior of the solutions by the generalized characteristic method and approximate conservation law theory as $t\to +\infty$.     

Global existence of smooth solutions to two-dimensional compressible isentropic Euler equations for Chaplygin gases

In this paper we investigate the two-dimensional compressible isentropic Euler equations for Chaplygin gases. Under the assumption that the initial data is close to a constant state and the vorticity of the initial velocity vanishes, we prove the global existence of the smooth solution to the Cauchy problem for twodimensional flow of Chaplygin gases.     

The compressible Gortler problem in two-dimensional boundary layers

In this paper the authors investigate the growth rates of Görtlervortices in a compressible flow in the inviscid limit of largeGörtler number. Numerical solutions are obtained for O(1)wavenumbers. The further limits of (i) large Mach number and(ii) large wavenumber with O(1) Mach number are considered.It is shown that two different types of disturbance mode canappear in this problem. The first is a wall layer mode, so namedas it has its eigenfunctions trapped in a thin layer near thewall. The other mode investigated is confined to a thin layeraway from the wall and termed a trapped-layer mode for largewavenumbers and an adjustment-layer mode for large Mach numbers,since then this mode has its eigenfunctions concentrated inthe temperature adjustment layer. It is possible to investigatethe near crossing of the modes which occurs in each of the limitsmentioned. The inviscid limit does not predict a fastest growingmode, but does enable a most dangerous mode to be identifiedfor O(1) Mach number. For hypersonic flow the most dangerousmode depends on the size of the Görtler number.     

On the statistical solutions of the two-dimensional,periodic Euler equation

We investigate the problem to define statistical solutions for the two-dimensional, periodic Euler Equation and prove a theorem of global existence and uniqueness for a regularized version. Moreover, we deduce the existence of the Euler flow in a weak form.     

Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle

In this paper, we study the stability of supersonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle of varying cross-sections. We formulate the problem as an initial–boundary value problem with the contact discontinuity as a free boundary. To deal with the free boundary value problem, we employ the Lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem. We develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain. Finally, by using the inverse Lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state, we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity.     

On an integral equation for the two-dimensional exterior Stokes problem

This paper is concerned with solving the two-dimensional exterior Stokes problem by a combination of double- and simple-layer potentials. This leads to a boundary integral equation, which is shown to be uniquely solvable. The present approach is a variation of a technique due to Leis, Brakhage and Werner, and Panich for treating the scalar Helmholtz equation. However, it contains extra parameter which may be chosen in an optimal manner from a computational point of view. Some numerical results in this connection are also included.     

The incompressible limit of solutions of the two-dimensional compressible Euler system with degenerating initial data

Using Strichartz estimates, it is possible to pass to the limit in the weakly compressible 2-D Euler system, when the Mach number ε tends to zero, even if the initial data are not uniformly smooth. This leads to results of convergence to solutions of the incompressible Euler system whose regularity is critical, such as vortex patches or Yudovich solutions. To cite this article: A. Dutrifoy, T. Hmidi, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Existence of a time periodic solution for the compressible Euler equation with a time periodic outer force

We are concerned with a time periodic supersonic flow through a bounded interval. This motion is described by the compressible Euler equation with a time periodic outer force. Our goal in this paper is to prove the existence of a time periodic solution. Although this is a fundamental problem for other equations, it has not been received much attention for the system of conservation laws until now.When we prove the existence of the time periodic solution, we face with two problems. One is to prove that initial data and the corresponding solutions after one period are contained in the same bounded set. To overcome this, we employ the generalized invariant region, which depends on the space variables. This enable us to investigate the behavior of solutions in detail. Second is to construct a continuous map. We apply a fixed point theorem to the map from initial data to solutions after one period. Then, the map needs to be continuous. To construct this, we introduce the modified Lax–Friedrichs scheme, which has a recurrence formula consisting of discretized approximate solutions. The formula yields the desired map. Moreover, the invariant region grantees that it maps a compact convex set to itself. In virtue of the fixed point theorem, we can prove a existence of a fixed point, which represents a time periodic solution. Finally, we apply the compensated compactness framework to prove the convergence of our approximate solutions.     

Axisymmetric solutions of the two-dimensional Euler equations with a two-constant equation of state

A family of self-similar and global bounded weak solutions are constructed rigorously for all positive time to the two-dimensional isentropic Euler equations with the equation of state $p=A_1{\rho }^{{\gamma }_1}+A_2{\rho }^{{\gamma }_2}$ for initial pure radial velocity. The main difficulty is that the equations cannot be directly reduced to an autonomous system of ordinary differential equations. To deal with it, we first establish the global existence and the detailed structures of solutions to a new system under the axisymmetric and self-similar assumptions.     

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.     

Exterior dirichlet problem for the reduced wave equation: asymptotic analysis of low frequencies

The order of convergence for the low frequency asymptotics of the exterior Dirichlet problem for the Helmholtz equation with variable, possibly non-smooth coefficients is shown to be Ω², the square of the frequency, except of some singular cases. In these cases the asymptotics are characterized completely by some lower order terms in the spherical harmonics expansion of the solution to the static 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.     

A two-dimensional inverse problem for an integro-differential equation of electrodynamics

An integro-differential equation corresponding to a two-dimensional problem of electrodynamics with dispersion is considered. It is assumed that the electrodynamic properties of a nonconducting medium with a constant magnetic permeability and the external current are independent of the x ₃ coordinate. In this case, the third component of the electric field vector satisfies a second-order scalar integro-differential equation with a variable permittivity of the medium. For this equation, we study the problem of finding the spatial part of the kernel entering the integral term. This corresponds to finding the part of the permittivity that depends on the electromagnetic frequency. It is assumed that the permittivity support is contained in some compact domain Ω ? ?². To find this coefficient inside Ω, we use information on the solution of the corresponding direct problem on the boundary of Ω on a finite time interval. An estimate for the conditional stability of the solution of the inverse problem is established under the assumption that the time interval is sufficiently large.