363 research outputs found
Global weak solutions to the density-dependent Hall-magnetohydrodynamics system
We are concerned with the global existence of finite energy weak solutions to
3D density-dependent magnetohydrodynamics (MHD) system with Hall-effect set in
a general smooth bounded domain. The perfectly conducting wall boundary
condition is imposed on the magnetic field. Due to the degeneracy of
Hall-effect term (a tri-linear term) in vacuum, we assumed initial density lies
in the bounded function space and having a positive lower bound. Particularly,
these bounds are preserved by the density transport equation which helps yield
a satisfying compactness argument of the magnetic field
Mathematics for 2d Interfaces
We present here a survey of recent results concerning the mathematical
analysis of instabilities of the interface between two incompressible, non
viscous, fluids of constant density and vorticity concentrated on the
interface. This configuration includes the so-called Kelvin-Helmholtz (the two
densities are equal), Rayleigh-Taylor (two different, nonzero, densities) and
the water waves (one of the densities is zero) problems. After a brief review
of results concerning strong and weak solutions of the Euler equation, we
derive interface equations (such as the Birkhoff-Rott equation) that describe
the motion of the interface. A linear analysis allows us to exhibit the main
features of these equations (such as ellipticity properties); the consequences
for the full, non linear, equations are then described. In particular, the
solutions of the Kelvin-Helmholtz and Rayleigh-Taylor problems are necessarily
analytic if they are above a certain threshold of regularity (a consequence is
the illposedness of the initial value problem in a non analytic framework). We
also say a few words on the phenomena that may occur below this regularity
threshold. Finally, special attention is given to the water waves problem,
which is much more stable than the Kelvin-Helmholtz and Rayleigh-Taylor
configurations. Most of the results presented here are in 2d (the interface has
dimension one), but we give a brief description of similarities and differences
in the 3d case.Comment: Survey. To appear in Panorama et Synth\`ese
Mathematical Aspects of Hydrodynamics
The workshop dealt with the partial differential equations that describe fluid motion, namely the Euler equations and the Navier-Stokes equations. This included topics in both inviscid and viscous fluids in two and three dimensions. A number of the talks were connected with issues of turbulence. Some talks addressed aspects of fluid dynamics such as magnetohydrodynamics, quantum and high energy physics, liquid crystals and the particle limit governed by the Boltzmann equations
Mini-Workshop: Interface Problems in Computational Fluid Dynamics
Multiple difficulties are encountered when designing algorithms to simulate flows having free surfaces, embedded particles, or elastic containers. One difficulty common to all of these problems is that the associated interfaces are Lagrangian in character, while the fluid equations are naturally posed in the Eulerian frame. This workshop explores different approaches and algorithms developed to resolve these issues
On unsteady internal flows of incompressible fluids characterized by implicit constitutive equations in the bulk and on the boundary
Long-time and large-data existence of weak solutions for initial- and
boundary-value problems concerning three-dimensional flows of
\emph{incompressible} fluids is nowadays available not only for Navier--Stokes
fluids but also for various fluid models where the relation between the Cauchy
stress tensor and the symmetric part of the velocity gradient is
\emph{nonlinear}. The majority of such studies however concerns models where
such a dependence is \emph{explicit} (the stress is a function of the velocity
gradient), which makes the class of studied models unduly restrictive. The same
concerns boundary conditions, or more precisely the slipping mechanisms on the
boundary, where the no-slip is still the most preferred condition considered in
the literature. Our main objective is to develop a robust mathematical theory
for unsteady internal flows of \emph{implicitly constituted} incompressible
fluids with implicit relations between the tangential projections of the
velocity and the normal traction on the boundary. The theory covers numerous
rheological models used in chemistry, biorheology, polymer and food industry as
well as in geomechanics. It also includes, as special cases, nonlinear slip as
well as stick-slip boundary conditions. Unlike earlier studies, the conditions
characterizing admissible classes of constitutive equations are expressed by
means of tools of elementary calculus. In addition, a fully constructive proof
(approximation scheme) is incorporated. Finally, we focus on the question of
uniqueness of such weak solutions
Cascades and Dissipative Anomalies in Nearly Collisionless Plasma Turbulence
We develop first-principles theory of kinetic plasma turbulence governed by
the Vlasov-Maxwell-Landau equations in the limit of vanishing collision rates.
Following an exact renormalization-group approach pioneered by Onsager, we
demonstrate the existence of a "collisionless range" of scales (lengths and
velocities) in 1-particle phase space where the ideal Vlasov-Maxwell equations
are satisfied in a "coarse-grained sense". Entropy conservation may
nevertheless be violated in that range by a "dissipative anomaly" due to
nonlinear entropy cascade. We derive "4/5th-law" type expressions for the
entropy flux, which allow us to characterize the singularities
(structure-function scaling exponents) required for its non-vanishing.
Conservation laws of mass, momentum and energy are not afflicted with anomalous
transfers in the collisionless limit. In a subsequent limit of small gyroradii,
however, anomalous contributions to inertial-range energy balance may appear
due both to cascade of bulk energy and to turbulent redistribution of internal
energy in phase space. In that same limit the "generalized Ohm's law" derived
from the particle momentum balances reduces to an "ideal Ohm's law", but only
in a coarse-grained sense that does not imply magnetic flux-freezing and that
permits magnetic reconnection at all inertial-range scales. We compare our
results with prior theory based on the gyrokinetic (high gyro-frequency) limit,
with numerical simulations, and with spacecraft measurements of the solar wind
and terrestrial magnetosphere.Comment: Several additions have been made that were requested by the referees
of the PRX submission. In particular, discussion previously relegated to
Supplemental Materials are now included in the main text as appendice
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