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1.
Remarks on a paper by J. T. Beale,T. Kato,and A. Majda 总被引:1,自引:0,他引:1
Gustavo Ponce 《Communications in Mathematical Physics》1985,98(3):349-353
We prove that the maximum norm of the deformation tensor controls the possible breakdown of smooth solutions for the 3-dimensional Euler equations. More precisely, the loss of regularity in a local smooth solution of the Euler equations implies the growth without bound of the deformation tensor as the critical time approaches; equivalently, if the deformation tensor remains bounded the existence of a smooth solution is guaranteed. 相似文献
2.
In this paper, we deal with global weak solutions of a non-conservative viscous compressible two-phase model in one space
dimension. This work extends in some sense the previous work, [Bresch et al., Arch Rat Mech Anal 196:599–629, 2010], which provides the global existence of weak solutions in the multi-dimensional framework with 1 < γ± < 6 assuming non-zero surface tension. In our study, we strongly improve the results by taking advantage of the one space
dimension. More precisely, we obtain global existence of weak solutions without using capillarity terms and for pressure laws
with the same range of coefficients as the degenerate barotropic mono-fluid system, namely γ± > 1. Then we prove that any possible vacuum state has to vanish within finite time after which densities are always away
from vacuum. This allows to prove that at least one phase corresponding to the global weak solution is a locally in time and
space (in a sense to be defined) strong solution after the vacuum states vanish. Our paper may be understood as a non-straightforward
generalization to the two-phase flow system of a previous paper [Li et al., Commun Math Phys 281(2):401–444, 2008], which treated the usual compressible barotropic Navier-Stokes equations for mono-fluid with a degenerate viscosity. Various
important mathematical difficulties occur when we want to generalize those results to the two-phase flows system since the
corresponding model is non-conservative. Far from vacuum, it involves a strong coupling between a nonlinear algebraic system
and a degenerate PDE system under constraint linked to fractions. Moreover, fractional densities may vanish if densities or
fractions vanish: A difficulty is to find estimates on the densities from estimates on fractional densities using the algebraic
system. Original approximate systems have also to be introduced compared to the works on the degenerate barotropic mono-fluid
system. Note that even if our result concerns “only” the one-dimensional case, it points out possible global weak solutions
(for such a non-conservative system) candidates to approach for instance shock structures and to define an appropriate a priori
family of paths in the phase space (in numerical schemes) at the zero dissipation limit. 相似文献
3.
H. Amirhashchi H. Zainuddin Anirudh Pradhan 《International Journal of Theoretical Physics》2011,50(8):2531-2545
Exact solution of Einstein’s field equations is obtained for massive string cosmological model of Bianchi III space-time using
the technique given by Letelier (Phys. Rev. D 28:2414, 1983) in presence of perfect fluid and decaying vacuum energy density Λ. To get the deterministic solution of the field equations
the expansion θ in the model is considered as proportional to the eigen value s2 2\sigma^{2}_{~2} of the shear tensor sj i\sigma^{j}_{~i} and also the fluid obeys the barotropic equation of state. It is observed that the particle density and the tension density
of the string are comparable at the two ends and they fall off asymptotically at similar rate. But in early stage as well
as at the late time of the evolution of the universe we have two types of scenario (i) universe is dominated by massive strings
and (ii) universe is dominated by strings depending on the nature of the two constants L and ℓ. The value of cosmological constant Λ for the model is found to be small and positive which is supported by the results from
recent supernovae Ia observations. Some physical and geometric properties of the model are also discussed. 相似文献
4.
This paper establishes a blowup criterion for the three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic (MHD) flows. It is essentially shown that for the Cauchy problem and the initial-boundary-value one of the three-dimensional compressible MHD flows with initial density allowed to vanish, the strong or smooth solution exists globally if the density is bounded from above and the velocity satisfies Serrin’s condition. Therefore, if the Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanishing or vacuum appearing in the non-vacuum region or even milder singularities) to form before the density becomes unbounded. This criterion is analogous to the well-known Serrin’s blowup criterion for the three-dimensional incompressible Navier-Stokes equations, in particular, it is independent of the temperature and magnetic field and is just the same as that of the barotropic compressible Navier-Stokes equations. As a direct application, it is shown that the same result also holds for the strong or smooth solutions to the three-dimensional full compressible Navier-Stokes system describing the motion of a viscous, compressible, and heat conducting fluid. 相似文献
5.
In this paper, we generalize the Kirchhoff-Sobolev parametrix of Klainerman and Rodnianski (Hyperbolic Equ. 4(3):401–433,
2007) to systems of tensor wave equations with additional first-order terms. We also present a different derivation, which better
highlights that such representation formulas are supported entirely on past null cones. This generalization of (Hyperbolic
Equ. 4(3):401–433, 2007) is a key component for extending Klainerman and Rodnianski’s breakdown criterion result for Einstein-vacuum spacetimes in
(J. Amer. Math. Soc. 23(2):345–382, 2009) to Einstein-Maxwell and Einstein-Yang-Mills spacetimes. 相似文献
6.
We study the partial regularity of a 3D model of the incompressible Navier-Stokes equations which was recently introduced
by the authors in [11]. This model is derived for axisymmetric flows with swirl using a set of new variables. It preserves
almost all the properties of the full 3D Euler or Navier-Stokes equations except for the convection term which is neglected
in the model. If we add the convection term back to our model, we would recover the full Navier-Stokes equations. In [11],
we presented numerical evidence which seems to support that the 3D model develops finite time singularities while the corresponding
solution of the 3D Navier-Stokes equations remains smooth. This suggests that the convection term play an essential role in
stabilizing the nonlinear vortex stretching term. In this paper, we prove that for any suitable weak solution of the 3D model
in an open set in space-time, the one-dimensional Hausdorff measure of the associated singular set is zero. The partial regularity
result of this paper is an analogue of the Caffarelli-Kohn-Nirenberg theory for the 3D Navier-Stokes equations. 相似文献
7.
Following Brown (Phys Rev D79:104029, 2009), in this paper we give an overview of how to modify standard hyperbolic formulations of the 3+1 evolution equations of General
Relativity in such a way that all auxiliary quantities are true tensors, thus allowing for these formulations to be used with
curvilinear sets of coordinates such as spherical or cylindrical coordinates. After considering the general case for both
the Nagy–Ortiz–Reula and the Baumgarte–Shapiro–Shibata–Nakamura (BSSN) formulations, we specialize to the case of spherical
symmetry and also discuss the issue of regularity at the origin. Finally, we show some numerical examples of the modified
BSSN formulation at work in spherical symmetry. 相似文献
8.
Xicheng Zhang 《Communications in Mathematical Physics》2012,311(1):133-155
In this article we study the fractal Navier-Stokes equations by using the stochastic Lagrangian particle path approach in
Constantin and Iyer (Comm Pure Appl Math LXI:330–345, 2008). More precisely, a stochastic representation for the fractal Navier-Stokes equations is given in terms of stochastic differential
equations driven by Lévy processes. Based on this representation, a self-contained proof for the existence of a local unique
solution for the fractal Navier-Stokes equation with initial data in
\mathbb W1,p{{\mathbb W}^{1,p}} is provided, and in the case of two dimensions or large viscosity, the existence of global solutions is also obtained. In
order to obtain the global existence in any dimensions for large viscosity, the gradient estimates for Lévy processes with
time dependent and discontinuous drifts are proved. 相似文献
9.
10.
Beltrami states for compressible barotropic flows are deduced by minimizing the total kinetic energy while keeping the total helicity constant. A Hamiltonian basis for these Beltrami states is sketched. An interesting physical application of the compressible barotropic Beltrami state arises with the Kuzmin-Oseledets formulation of compressible Euler equations. Further, Ertel's invariant is shown to become degenerate in the compressible barotropic Beltrami state. 相似文献
11.
12.
13.
Gautam Iyer 《Communications in Mathematical Physics》2006,266(3):631-645
We prove existence and regularity of the stochastic flows used in the stochastic Lagrangian formulation of the incompressible Navier-Stokes equations (with periodic boundary conditions), and consequently obtain a C
k,α local existence result for the Navier-Stokes equations. Our estimates are independent of viscosity, allowing us to consider the inviscid limit. We show that as ν → 0, solutions of the stochastic Lagrangian formulation (with periodic boundary conditions) converge to solutions of the Euler equations at the rate of
. 相似文献
14.
A stochastic analysis of an elastostatics problem for a half-plane under random white noise excitations of the displacement
vector prescribed on the boundary is given. Solutions of the problem are inhomogeneous random fields governed by the Lamé
equation with random boundary conditions. This is used to model the displacements, strain tensor, vorticity, and the deformation
energy, and to give exact representations for their correlation tensors, as well as the corresponding Karhunen-Loève (K-L)
expansions. Numerical calculations illustrating the rate of convergence of the spectral and K-L expansions are also given.
An interesting behaviour of the strain correlation tensor for the increasing value of the elasticity constant is found theoretically
and confirmed by calculations. The paper presents the second part of our study, the first being published recently in Sabelfeld
and Shalimova (J. Stat. Phys. 132(6):1071–1095, 2008) where only the displacement correlation tensor was derived and analyzed. 相似文献
15.
Combustion phenomena are of high scientific and technological interest, in particular for energy generation and transportation
systems. Direct Numerical Simulations (DNS) have become an essential and well established research tool to investigate the
structure of turbulent flames, since they do not rely on any approximate turbulence models. In this work two complementary
DNS codes are employed to investigate different types of fuels and flame configurations. The code is π3 is a 3-dimensional DNS code using a low-Mach number approximation. Chemistry is described through a tabulation, using two
coordinates to enter a database constructed for example with 29 species and 141 reactions for methane combustion. It is used
here to investigate the growth of a turbulent premixed flame in a methane-air mixture (Case 1). The second code,Sider is an explicit three-dimensional DNS code solving the fully compressible reactive Navier-Stokes equations, where the chemical
processes are computed using a complete reaction scheme, taking into account accurate diffusion properties. It is used here
to compute a hydrogen/air turbulent diffusion flame (Case 2), considering 9 chemical species and 38 chemical reactions. 相似文献
16.
We study the global regularity, for all time and all initial data in H 1/2, of a recently introduced decimated version of the incompressible 3D Navier-Stokes (dNS) equations. The model is based on a projection of the dynamical evolution of Navier-Stokes (NS) equations into the subspace where helicity (the L 2-scalar product of velocity and vorticity) is sign-definite. The presence of a second (beside energy) sign-definite inviscid conserved quadratic quantity, which is equivalent to the H 1/2-Sobolev norm, allows us to demonstrate global existence and uniqueness, of space-periodic solutions, together with continuity with respect to the initial conditions, for this decimated 3D model. This is achieved thanks to the establishment of two new estimates, for this 3D model, which show that the H 1/2 and the time average of the square of the H 3/2 norms of the velocity field remain finite. Such two additional bounds are known, in the spirit of the work of H. Fujita and T. Kato (Arch. Ration. Mech. Anal. 16:269–315, 1964; Rend. Semin. Mat. Univ. Padova 32:243–260, 1962), to be sufficient for showing well-posedness for the 3D NS equations. Furthermore, they are directly linked to the helicity evolution for the dNS model, and therefore with a clear physical meaning and consequences. 相似文献
17.
We prove a Beale-Kato-Majda type criterion for the loss of regularity for solutions of the incompressible Euler equations in , for . Instead of double exponential estimates of Beale-Kato-Majda type, we obtain a single exponential bound on involving the length parameter introduced by Constantin in (SIAM Rev. 36(1):73–98, 1994). In particular, we derive lower bounds on the blowup rate of such solutions. 相似文献
18.
We study a multidimensional hyperbolic quasi-gasdynamic (HQGD) system of equations containing terms with a regularizing parameter τ > 0 and 2nd order space and time derivatives; the body force is taken into account. We transform it to a form close to the compressible Navier–Stokes system of equations. Then we derive the entropy balance equation and show that the entropy production is similar to the latter system plus a term of the order of O(τ2). We analyze an equation for the total entropy as well. We also show that the corresponding residuals in the HQGD equations with respect to the compressible Navier–Stokes ones are of the order of O(τ2) too. Finally we treat the simplified barotropic HQGD system of equations with the general state equation and the stationary potential body force and obtain the corresponding results for it. 相似文献
19.
A combined incompressible and vanishing capillarity limit in the barotropic compressible Navier–Stokes equations for smooth solutions is proved. The equations are considered on the two-dimensional torus with well prepared initial data. The momentum equation contains a rotational term originating from a Coriolis force, a general Korteweg-type tensor modeling capillary effects, and a density-dependent viscosity. The limiting model is the viscous quasi-geostrophic equation for the “rotated” velocity potential. The proof of the singular limit is based on the modulated energy method with a careful choice of the correction terms. 相似文献
20.
Dong Li 《Journal of statistical physics》2009,134(4):681-700
We study the limiting behavior for the solutions of a nonlinear recurrent relation which arises from the study of Navier-Stokes
equations (Li and Sinai in J. Eur. Math. Soc. 10(2):267–313, 2008). Some stability theorems are also shown concerning a related class of linear recurrent relations.
This material is based upon work supported by the National Science Foundation under agreement No. DMS-0111298. Any options,
findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect
the views of the National Science Foundation. 相似文献