Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line

We study a large time behavior of a solution to the initial boundary value problem for an isentropic and compressible viscous fluid in a one-dimensional half space. The unique existence and the asymptotic stability of a stationary solution are proved by S. Kawashima, S. Nishibata and P. Zhu for an outflow problem where the fluid blows out through the boundary. The main concern of the present paper is to investigate a convergence rate of a solution toward the stationary solution. For the supersonic flow at spatial infinity, we obtain an algebraic or an exponential decay rate. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in the spatial asymptotic point, the solution converges to the corresponding stationary solution with the same rate in time as time tends to infinity. An algebraic convergence rate is also obtained for the transonic flow. These results are proved by the weighted energy method.     

Effect of bulk viscosity in supersonic flow past spacecraft

In this paper, we consider the effect of bulk viscosity in various hydrodynamic problems. We numerically study this effect on the front structure of the one-dimensional stationary shock wave and on the flow past blunt body. We estimate the effect of the bulk viscosity coefficient (BVC) on the heat transfer and drag of a sphere in a supersonic flow, apparently for the first time, by the numerical solution of parabolized Navier–Stokes equations. The solution is obtained by an original fast convergent method of global iterations of the longitudinal pressure gradient. The directions of further investigations of bulk viscosity are suggested.     

Multilevel optimization of a supersonic aircraft

This paper investigates the application of a multilevel preconditioned algorithm for the sonic boom reduction of a supersonic business jet. The optimization algorithm relies on a gradient approach with an adjoint state evaluation. The multi-level preconditioner is designed from an analysis of the gradient regularity loss. The sonic boom reduction is achieved in an indirect way by minimizing what we call the sonic boom downwards emission, which is computed in the near field. Additional aerodynamic performances like lift and drag forces are also guaranteed by including their evaluation in the problem's cost functional. Applications to 3D geometries are presented.     

Stability of oblique shock front

The stability of the weak planar oblique shock front with respect to the perturbation of the wall is discussed. By the analysis of the formation and the global construction of shock and its asymptotic behaviour for stationary supersonic flow along a smooth rigid wall we obtain the stability of the solution containing a weak planar shock front. The stability can be used to single out a physically reasonable solution together with the entropy condition     

An inverse problem for supersonic flow past a curved wedge

This paper studies an inverse problem for supersonic potential flow past a curved wedge, in which we design a suitable curved wedge such that the shock produced by the curved wedge can be controlled to the given position. Under suitable conditions, by characteristic method, we prove the existence of the global classical solution to this inverse problem and develop an optimal decay rate on the given shock’s second order derivatives. We finally construct a specific wedge such that the shock generated by the wedge is a convex combined one.     

Supersonic flow onto a solid wedge

We consider the problem of two‐dimensional supersonic flow onto a solid wedge, or equivalently in a concave corner formed by two solid walls. For mild corners, there are two possible steady state solutions, one with a strong and one with a weak shock emanating from the corner. The weak shock is observed in supersonic flights. A longstanding natural conjecture is that the strong shock is unstable in some sense. We resolve this issue by showing that a sharp wedge will eventually produce weak shocks at the tip when accelerated to a supersonic speed. More precisely, we prove that for upstream state as initial data in the entire domain, the time‐dependent solution is self‐similar, with a weak shock at the tip of the wedge. We construct analytic solutions for self‐similar potential flow, both isothermal and isentropic with arbitrary γ ≥ 1. In the process of constructing the self‐similar solution, we develop a large number of theoretical tools for these elliptic regions. These tools allow us to establish large‐data results rather than a small perturbation. We show that the wave pattern persists as long as the weak shock is supersonic‐supersonic; when this is no longer true, numerics show a physical change of behavior. In addition, we obtain rather detailed information about the elliptic region, including analyticity as well as bounds for velocity components and shock tangents. © 2007 Wiley Periodicals, Inc.     

Stability of a supersonic flow past a wedge with adjoint weak neutrally stable shock wave

We study the classical problem of a supersonic stationary flow of a nonviscous nonheat-conducting gas in local thermodynamic equilibrium past an infinite plane wedge. Under the Lopatinski? condition on the shock wave (neutral stability), we prove the well-posedness of the linearized mixed problem (the main solution is a weak shock wave), obtain a representation of the classical solution, where, in this case (in contrast to the case of the uniform Lopatinski? condition—an absolutely stable shock wave), plane waves additionally appear in the representation. If the initial data have compact support, the solution reaches the given regime in infinite time.     

Long memory in a linear stochastic Volterra differential equation

In this paper we consider a linear stochastic Volterra equation which has a stationary solution. We show that when the kernel of the fundamental solution is regularly varying at infinity with a log-convex tail integral, then the autocovariance function of the stationary solution is also regularly varying at infinity and its exact pointwise rate of decay can be determined. Moreover, it can be shown that this stationary process has either long memory in the sense that the autocovariance function is not integrable over the reals or is subexponential. Under certain conditions upon the kernel, even arbitrarily slow decay rates of the autocovariance function can be achieved. Analogous results are obtained for the corresponding discrete equation.     

Global well-posedness and large time behavior to the two-phase compressible-incompressible flow with free boundary

In present paper, the free boundary value problem to the two-phase compressible-incompressible flow with surface tension is investigated. The global well-posedness and the exponential decay to the steady state are obtained if the initial data are perturbated around the stationary state.     

Nonstationary interaction of supersonic inviscid gas flows

We consider the flow of supersonic homogeneous gas past a supersonic spherical source. This problem provides a gas-dynamic model of the interaction of interstellar wind with solar wind, and is thus also of independent interest. It is solved using an explicit through divergence scheme of third-order approximation. The analysis focuses on formation and stability of the structure of discontinuity surfaces and convergence to the stationary solution. The results are compared qualitatively and quantitatively with solutions obtained by other methods. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996, pp. 125–128.     

An exponential decay estimate for the stationary axisymmetric perturbation of Poiseuille flow in a circular pipe

We prove a decay estimate for the steady state incompressible Navier-Stokes equations. The estimate describes the exponential decay, in the axial direction of a semi-infinite circular tube, for an energy-type functional in terms of the axisymmetric perturbation of Poiseuille flow, provided that the Reynolds number does not exceed a critical value, for which we exhibit a lower and an upper bound. Since the motion is considered axisymmetric we use a stream function formulation, and the results are similar to those obtained by Horgan [8], for a two-dimensional channel flow problem. For the Stokes problem our estimate for the rate of decay is a lower bound to the actual rate of decay which is obtained from an asymptotic solution to the Stokes equations. Finally we describe a numerical approach to computing bounds to the energy functionalE(0).     

Convergence rates to discrete shocks for nonconvex conservation laws

Summary. This paper is concerned with polynomial decay rates of perturbations to stationary discrete shocks for the Lax-Friedrichs scheme approximating non-convex scalar conservation laws. We assume that the discrete initial data tend to constant states as , respectively, and that the Riemann problem for the corresponding hyperbolic equation admits a stationary shock wave. If the summation of the initial perturbation over is small and decays with an algebraic rate as , then the perturbations to discrete shocks are shown to decay with the corresponding rate as . The proof is given by applying weighted energy estimates. A discrete weight function, which depends on the space-time variables for the decay rate and the state of the discrete shocks in order to treat the non-convexity, plays a crucial role. Received November 25, 1998 / Published online November 8, 2000     

扰动的Couette－Poiseuille流的特征值的计算

本文考虑的问题是二维粘性渠流。对0到2000之间的雷诺数,计算了平稳扰动的Couette-Poiseuille流的下游特征值,其特征方程类似于Orr-Sommerfeld方程。所用的方法是谱方法和初值方法(复合矩阵方法).就几种有趣的流量,给出了相应的特征值的计算结果。这些特征值确定了扰动的衰减率。     

Justification of the Courant-Friedrich's hypothesis in the case of a weak shock. Part I. Presentation of solution to linear problem

As is well known, two solutions of the problem of a supersonic stationary inviscid nonheatconducting gas flow onto a planar infinite wedge are theoretically possible: the solution with a strong shock (the flow speed behind the shock is subsonic) and the solution with a weak shock (the flow speed behind the shock is supersonic). Unlike the well-studied case of a strong shock that is generically unstable [A.M. Blokhin, D.L. Tkachev, L.O. Baldan, Study of the stability in the problem on flowing around a wedge. The case of strong wave, J. Math. Anal. Appl. 319 (2006) 248-277; A.M. Blokhin, D.L. Tkachev, Yu.Yu. Pashinin, Stability condition for strong shock waves in the problem of flow around an infinite plane wedge, Nonlinear Anal. Hybrid Syst. 2 (2008) 1-17], R. Courant and K.O. Friedrichs [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, New York, 1948] assumed that the solution with a weak shock is asymptotically stable by Lyapunov. Presentation of classical solution to the corresponding problem which is found in the present paper is the first step on the way to justification of Courant-Friedrichs hypothesis on linear level.     

The condition on the stability of stationary lines in a curvature flow in the whole plane

The long time behavior of a curve in the whole plane moving by a curvature flow is studied. Studying the Cauchy problem, we deal with moving curves represented by entire graphs on the x-axis. Here the initial curves are given by bounded functions on the x-axis. It is proved that the solution converges uniformly to the solution of the Cauchy problem of the heat equation with the same initial value. The difference is of order O(t−1/2) as time goes to infinity. The proof is based on the decay estimates for the derivatives of the solution. By virtue of the stability results for the heat equation, our result gives the sufficient and necessary condition on the stability of constant solutions that represent stationary lines of the curvature flow in the whole plane.     

Nodal Type Bound States for Nonlinear Schrödinger Equations with Decaying Potentials

In this paper, we are concerned with the existence of nodal type bound state for the following stationary nonlinear Schrödinger equation $$-Δu(x)+V(x)u(x)=|u|^{p-1}u, x∈ R^N, N ≥ 3,$$ where 1 < p < (N+2)/(N-2) and the potential V(x) is a positive radial function and may decay to zero at infinity. Under appropriate assumptions on the decay rate of V(x), Souplet and Zhang [1] proved the above equation has a positive bound state. In this paper, we construct a nodal solution with precisely two nodal domains and prove that the above equation has a nodal type bound state under the same conditions on V(x) as in [1].     

Shock boundary layer interactions in dense gases

The concept of triple deck theory is applied to study laminar interacting boundary layers of dense gases in external purely supersonic flow. An impinging shock is generated at distances which are large compared to the upper deck thickness. As predicted by weakly nonlinear theory such a discontinuity may disintegrate into a sonic shock and an associated wave fan depending on its amplitude and the magnitude of the so called fundamental derivative. Incoming and outgoing waves are computed analytically by means of the method of multiple scales taking into account that mutual interaction effects between them are restricted to the linear upper deck region. The lower deck problem is solved numerically. The results show that it is possible to reduce the size of the separation bubble or even to avoid the occurrence of flow separation by choosing an optimal thermodynamic state. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Long-time asymptotics for a classical particle interacting with a scalar wave field

We consider the Harniltonian system consisting of scalar wave field and a single particle coupled in a translation invariant manner. The point particle is subject to a confining external potential. The stationary solutions of the system are a Coulomb type wave field centered at those particle positions for which the external force vanishes. We prove that solutions of finite energy converge, in suitable local energy seminorms, to the set of stationary solutions in the long time limit t f oo. The rate of relaxation to a stable stationary solution is determined by spatial decay of initial data. 'Supported partly by French-Russian A.M.Liapunov Center of Moscow State University, by research grants of RFBR (9601-00527) and of Volkswagen-Stiftung.     

Stability of stationary solutions for the non-isentropic Euler-Maxwell system in the whole space

In this paper we discuss the asymptotic stability of stationary solutions for the non-isentropic Euler-Maxwell system in R³. It is known in the authors’ previous works [17, 18, 19] that the Euler-Maxwell system verifies the decay property of the regularity-loss type. In this paper we first prove the existence and uniqueness of a small stationary solution. Then we show that the non-stationary problemhas a global solution in a neighborhood of the stationary solution under smallness condition on the initial perturbation. Moreover, we show the asymptotic convergence of the solution toward the stationary solution as time tends to infinity. The crucial point of the proof is to derive a priori estimates by using the energy method.     

Properties of the solution of the adjoint problem describing the motion of viscous fluids in a tube

For the Navier-Stokes equations, we study a solution invariant with respect to a oneparameter group and modeling a nonstationary motion of two viscous fluids in a cylindrical tube; the fluid layer near the tube wall can be viewed as a lubricant. The motion is due to a nonstationary pressure drop. We obtain a priori estimates for the velocities in the layers. We find a stationary state of the system and show that it is the limit state as t → ∞ provided that the pressure gradient in one of the fluids stabilizes with time. We solve the inverse problem of finding the pressure gradients and the velocity field from a known flow rate.