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1.
The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations,
coupled to one another along the moving material interface through the continuity of the velocity and traction vectors. We
prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics
coupled to the incompressible Navier-Stokes equations. Unlike our approach in [5] for the case of linear elastodynamics, we
cannot employ a fixed-point argument on the nonlinear system itself, and are instead forced to regularize it by a particular parabolic artificial viscosity term. We proceed to show that with this specific regularization, we obtain a time interval
of existence which is independent of the artificial viscosity; together with a priori estimates, we identify the global solution
(in both phases), as well as the interface motion, as a weak limit in strong norms of our sequence of regularized problems. 相似文献
2.
Laure Saint-Raymond 《Archive for Rational Mechanics and Analysis》2000,155(3):171-199
Using the stability results of Bressan & Colombo [BC] for strictly hyperbolic 2 × 2 systems in one space dimension, we prove that the solutions of isentropic and non-isentropic
Euler equations in one space dimension with the respective initial data (ρ0, u
0) and (ρ0, u
0, &\theta;0=ρ0
γ− 1) remain close as soon as the total variation of (ρ0, u
0) is sufficiently small.
Accepted April 25, 2000?Published online November 24, 2000 相似文献
3.
4.
Gui-Qiang Chen Cleopatra Christoforou Yongqian Zhang 《Archive for Rational Mechanics and Analysis》2008,189(1):97-130
We establish the L
1-estimates for continuous dependence of entropy solutions to the full Euler equations away from the vacuum on two physical
parameters: the adiabatic exponent γ → 1 that passes from the non-isentropic to isothermal Euler equations and the Mach number
that passes from the compressible to incompressible Euler equations. Our analysis involves the effective approach developed
in our earlier work and additional new techniques that generalize this approach to the setting of the full Euler equations. 相似文献
5.
Global Behavior of Compressible Navier-Stokes Equations with a Degenerate Viscosity Coefficient 总被引:2,自引:0,他引:2
In this paper, we study a free boundary problem for compressible Navier-Stokes equations with density-dependent viscosity.
Precisely, the viscosity coefficient μ is proportional to ρ
θ
with
, where ρ is the density, and γ > 1 is the physical constant of polytropic gas. Under certain assumptions imposed on the initial data, we obtain the global
existence and uniqueness of the weak solution, give the uniform bounds (with respect to time) of the solution and show that
it converges to a stationary one as time tends to infinity. Moreover, we estimate the stabilization rate in L
∞ norm, (weighted) L
2 norm and weighted H
1 norm of the solution as time tends to infinity. 相似文献
6.
We study the vanishing viscosity limit of the compressible Navier–Stokes equations to the Riemann solution of the Euler equations
that consists of the superposition of a shock wave and a rarefaction wave. In particular, it is shown that there exists a
family of smooth solutions to the compressible Navier–Stokes equations that converges to the Riemann solution away from the
initial and shock layers at a rate in terms of the viscosity and the heat conductivity coefficients. This gives the first
mathematical justification of this limit for the Navier–Stokes equations to the Riemann solution that contains these two typical
nonlinear hyperbolic waves. 相似文献
7.
In this paper, we study the existence and uniqueness of a degenerate parabolic equation, with nonhomogeneous boundary conditions,
coming from the linearization of the Crocco equation [12]. The Crocco equation is a nonlinear degenerate parabolic equation
obtained from the Prandtl equations with the so-called Crocco transformation. The linearized Crocco equation plays a major
role in stabilization problems of fluid flows described by the Prandtl equations [5]. To study the infinitesimal generator
associated with the adjoint linearized Crocco equation – with homogeneous boundary conditions – we first study degenerate
parabolic equations in which the x-variable plays the role of a time variable. This equation is doubly degenerate: the coefficient in front of ∂x vanishes on a part of the boundary, and the coefficient of the elliptic operator vanishes in another part of the boundary.
This makes very delicate the proof of uniqueness of solution. To overcome this difficulty, a uniqueness result is first obtained
for an equation in which the elliptic operator is symmetric, and it is next extended to the original equation by combining
an iterative process and a fixed point argument (see Th. 4.9). This kind of argument is also used to prove estimates, which
cannot be obtained in a classical way. 相似文献
8.
Extending our previous results for artificial viscosity systems, we show, under suitable spectral hypotheses, that shock wave
solutions of compressible Navier–Stokes and magnetohydrodynamics equations undergo Hopf bifurcation to nearby time-periodic
solutions. The main new difficulty associated with physical viscosity and the corresponding absence of parabolic smoothing
is the need to show that the difference between nonlinear and linearized solution operators is quadratically small in H
s
for data in H
s
. We accomplish this by a novel energy estimate carried out in Lagrangian coordinates; interestingly, this estimate is false
in Eulerian coordinates. At the same time, we greatly sharpen and simplify the analysis of the previous work.
Research of B.T. was partially supported under NSF grant number DMS-0505780.
Research of K.Z. was partially supported under NSF grant number DMS-0300487. 相似文献
9.
We study a moving boundary problem modeling the growth of in vitro tumors. This problem consists of two elliptic equations
describing the distribution of the nutrient and the internal pressure, respectively, and a first-order partial differential
equation describing the evolution of the moving boundary. An important feature is that the effect of surface tension on the
moving boundary is taken into account. We show that this problem is locally well-posed for a large class of initial data by
using analytic semi-group theory. We also prove that if the surface tension coefficient γ is larger than a threshold value
γ
* then the unique flat equilibrium is asymptotically stable, whereas in the case γ < γ
* this flat equilibrium is unstable. 相似文献
10.
We study the Euler equations for slightly compressible fluids, that is, after rescaling, the limits of the Euler equations
of fluid dynamics as the Mach number tends to zero. In this paper, we consider the general non-isentropic equations and general
data. We first prove the existence of classical solutions for a time independent of the small parameter. Then, on the whole
space ℝ
d
, we prove that the solution converges to the solution of the incompressible Euler equations.
Accepted December 1, 2000?Published online April 23, 2001 相似文献
11.
We consider the 3-D evolutionary Navier–Stokes equations with a Navier slip-type boundary condition, see (1.2), and study
the problem of the strong convergence of the solutions, as the viscosity goes to zero, to the solution of the Euler equations
under the zero-flux boundary condition. We prove here, in the flat boundary case, convergence in Sobolev spaces W
k, p
(Ω), for arbitrarily large k and p (for previous results see Xiao and Xin in Comm Pure Appl Math 60:1027–1055, 2007 and Beir?o da Veiga and Crispo in J Math
Fluid Mech, 2009, doi:). However this problem is still open for non-flat, arbitrarily smooth, boundaries. The main obstacle consists in some boundary
integrals, which vanish on flat portions of the boundary. However, if we drop the convective terms (Stokes problem), the inviscid,
strong limit result holds, as shown below. The cause of this different behavior is quite subtle. As a by-product, we set up
a very elementary approach to the regularity theory, in L
p
-spaces, for solutions to the Navier–Stokes equations under slip type boundary conditions. 相似文献
12.
. This paper is concerned with the initial‐boundary‐value problem for a nonlinear hyperbolic system of conservation laws.
We study the boundary layers that may arise in approximations of entropy discontinuous solutions. We consider both the vanishing‐viscosity
method and finite‐difference schemes (Lax‐Friedrichs‐type schemes and the Godunov scheme). We demonstrate that different regularization methods generate different boundary layers. Hence, the boundary condition can be formulated only if an approximation scheme is selected first. Assuming
solely uniform bounds on the approximate solutions and so dealing with solutions, we derive several entropy inequalities satisfied by the boundary layer in each case under consideration. A Young
measure is introduced to describe the boundary trace. When a uniform bound on the total variation is available, the boundary
Young measure reduces to a Dirac mass.
From the above analysis, we deduce several formulations for the boundary condition which apply whether the boundary is characteristic
or not. Each formulation is based on a set of admissible boundary values, following the terminology of Dubois & LeFloch[15]. The local structure of these sets and the well‐posedness of the corresponding initial‐boundary‐value problem are investigated.
The results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics.
(Accepted July 2, 1998) 相似文献
13.
Charles M. Elliott Reiner Schätzle Barbara E. E. Stoth 《Archive for Rational Mechanics and Analysis》1998,145(2):99-127
In a Type‐II superconductor the magnetic field penetrates the superconducting body through the formation of vortices. In
an extreme Type‐II superconductor these vortices reduce to line singularities. Because the number of vortices is large it
seems feasible to model their evolution by an averaged problem, known as the mean-field model of superconductivity. We assume
that the evolution law of an individual vortex, which underlies the averaging process, involves the current of the generated
magnetic field as well as the curvature vector. In the present paper we study a two‐dimensional reduction, assuming all vortices
to be perpendicular to a given direction. Since both the magnetic field H and the averaged vorticity ω are curl‐free, we may represent them via a scalar magnetic potential q and a scalar stream function ψ, respectively. We study existence, uniqueness and asymptotic behaviour of solutions (ψ, q) of the resulting degenerate elliptic‐parabolic system (with curvature taken into account or not) by means of viscosity and
weak solutions. In addition we relate (ψ, q) to solutions (ω, H) of the mean‐field equations without curvature. Finally we construct special solutions of the corresponding stationary equations
with two or more superconducting phases.
(Accepted August 8, 1997) 相似文献
14.
Jeffrey Humpherys Gregory Lyng Kevin Zumbrun 《Archive for Rational Mechanics and Analysis》2009,194(3):1029-1079
Extending recent results in the isentropic case, we use a combination of asymptotic ODE estimates and numerical Evans-function
computations to examine the spectral stability of shock-wave solutions of the compressible Navier–Stokes equations with ideal
gas equation of state. Our main results are that, in appropriately rescaled coordinates, the Evans function associated with
the linearized operator about the wave (i) converges in the large-amplitude limit to the Evans function for a limiting shock
profile of the same equations, for which internal energy vanishes at one end state; and (ii) has no unstable (positive real
part) zeros outside a uniform ball |λ| ≦ Λ. Thus, the rescaled eigenvalue ODE for the set of all shock waves, augmented with
the (nonphysical) limiting case, form a compact family of boundary-value problems that can be conveniently investigated numerically.
An extensive numerical Evans-function study yields one-dimensional spectral stability, independent of amplitude, for gas constant
γ in [1.2, 3] and ratio ν/μ of heat conduction to viscosity coefficient within [0.2, 5] (γ ≈ 1.4, ν/μ ≈ 1.47 for air). Other values may be treated similarly but were not considered. The method of analysis extends also to the
multi-dimensional case, a direction that we shall pursue in a future work. 相似文献
15.
Franck Sueur 《Journal of Mathematical Fluid Mechanics》2014,16(1):163-178
In this paper we investigate the issue of the inviscid limit for the compressible Navier–Stokes system in an impermeable fixed bounded domain. We consider two kinds of boundary conditions. The first one is the no-slip condition. In this case we extend the famous conditional result (Kato, T.: Remarks on zero viscosity limit for nonstationary Navier–Stokes flows with boundary. In: Seminar on nonlinear partial differential equations, vol. 2, pp. 85–98. Math. Sci. Res. Inst. Publ., Berkeley 1984) obtained by Kato in the homogeneous incompressible case. Kato proved that if the energy dissipation rate of the viscous flow in a boundary layer of width proportional to the viscosity vanishes then the solutions of the incompressible Navier–Stokes equations converge to some solutions of the incompressible Euler equations in the energy space. We provide here a natural extension of this result to the compressible case. The other case is the Navier condition which encodes that the fluid slips with some friction on the boundary. In this case we show that the convergence to the Euler equations holds true in the energy space, as least when the friction is not too large. In both cases we use in a crucial way some relative energy estimates proved recently by Feireisl et al. in J. Math. Fluid Mech. 14:717–730 (2012). 相似文献
16.
In this paper, we study the well-posedness problem on transonic shocks for steady ideal compressible flows through a two-dimensional
slowly varying nozzle with an appropriately given pressure at the exit of the nozzle. This is motivated by the following transonic
phenomena in a de Laval nozzle. Given an appropriately large receiver pressure P
r
, if the upstream flow remains supersonic behind the throat of the nozzle, then at a certain place in the diverging part of
the nozzle, a shock front intervenes and the flow is compressed and slowed down to subsonic speed, and the position and the
strength of the shock front are automatically adjusted so that the end pressure at exit becomes P
r
, as clearly stated by Courant and Friedrichs [Supersonic flow and shock waves, Interscience Publishers, New York, 1948 (see
section 143 and 147)]. The transonic shock front is a free boundary dividing two regions of C
2,α flow in the nozzle. The full Euler system is hyperbolic upstream where the flow is supersonic, and coupled hyperbolic-elliptic
in the downstream region Ω+ of the nozzle where the flow is subsonic. Based on Bernoulli’s law, we can reformulate the problem by decomposing the 3 ×
3 Euler system into a weakly coupled second order elliptic equation for the density ρ with mixed boundary conditions, a 2 × 2 first order system on u
2 with a value given at a point, and an algebraic equation on (ρ, u
1, u
2) along a streamline. In terms of this reformulation, we can show the uniqueness of such a transonic shock solution if it
exists and the shock front goes through a fixed point. Furthermore, we prove that there is no such transonic shock solution
for a class of nozzles with some large pressure given at the exit.
This research was supported in part by the Zheng Ge Ru Foundation when Yin Huicheng was visiting The Institute of Mathematical
Sciences, The Chinese University of Hong Kong. Xin is supported in part by Hong Kong RGC Earmarked Research Grants CUHK-4028/04P,
CUHK-4040/06P, and Central Allocation Grant CA05-06.SC01. Yin is supported in part by NNSF of China and Doctoral Program of
NEM of China. 相似文献
17.
We prove a blow-up criterion in terms of the upper bound of (ρ, ρ
−1, θ) for a strong solution to three dimensional compressible viscous heat-conductive flows. The main ingredient of the proof
is an a priori estimate for a quantity independently introduced in Haspot (Regularity of weak solutions of the compressible isentropic Navier–Stokes equation, arXiv:1001.1581, 2010) and Sun et al. (J Math Pure Appl 95:36–47, 2011), whose divergence can be viewed as the effective viscous flux. 相似文献
18.
We consider the question of stability for planar wave solutions that arise in multidimensional conservation laws with only
fourth-order regularization. Such equations arise, for example, in the study of thin films, for which planar waves correspond
to fluid coating a pre-wetted surface. An interesting feature of these equations is that both compressive, and undercompressive,
planar waves arise as solutions (compressive or undercompressive with respect to asymptotic behavior relative to the un-regularized
hyperbolic system), and numerical investigation by Bertozzi, Münch, and Shearer indicates that undercompressive waves can
be nonlinearly stable. Proceeding with pointwise estimates on the Green's function for the linear fourth-order convection–regularization
equation that arises upon linearization of the conservation law about the planar wave solution, we establish that under general
spectral conditions, such as appear to hold for shock fronts arising in our motivating thin films equations, compressive waves
are stable for all dimensions d≧2 and undercompressive waves are stable for dimensions d≧3. (In the special case d=1, compressive waves are stable under a very general spectral condition.) We also consider an alternative spectral criterion
(valid, for example, in the case of constant-coefficient regularization), for which we can establish nonlinear stability for
compressive waves in dimensions d≧3 and undercompressive waves in dimensions d≧5. The case of stability for undercompressive waves in the thin films equations for the critical dimensions d=1 and d=2 remains an interesting open problem. 相似文献
19.
《European Journal of Mechanics - B/Fluids》2005,24(1):1-17
This paper presents a general method for imposing boundary conditions in the context of hyperbolic systems of conservation laws. This method is particularly well suited for approximations in the framework of Finite Volume Methods in the sense that it computes directly the normal flux at the boundary. We generalize our approach to nonconservative hyperbolic systems and discuss both the characteristic and the noncharacteristic cases. We present several applications to models occurring in Computational Fluid Mechanics like the Euler equations for compressible inviscid fluids with real equation of state, shallow water equations, magnetohydrodynamics equations and two fluid models. 相似文献
20.
Zhi-Qiang Shao 《Journal of Elasticity》2010,98(1):25-64
In this paper, we investigate the asymptotic behavior of global classical solutions to the mixed initial-boundary value problem
with small BV data for linearly degenerate quasilinear hyperbolic systems with general nonlinear boundary conditions in the
half space {(t,x)|t≥0,x≥0}. Based on the existence result on the global classical solution, we prove that when t tends to the infinity, the solution approaches a combination of C
1 traveling wave solutions, provided that the C
1 norm of the initial and boundary data is bounded and the BV norm of the initial and boundary data is sufficiently small.
Applications to quasilinear hyperbolic systems arising in physics and mechanics, particularly to the system describing the
motion of the relativistic string in the Minkowski space-time R
1+n
, are also given. 相似文献