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1.
The Navier-Stokes systems for compressible fluids with density-dependent viscosities are considered in the present paper.
These equations, in particular, include the ones which are rigorously derived recently as the Saint-Venant system for the
motion of shallow water, from the Navier-Stokes system for incompressible flows with a moving free surface [14]. These compressible
systems are degenerate when vacuum state appears. We study initial-boundary-value problems for such systems for both bounded
spatial domains or periodic domains. The dynamics of weak solutions and vacuum states are investigated rigorously.
First, it is proved that the entropy weak solutions for general large initial data satisfying finite initial entropy exist
globally in time. Next, for more regular initial data, there is a global entropy weak solution which is unique and regular
with well-defined velocity field for short time, and the interface of initial vacuum propagates along the particle path during
this time period. Then, it is shown that for any global entropy weak solution, any (possibly existing) vacuum state must vanish
within finite time. The velocity (even if regular enough and well-defined) blows up in finite time as the vacuum states vanish.
Furthermore, after the vanishing of vacuum states, the global entropy weak solution becomes a strong solution and tends to
the non-vacuum equilibrium state exponentially in time. 相似文献
2.
We prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth(strong)
solutions for the 3-dimensional (3D) barotropic compressible Navier-Stokes equations. More precisely, if a solution of the
3D barotropic compressible Navier-Stokes equations is initially regular and loses its regularity at some later time, then
the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches. Our result
is the same as Ponce’s criterion for 3-dimensional incompressible Euler equations (Ponce in Commun Math Phys 98:349–353, 1985). In addition, initial vacuum states are allowed in our cases. 相似文献
3.
We prove that weak solutions of the Navier–Stokes equations for compressible fluid flow in one space dimension do not exhibit
vacuum states, provided that no vacuum states are present initially. The solutions and external forces that we consider are
quite general: the essential requirements are that the mass and energy densities of the fluid be locally integrable at each
time, and that the L
2
loc-norm of the velocity gradient be locally integrable in time. Our analysis shows that, if a vacuum state were to occur, the
viscous force would impose an impulse of infinite magnitude on the adjacent fluid, thus violating the hypothesis that the
momentum remains locally finite.
Received: 20 March 2000 / Accepted: 16 July 2000 相似文献
4.
Federico Bassetti Lucia Ladelli Giuseppe Toscani 《Journal of statistical physics》2011,142(4):686-709
We introduce a class of Kac-like kinetic equations on the real line, with general random collisional rules which, in some
special cases, identify models for granular gases with a background heat bath (Carrillo et al. in Discrete Contin. Dyn. Syst.
24(1):59–81, 2009), and models for wealth redistribution in an agent-based market (Bisi et al. in Commun. Math. Sci. 7:901–916, 2009). Conditions on these collisional rules which guarantee both the existence and uniqueness of equilibrium profiles and their
main properties are found. The characterization of these stationary states is of independent interest, since we show that
they are stationary solutions of different evolution problems, both in the kinetic theory of rarefied gases (Cercignani et al.
in J. Stat. Phys. 105:337–352, 2001; Villani in J. Stat. Phys. 124:781–822, 2006) and in the econophysical context (Bisi et al. in Commun. Math. Sci. 7:901–916, 2009). 相似文献
5.
Gülçin Uluyazı 《General Relativity and Gravitation》2012,44(4):875-886
We investigate the relation between the brane-based and the bulk-based approaches for anisotropic case in brane-world models.
In the brane-based approach, the brane is chosen to be fixed on a coordinate system, whereas in the bulk-based approach it
is no longer static as it moves along the extra dimension. It was shown that these two approaches are equivalent for specific
models in Mukohyama et al. (Phys Rev D 62:024028, 2000), Bowcock et al. (Class Quant Gravit 17:4745–4764, 2000). In this paper, it is aimed to get general formalism of the equivalence obtained in Mukohyama et al. (Phys Rev D 62:024028,
2000). We found that calculations driven by a general anisotropic bulk-based metric yield a brane-based metric in Gaussian Normal
Coordinates by conserving spatial anisotropy. We also derive solutions for an anisotropic bulk-based model and get the corresponding
brane-based metric of the model. 相似文献
6.
The relationship between the diffusivity D
n
and the mobility μ
n
of chemically doped organic n-type semiconductors exhibiting a disordered band structure is presented. These semiconductors
have a Gaussian-type density of states. So, calculations have been performed to elucidate the dependence of D
n
/μ
n
on the various parameters of this Gaussian density of states. Y. Roichman and N. Tessler (Appl. Phys. Lett. 80:1948, 2002), and subsequently Peng et al. (Appl. Phys. A 86:225, 2007), conducted numerical simulations to study this diffusivity–mobility relationship in organic semiconductors. However, almost
all other previous studies of the diffusivity–mobility relationship for inorganic semiconductors are based on Fermi–Dirac
integrals. An analytical formulation has therefore been developed for the diffusivity/mobility relationship for organic semiconductors
based on Fermi–Dirac integrals. The D
n
/μ
n
relationship is general enough to be applicable to both non-degenerate and degenerate organic semiconductors. It may be an
important tool to study electrical transport in these semiconductors. 相似文献
7.
Xianwen Zhang 《Journal of statistical physics》2010,141(3):566-588
Global-in-time existence of weak solutions to the Cauchy problem of the three dimensional Vlasov-Poisson-BGK system is shown
for initial data belonging to the space L
p
(ℝ3×ℝ3) with p>9 and having finite second order velocity moments. This result solves partially the well-posed problem for the Vlasov-Poisson-BGK
system proposed by B. Perthame: “Higher moments for kinetic equations: the Vlasov-Poisson and Fokker-Planck cases,” Math.
Meth. Appl. Sci. 13:441–452, 1990. 相似文献
8.
In this paper we have given a generalization of the earlier work by Prigogine et al. (Gen. Relativ. Gravit. 19:1, 1989; Gen. Relativ. Gravit. 21(8):767–776, 1989) who have constructed a phenomenological model of entropy production via particle creation in the very early universe generated out of the vacuum rather than from a singularity, by including radiation also as the energy source and tried to develop an alternative cosmological model in which particle creation prevents the big bang. We developed Radiation dominated model of the universe which shows a general tendency that (i) it originates from instability of vacuum rather than from a singularity. (ii) Up to a characteristic time t c cosmological quantities like density, pressure, Hubble constant and expansion parameter vary rapidly with time. (iii) After the characteristic time these quantities settles down and the models are turned into de-Sitter type model with uniform matter, radiation, creation densities and Hubble’s constant H. The de-Sitter regime survives during a decay time t d then connects continuously to a usual adiabatic matter radiation RW universe. The interesting thing in the paper is that we have related the phenomenological radiation dominated model to macroscopic model of quantum particle creation in the early universe giving rise to the present observed value of cosmic background radiation. It is also found that the dust filled model tallies exactly with that of the Prigogine’s one, which justifies that our model is generalized Prigogine’s model. Although the model originates from instability of vacuum rather than from a singularity, still there is a couple of unavoidable singularities in the model. 相似文献
9.
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
10.
The “inflow problem” for a one-dimensional compressible barotropic flow on the half-line R
+= (0,+∞) is investigated. Not only classical waves but also the new wave, which is called the “boundary layer solution”, arise.
Large time behaviors of the solutions to be expected have been classified in terms of the boundary values by [A. Matsumura,
Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas, to
appear in Proceedings of IMS Conference on Differential Equations from Mechanics, Hong Kong, 1999]. In this paper we give
the rigorous proofs of the stability theorems on both the boundary layer solution and a superposition of the boundary layer
solution and the rarefaction wave.
Received: 25 April 2000 / Accepted: 20 April 2001 相似文献
11.
Inflationary models of the early universe provide a natural mechanism for the formation of large scale structure. This success
brings to forefront the question of naturalness: Does a sufficiently long slow roll inflation occur generically or does it
require a careful fine tuning of initial parameters? In recent years there has been considerable controversy on this issue
(Hollands and Wald in Gen Relativ Gravit, 34:2043, 2002; Kofman et al. in J High Energy Phys 10:057, 2002); (Gibbons and Turok in Phys Rev D 77:063516, 2008). In particular, for a quadratic potential, Kofman et al. (J High Energy Phys 10:057, 2002) have argued that the probability of inflation with at least 65 e-foldings is close to one, while Gibbons and Turok (Phys
Rev D 77:063516, 2008) have argued that this probability is suppressed by a factor of ~10−85. We first clarify that such dramatically different predictions can arise because the required measure on the space of solutions
is intrinsically ambiguous in general relativity. We then show that this ambiguity can be naturally resolved in loop quantum
cosmology (LQC) because the big bang is replaced by a big bounce and the bounce surface can be used to introduce the structure
necessary to specify a satisfactory measure. The second goal of the paper is to present a detailed analysis of the inflationary
dynamics of LQC using analytical and numerical methods. By combining this information with the measure on the space of solutions,
we address a sharper question than those investigated in Kofman et al. (J High Energy Phys 10:057, 2002), Gibbons and Turok (Phys Rev D 77:063516, 2008), Ashtekar and Sloan (Phys Lett B 694:108, 2010): What is the probability of a sufficiently long slow roll inflation which is compatible with the seven year WMAP data? We show that the probability is very close to 1. The material is so organized that cosmologists who may be more interested
in the inflationary dynamics in LQC than in the subtleties associated with measures can skip that material without loss of
continuity. 相似文献
12.
András Sütő 《Communications in Mathematical Physics》2011,305(3):657-710
Weak limits as the density tends to infinity of classical ground states of integrable pair potentials are shown to minimize
the mean-field energy functional. By studying the latter we derive global properties of high-density ground state configurations
in bounded domains and in infinite space. Our main result is a theorem stating that for interactions having a strictly positive
Fourier transform the distribution of particles tends to be uniform as the density increases, while high-density ground states
show some pattern if the Fourier transform is partially negative. The latter confirms the conclusion of earlier studies by
Vlasov (in J. Phys. (USSR) IX:25–40, 1945), Kirzhnits and Nepomnyashchii (in Sov. Phys. JETP 32:1191–1197, 1971), and Likos et al. (in J. Chem. Phys. 126:224502, 2007). Other results include the proof that there is no Bravais lattice among high-density ground states of interactions whose
Fourier transform has a negative part and the potential diverges or has a cusp at zero. We also show that in the ground state
configurations of the penetrable sphere model particles are superimposed on the sites of a close-packed lattice. 相似文献
13.
Fang et al. (J. Diff. Geom. 80(2):175–259, 2008, Sect. 4, Conj. 4.17) conjecture that a certain spectral string-theoretic invariant of Calabi-Yau threefolds (the BCOV torsion)
is a birational invariant. We prove a weak form of this conjecture. The proof combines the arithmetic Riemann-Roch theorem
in Arakelov geometry with some inputs from motivic integration theory. 相似文献
14.
We prove the global existence of weak solutions to the Cauchy problem for the compressible isentropic Navier–Stokes equations
in ℝ
n
(n= 2, 3) when the Cauchy data are spherically symmetric. The proof is based on the exploitation of the one-dimensional feature
of symmetric solutions and use of a new (multidimensional) property induced by the viscous flux. The present paper extends
Lions' existence theorem [15] to the case 1< γ <γ
n
for spherically symmetric initial data, where γ is the specific heat ratio in the pressure, γ
n
= 3/2 for n= 2 and γ
n
= 9/5 for n= 3.
Dedicated to Professor Rolf Leis on the occasion of his 70th birthday
Received: 17 January 2000 / Accepted: 3 July 2000 相似文献
15.
Anatoliy K. Prykarpatsky Nikolai N. Bogolubov 《International Journal of Theoretical Physics》2012,51(1):237-245
R. Feynman’s “heretical” approach (Dyson in Am. J. Phys. 58:209–211, 1990; Dyson in Phys. Today 42(2):32–38, 1989) to deriving the Lorentz force based Maxwell electromagnetic equations is revisited, the its complete legacy is argued both
by means of the geometric considerations and its deep relation with the vacuum field theory approach devised (Prykarpatsky
et al. in Int. J. Theor. Phys. 49:798–820, 2010; Prykarpatsky et al. in Preprint ICTP, 2008, ). Being completely classical, we reanalyze the Feynman’s derivation from the classical Lagrangian and Hamiltonian points
of view and construct its nontrivial relativistic generalization compatible with the vacuum field theory approach. 相似文献
16.
Yaacov Kopeliovich 《Letters in Mathematical Physics》2010,94(3):313-333
Let X be a general cyclic cover of
\mathbbCP1{\mathbb{CP}^{1}} ramified at m points, λ1... λ
m
. we define a class of non-positive divisors on X of degree g −1 supported in the pre images of the branch points on X, such that the Riemann theta function does not vanish on their image in J(X). We generalize the results of Bershadsky and Radul (Commun Math Phys 116:689–700, 1988), Nakayashiki (Publ Res Inst Math Sci 33(6):987–1015, 1997) and Enolskii and Grava (Lett Math Phys 76(2–3):187–214, 2006) and prove that up to a certain determinant of the non-standard periods of X, the value of the Riemann theta function at these divisors raised to a high enough power is a polynomial in the branch point
of the curve X. Our approach is based on a refinement of Accola’s results for 3 cyclic sheeted cover (Accola, in Trans Am Math Soc 283:423–449,
1984) and a generalization of Nakayashiki’s approach explained in Nakayashiki (Publ Res Inst Math Sci 33(6):987–1015, 1997) for general cyclic covers. 相似文献
17.
In a recent paper (Vigoureux et al. in Int. J. Theor. Phys. 47:928, 2007) it has been suggested that the velocity of light and the expansion of the universe are two aspects of one single concept
connecting space and time in the expanding universe. It has then be shown that solving Friedmann’s equations with that interpretation
(and keeping c=constant) can explain number of unnatural features of the standard cosmology (for example: the flatness problem, the problem
of the observed uniformity in term of temperature and density of the cosmological background radiation, the small-scale inhomogeneity
problem…) and leads to reconsider the Hubble diagram of distance moduli and redshifts as obtained from recent observations
of type Ia supernovae without having to need an accelerating universe. In the present work we examine the problem of the cosmological
constant. We show that our model can exactly generate Λ (equation of state P
φ
=−ρ
φ
c
2 with Λ∝
R
−2) contrarily to the standard model which cannot generate it exactly. We also show how it can solve the so-called cosmic coincidence
problem. 相似文献
18.
Andrea Ferrario Carlo Antonio Rossi Thomas Willwacher 《Letters in Mathematical Physics》2011,95(1):27-39
The aim of this short note is to present a proof of the existence of an A
∞-quasi-isomorphism between the A
∞-S(V
*)-ù(V){\wedge(V)} -bimodule K, introduced in Calaque et al. (Bimodules and branes in deformation quantization, 2009), and the Koszul complex K(V) of S(V
*), viewed as an A
∞-S(V
*)-ù(V){\wedge(V)} -bimodule, for V a finite-dimensional (complex or real) vector space. 相似文献
19.
Yuri P. Kalmykov Bernard P. J. Mulligan Serguey V. Titov William T. Coffey 《Journal of statistical physics》2010,141(3):589-606
The time evolution equation for the probability density function of spin orientations in the phase space representation of
the polar and azimuthal angles is derived for the nonaxially symmetric problem of a quantum paramagnet subjected to a uniform
magnetic field of arbitrary direction. This is accomplished by first rotating the coordinate system into one in which the polar axis is collinear with
the field vector, then writing the reduced density matrix equation in the new coordinate system as an explicit inverse Wigner-Stratonovich
transformation so that the phase space master equation may be derived just as in the axially symmetric case [Yu.P. Kalmykov
et al., J. Stat. Phys. 131:969, 2008]. The properties of this equation, resembling the corresponding Fokker-Planck equation, are investigated. In particular,
in the large spin limit, S→∞, the master equation becomes the classical Fokker-Planck equation describing the magnetization dynamics of a classical
paramagnet in an arbitrarily directed uniform external field. 相似文献
20.
Daniel Coutand Hans Lindblad Steve Shkoller 《Communications in Mathematical Physics》2010,296(2):559-587
We prove a priori estimates for the three-dimensional compressible Euler equations with moving physical vacuum boundary, with an equation of state given by p(ρ) = C
γ
ρ
γ for γ > 1. The vacuum condition necessitates the vanishing of the pressure, and hence density, on the dynamic boundary, which
creates a degenerate and characteristic hyperbolic free-boundary system to which standard methods of symmetrizable hyperbolic equations cannot be applied. 相似文献