538 research outputs found
Non-conservation of dimension in divergence-free solutions of passive and active scalar systems
For any , we give an explicit construction of a compactly
supported, uniformly continuous, and (weakly) divergence-free velocity field in
that weakly advects a measure whose support is initially the
origin but for positive times has Hausdorff dimension .
These velocities are uniformly continuous in space-time and compactly
supported, locally Lipschitz except at one point and satisfy the conditions for
the existence and uniqueness of a Regular Lagrangian Flow in the sense of Di
Perna and Lions theory.
We then construct active scalar systems in and
with measure-valued solutions whose initial support has co-dimension 2 but such
that at positive times it only has co-dimension 1. The associated velocities
are divergence free, compactly supported, continuous, and sufficiently regular
to admit unique Regular Lagrangian Flows.
This is in part motivated by the investigation of dimension conservation for
the support of measure-valued solutions to active scalar systems. This question
occurs in the study of vortex filaments in the three-dimensional Euler
equations.Comment: 32 pages, 3 figures. This preprint has not undergone peer review
(when applicable) or any post-submission improvements or corrections. The
Version of Record of this article is published in Arch Rational Mech Anal,
and is available online at https://doi.org/10.1007/s00205-021-01708-
The Ground State Energy of Heavy Atoms According to Brown and Ravenhall: Absence of Relativistic Effects in Leading Order
It is shown that the ground state energy of heavy atoms is, to leading order,
given by the non-relativistic Thomas-Fermi energy. The proof is based on the
relativistic Hamiltonian of Brown and Ravenhall which is derived from quantum
electrodynamics yielding energy levels correctly up to order Ry
Turning waves and breakdown for incompressible flows
We consider the evolution of an interface generated between two immiscible
incompressible and irrotational fluids. Specifically we study the Muskat and
water wave problems. We show that starting with a family of initial data given
by (\al,f_0(\al)), the interface reaches a regime in finite time in which is
no longer a graph. Therefore there exists a time where the solution of
the free boundary problem parameterized as (\al,f(\al,t)) blows-up: \|\da
f\|_{L^\infty}(t^*)=\infty. In particular, for the Muskat problem, this result
allows us to reach an unstable regime, for which the Rayleigh-Taylor condition
changes sign and the solution breaks down.Comment: 15 page
Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces
This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in RdRd , where d = 2, 3, with initial data B0∈Hs(Rd)B0∈Hs(Rd) and u0∈Hs−1+ϵ(Rd)u0∈Hs−1+ϵ(Rd) for s>d/2s>d/2 and any 0<ϵ<10<ϵ<1 . The proof relies on maximal regularity estimates for the Stokes equation. The obstruction to taking ϵ=0ϵ=0 is explained by the failure of solutions of the heat equation with initial data u0∈Hs−1u0∈Hs−1 to satisfy u∈L1(0,T;Hs+1)u∈L1(0,T;Hs+1) ; we provide an explicit example of this phenomenon
Atoms and Quantum Dots With a Large Number of Electrons: the Ground State Energy
We compute the ground state energy of atoms and quantum dots with a large
number N of electrons. Both systems are described by a non-relativistic
Hamiltonian of electrons in a d-dimensional space. The electrons interact via
the Coulomb potential. In the case of atoms (d=3), the electrons are attracted
by the nucleus, via the Coulomb potential. In the case of quantum dots (d=2),
the electrons are confined by an external potential, whose shape can be varied.
We show that the dominant terms of the ground state energy are those given by a
semiclassical Hartree-exchange energy, whose N to infinity limit corresponds to
Thomas-Fermi theory. This semiclassical Hartree-exchange theory creates
oscillations in the ground state energy as a function of N. These oscillations
reflect the dynamics of a classical particle moving in the presence of the
Thomas-Fermi potential. The dynamics is regular for atoms and some dots, but in
general in the case of dots, the motion contains a chaotic component. We
compute the correlation effects. They appear at the order N ln N for atoms, in
agreement with available data. For dots, they appear at the order N.Comment: 30 pages, 1 figur
Coercivity and stability results for an extended Navier-Stokes system
In this article we study a system of equations that is known to {\em extend}
Navier-Stokes dynamics in a well-posed manner to velocity fields that are not
necessarily divergence-free. Our aim is to contribute to an understanding of
the role of divergence and pressure in developing energy estimates capable of
controlling the nonlinear terms. We address questions of global existence and
stability in bounded domains with no-slip boundary conditions. Even in two
space dimensions, global existence is open in general, and remains so,
primarily due to the lack of a self-contained energy estimate. However,
through use of new coercivity estimates for the linear equations, we
establish a number of global existence and stability results, including results
for small divergence and a time-discrete scheme. We also prove global existence
in 2D for any initial data, provided sufficient divergence damping is included.Comment: 29 pages, no figure
Holographic formula for the determinant of the scattering operator in thermal AdS
A 'holographic formula' expressing the functional determinant of the
scattering operator in an asymptotically locally anti-de Sitter(ALAdS) space
has been proposed in terms of a relative functional determinant of the scalar
Laplacian in the bulk. It stems from considerations in AdS/CFT correspondence
of a quantum correction to the partition function in the bulk and the
corresponding subleading correction at large N on the boundary. In this paper
we probe this prediction for a class of quotients of hyperbolic space by a
discrete subgroup of isometries. We restrict to the simplest situation of an
abelian group where the quotient geometry describes thermal AdS and also the
non-spinning BTZ instanton. The bulk computation is explicitly done using the
method of images and the answer can be encoded in a (Patterson-)Selberg
zeta-function.Comment: 11 pages, published JPA versio
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