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1.
We prove an inequality on the Kantorovich-Rubinstein distance–which can be seen as a particular case of a Wasserstein metric–between
two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, but with a moderate angular singularity.
Our method is in the spirit of [7]. We deduce some well-posedness and stability results in the physically relevant cases of
hard and moderately soft potentials. In the case of hard potentials, we relax the regularity assumption of [6], but we need
stronger assumptions on the tail of the distribution (namely some exponential decay). We thus obtain the first uniqueness
result for measure initial data. In the case of moderately soft potentials, we prove existence and uniqueness assuming only
that the initial datum has finite energy and entropy (for very moderately soft potentials), plus sometimes an additionnal
moment condition. We thus improve significantly on all previous results, where weighted Sobolev spaces were involved. 相似文献
2.
We study the singularity formation of smooth solutions of the relativistic Euler equations in (3 + 1)-dimensional spacetime
for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any smooth solution,
with compactly supported non-trivial initial data, blows up in finite time. For the case of infinite initial energy, we first
prove the existence, uniqueness and stability of a smooth solution if the initial data is in the subluminal region away from
the vacuum. By further assuming the initial data is a smooth compactly supported perturbation around a non-vacuum constant
background, we prove the property of finite propagation speed of such a perturbation. The smooth solution is shown to blow
up in finite time provided that the radial component of the initial ``generalized' momentum is sufficiently large. 相似文献
3.
We study the massless scalar field on asymptotically flat spacetimes with closed timelike curves (CTC’s), in which all future-directed
CTC’s traverse one end of a handle (wormhole) and emerge from the other end at an earlier time. For a class of static geometries
of this type, and for smooth initial data with all derivatives in L
2
on ℐ
−
, we prove existence of smooth solutions which are regular at null and spatial infinity (have finite energy and finite L
2
-norm) and have the given initial data on ℐ
−
. A restricted uniqueness theorem is obtained, applying to solutions that fall off in time at any fixed spatial position.
For a complementary class of spacetimes in which CTC’s are confined to a compact region, we show that when solutions exist
they are unique in regions exterior to the CTC’s. (We believe that more stringent uniqueness theorems hold, and that the present
limitations are our own.) An extension of these results to Maxwell fields and massless spinor fields is sketched. Finally,
we discuss a conjecture whose meaning is essentially that the Cauchy problem for free fields is well defined in the presence
of CTC’s whenever the problem is well-posed in a geometric-optics limit. We provide some evidence in support of this conjecture,
and we present counterexamples that show that neither existence nor uniqueness is guaranteed under weaker conditions. In particular,
both existence and uniqueness can fail in smooth, asymptotically flat spacetimes with a compact nonchronal region.
Received: 28 November 1994/Accepted: 20 May 1996 相似文献
4.
Relation between Fresnel transform of input light field and the two-parameter Radon transform of Wigner function of the field
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This paper proves a new theorem on the relationship between optical
field Wigner function's two-parameter Radon transform and optical
Fresnel transform of the field, i.e., when an input field ψ( x') propagates through an optical [ D( -B) ( -C) A] system, the energy density of the output field is equal to the Radon transform
of the Wigner function of the input field, where the Radon transform
parameters are D,B. It prove this theorem in both spatial-domain
and frequency-domain, in the latter case the Radon transform
parameters are A,C. 相似文献
5.
Two methods are described to determine local parameters of the unsteady asymmetric plasma. The basis of one method is the parametric approach. This method is used to determine plasma parameters in the case when an assumption can be made on the shape of the intensity isolines. The temperature field was calculated according to the intensity distributions in X direction obtained simultaneously from two lines of sight. The second method was applied for measurements in arbitrary plasma configuration. In this case the temperature fields were calculated according to measurements from lines of sight simultaneously, and the obtained data were used in the Radon transformation inversion. 相似文献
6.
José A. Carrillo Young-Pil Choi Seung-Yeal Ha Moon-Jin Kang Yongduck Kim 《Journal of statistical physics》2014,156(2):395-415
We present synchronization and contractivity estimates for the kinetic Kuramoto model obtained from the Kuramoto phase model in the mean-field limit. For identical Kuramoto oscillators, we present an admissible class of initial data leading to time-asymptotic complete synchronization, that is, all measure valued solutions converge to the traveling Dirac measure concentrated on the initial averaged phase. In the case of non-identical oscillators, we show that the velocity field converges to the average natural frequency proving that the oscillators move asymptotically with the same frequency under suitable assumptions on the initial configuration. If two initial Radon measures have the same natural frequency density function and strength of coupling, we show that the Wasserstein \(p\) -distance between corresponding measure valued solutions is exponentially decreasing in time. This contraction principle is more general than previous \(L^1\) -contraction properties of the Kuramoto phase model. 相似文献
7.
Demetrios Christodoulou 《Communications in Mathematical Physics》1986,105(3):337-361
In this paper we begin the study of the global initial value problem for Einstein's equations in the spherically symmetric case with a massless scalar field as the material model. We reduce the problem to a single nonlinear evolution equation. Taking as initial hypersurface a future light cone with vertex at the center of symmetry, we prove, the local, in retarded time, existence and global uniqueness of classical solutions. We also prove that if the initial data is sufficiently small there exists a global classical solution which disperses in the infinite future.Research supported in part by National Science Foundation grants MCS-8201599 to the Courant Institute and PHY-8318350 to Syracuse University 相似文献
8.
The fractional cable equation is studied on a bounded space domain. One of the prescribed boundary conditions is of Dirichlet type, the other is of a general form, which includes the case of nonlocal boundary conditions. In real problems nonlocal boundary conditions are prescribed when the data on the boundary can not be measured directly. We apply spectral projection operators to convert the problem to a system of integral equations in any generalized eigenspace. In this way we prove uniqueness of the solution and give an algorithm for constructing the solution in the form of an expansion in terms of the generalized eigenfunctions and three-parameter Mittag-Leffler functions. Explicit representation of the solution is given for the case of double eigenvalues. We consider some examples and as a particular case we recover a recent result. The asymptotic behavior of the solution is also studied. 相似文献
9.
We study the Boltzmann equation for a space-homogeneous gas of inelastic hard spheres, with a diffusive term representing a random background forcing. Under the assumption that the initial datum is a nonnegative L2(N) function, with bounded mass and kinetic energy (second moment), we prove the existence of a solution to this model, which instantaneously becomes smooth and rapidly decaying. Under a weak additional assumption of bounded third moment, the solution is shown to be unique. We also establish the existence (but not uniqueness) of a stationary solution. In addition we show that the high-velocity tails of both the stationary and time-dependent particle distribution functions are overpopulated with respect to the Maxwellian distribution, as conjectured by previous authors, and we prove pointwise lower estimates for the solutions. 相似文献
10.
A class of extended vector fields, called extended divergence-measure fields, is analyzed. These fields include vector fields
in L
p
and vector-valued Radon measures, whose divergences are Radon measures. Such extended vector fields naturally arise in the
study of the behavior of entropy solutions of the Euler equations for gas dynamics and other nonlinear systems of conservation
laws. A new notion of normal traces over Lipschitz deformable surfaces is developed under which a generalized Gauss-Green
theorem is established even for these extended fields. An explicit formula is obtained to calculate the normal traces over
any Lipschitz deformable surface, suitable for applications, by using the neighborhood information of the fields near the
surface and the level set function of the Lipschitz deformation surfaces. As an application, we prove the uniqueness and stability
of Riemann solutions that may contain vacuum in the class of entropy solutions of the Euler equations for gas dynamics.
Received: 7 May 2002 / Accepted: 2 December 2002
Published online: 2 April 2003
Communicated by P. Constantin 相似文献
11.
We prove the uniqueness of Riemann solutions in the class of entropy solutions in with arbitrarily large oscillation for the 3 × 3 system of Euler equations in gas dynamics. The proof for solutions with
large oscillation is based on a detailed analysis of the global behavior of shock curves in the phase space and the singularity
of centered rarefaction waves near the center in the physical plane. The uniqueness of Riemann solutions yields their inviscid
large-time stability under arbitrarily large
perturbation of the Riemann initial data, as long as the corresponding solutions are in L
∞ and have local bounded total variation satisfying a natural condition on its growth with time. No specific reference to any
particular method for constructing the entropy solutions is needed.
The uniqueness result for Riemann solutions can easily be extended to entropy solutions U(x,t), piecewise Lipschitz in x, for any t > 0, with arbitrarily large oscillation.
Received: 23 April 2001 / Accepted: 20 September 2001 相似文献
12.
The equations for the three-dimensional incompressible flow of liquid crystals are considered in a smooth bounded domain. The existence and uniqueness of the global strong solution with small initial data are established. It is also proved that when the strong solution exists, all the global weak solutions constructed in [16] must be equal to the unique strong solution. 相似文献
13.
A. Heintz 《Journal of statistical physics》1998,90(3-4):663-695
The paper is concerned with the Enskog equation with a constant high density factor for large initial data in L
1(R
n). The initial boundary value problem is investigated for bounded domains with irregular boundaries. The proof of an H-theorem for the case of general domains and boundary conditions is given. The main result guarantees the existence of global solutions of boundary value problems for large initial data with all v-moments initially finite and domains having boundary with finite Hausdorff measure and satisfying a cone condition. Existence and uniqueness are first proved for the case of bounded velocities. The solution has finite norm
where q = (t
0, x) is taken on all possible n-dimensional planes Q(v) in R
n+l intersecting a fixed point and orthogonal to vectors (1, v), v R
n. 相似文献
14.
The Brownian loop measure is a conformally invariant measure on loops in the plane that arises when studying the Schramm–Loewner evolution (SLE). When an SLE curve in a domain evolves from an interior point, it is natural to consider the loops that hit the curve and leave the domain, but their measure is infinite. We show that there is a related normalized quantity that is finite and invariant under Möbius transformations of the plane. We estimate this quantity when the curve is small and the domain simply connected. We then use this estimate to prove a formula for the Radon–Nikodym derivative of reversed radial SLE with respect to whole-plane SLE. 相似文献
15.
Robert M. Strain 《Communications in Mathematical Physics》2010,300(2):529-597
In this paper it is shown that unique solutions to the relativistic Boltzmann equation exist for all time and decay with any
polynomial rate towards their steady state relativistic Maxwellian provided that the initial data starts out sufficiently
close in L¥l{L^\infty_\ell}. If the initial data are continuous then so is the corresponding solution. We work in the case of a spatially periodic box.
Conditions on the collision kernel are generic in the sense of Dudyński and Ekiel-Jeżewska (Commun Math Phys 115(4):607–629,
1985); this resolves the open question of global existence for the soft potentials. 相似文献
16.
Multiplicity of Phase Transitions and Mean-Field Criticality on Highly Non-Amenable Graphs 总被引:1,自引:0,他引:1
Roberto H. Schonmann 《Communications in Mathematical Physics》2001,219(2):271-322
We consider independent percolation, Ising and Potts models, and the contact process, on infinite, locally finite, connected
graphs.
It is shown that on graphs with edge-isoperimetric Cheeger constant sufficiently large, in terms of the degrees of the vertices
of the graph, each of the models exhibits more than one critical point, separating qualitatively distinct regimes. For unimodular
transitive graphs of this type, the critical behaviour in independent percolation, the Ising model and the contact process
are shown to be mean-field type.
For Potts models on unimodular transitive graphs, we prove the monotonicity in the temperature of the property that the free
Gibbs measure is extremal in the set of automorphism invariant Gibbs measures, and show that the corresponding critical temperature
is positive if and only if the threshold for uniqueness of the infinite cluster in independent bond percolation on the graph
is less than 1.
We establish conditions which imply the finite-island property for independent percolation at large densities, and use those
to show that for a large class of graphs the q-state Potts model has a low temperature regime in which the free Gibbs measure decomposes as the uniform mixture of the q ordered phases.
In the case of non-amenable transitive planar graphs with one end, we show that the q-state Potts model has a critical point separating a regime of high temperatures in which the free Gibbs measure is extremal
in the set of automorphism-invariant Gibbs measures from a regime of low temperatures in which the free Gibbs measure decomposes
as the uniform mixture of the q ordered phases.
Received: 27 March 2000 / Accepted: 7 December 2000 相似文献
17.
When solving the inverse scattering problem for a discrete Sturm–Liouville operator with a rapidly decreasing potential,
one gets reflection coefficients s
± and invertible operators , where is the Hankel operator related to the symbol s
±. The Marchenko–Faddeev theorem [8] (in the continuous case, for the discrete case see [4, 6]), guarantees the uniqueness
of the solution of the inverse scattering problem. In this article we ask the following natural question – can one find a
precise condition guaranteeing that the inverse scattering problem is uniquely solvable and that operators are invertible? Can one claim that uniqueness implies invertibility or vise versa?
Moreover, we are interested here not only in the case of decreasing potential but also in the case of asymptotically almost
periodic potentials. So we merge here two mostly developed cases of the inverse problem for Sturm–Liouville operators: the
inverse problem with (almost) periodic potential and the inverse problem with the fast decreasing potential.
Received: 7 September 2001 / Accepted: 3 December 2001 相似文献
18.
19.
We find an integral transform realizing the connection between the s-parameterized quasidistributions of a quantum state and its corresponding tomogram, which looks like a Radon transform. We
show that the kernel of the new Radon transform is a Gaussian function with s-parameterized dispersion. It can be considered as a broadened delta-function appeared in the standard Radon transform. 相似文献
20.
Fei Ran Tian 《Communications in Mathematical Physics》1994,166(1):79-115
We study the initial value problem for the Whitham averaged system which is important in determining the KdV zero dispersion limit. We use the hodograph method to show that, for a generic non-trivial monotone initial data, the Whitham averaged system has a solution within a region in thex-t plane for all time bigger than a large time. Furthermore, the Whitham solution matches the Burgers solution on the boundaries of the region. For hump-like initial data, the hodograph method is modified to solve the non-monotone (inx) solutions of the Whitham averaged system. In this way, we show that, for a hump-like initial data, the Whitham averaged system has a solution within a cusp for a short time after the increasing and decreasing parts of the initial data beging to interact. On the cusp, the Whitham and Burgers solutions are matched. 相似文献