783 research outputs found
Global Well-Posedness for the Microscopic FENE Model with a Sharp Boundary Condition
We prove global well-posedness for the microscopic FENE model under a sharp
boundary requirement. The well-posedness of the FENE model that consists of the
incompressible Navier-Stokes equation and the Fokker-Planck equation has been
studied intensively, mostly with the zero flux boundary condition. Recently it
was illustrated by C. Liu and H. Liu [2008, SIAM J. Appl. Math.,
68(5):1304--1315] that any preassigned boundary value of a weighted
distribution will become redundant once the non-dimensional parameter . In
this article, we show that for the well-posedness of the microscopic FENE model
() the least boundary requirement is that the distribution near boundary
needs to approach zero faster than the distance function. Under this condition,
it is shown that there exists a unique weak solution in a weighted Sobolev
space. Moreover, such a condition still ensures that the distribution is a
probability density. The sharpness of this boundary requirement is shown by a
construction of infinitely many solutions when the distribution approaches zero
as fast as the distance function.Comment: pages 20; added a proof that 'solution is still a probability density
under the sharp boundary requirement
An Invariant-region-preserving (IRP) Limiter to DG Methods for Compressible Euler Equations
We introduce an explicit invariant-region-preserving limiter applied to DG
methods for compressible Euler equations. The invariant region considered
consists of positivity of density and pressure and a maximum principle of a
specific entropy. The modified polynomial by the limiter preserves the cell
average, lies entirely within the invariant region and does not destroy the
high order of accuracy for smooth solutions. Numerical tests are presented to
illustrate the properties of the limiter. In particular, the tests on Riemann
problems show that the limiter helps to damp the oscillations near
discontinuities.Comment: 12 pages, 3 figures, 2 tables, XVI International Conference on
Hyperbolic Problems Theory, Numerics, Application
Invariant-region-preserving DG methods for multi-dimensional hyperbolic conservation law systems, with an application to compressible Euler equations
An invariant-region-preserving (IRP) limiter for multi-dimensional hyperbolic
conservation law systems is introduced, as long as the system admits a global
invariant region which is a convex set in the phase space. It is shown that the
order of approximation accuracy is not destroyed by the IRP limiter, provided
the cell average is away from the boundary of the convex set. Moreover, this
limiter is explicit, and easy for computer implementation. A generic algorithm
incorporating the IRP limiter is presented for high order finite volume type
schemes. For arbitrarily high order discontinuous Galerkin (DG) schemes to
hyperbolic conservation law systems, sufficient conditions are obtained for
cell averages to remain in the invariant region provided the projected
one-dimensional system shares the same invariant region as the full
multi-dimensional hyperbolic system {does}. The general results are then
applied to both one and two dimensional compressible Euler equations so to
obtain high order IRP DG schemes. Numerical experiments are provided to
validate the proven properties of the IRP limiter and the performance of IRP DG
schemes for compressible Euler equations.Comment: 33 pages, 8 tables, 5 figures, accepted for publication in Journal of
Computational Physics, 201
Rigorous derivation of the hydrodynamical equations for rotating superfluids
Using a modified WKB approach, we present a rigorous semi-classical analysis
for solutions of nonlinear Schroedinger equations with rotational forcing. This
yields a rigorous justification for the hydrodynamical system of rotating
superfluids. In particular it is shown that global-in-time semi-classical
convergence holds whenever the limiting hydrodynamical system has global smooth
solutions and we also discuss the semi-classical dynamics of several physical
quantities describing rotating superfluids.Comment: to appear in Math. Mod. Methods Appl. Sc
Selection dynamics for deep neural networks
This paper presents a partial differential equation framework for deep
residual neural networks and for the associated learning problem. This is done
by carrying out the continuum limits of neural networks with respect to width
and depth. We study the wellposedness, the large time solution behavior, and
the characterization of the steady states of the forward problem. Several
useful time-uniform estimates and stability/instability conditions are
presented. We state and prove optimality conditions for the inverse deep
learning problem, using standard variational calculus, the
Hamilton-Jacobi-Bellmann equation and the Pontryagin maximum principle. This
serves to establish a mathematical foundation for investigating the algorithmic
and theoretical connections between neural networks, PDE theory, variational
analysis, optimal control, and deep learning.Comment: 27. arXiv admin note: text overlap with arXiv:1807.01083 by other
author
Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics
We investigate a class of nonlocal conservation laws with the nonlinear
advection coupling both local and nonlocal mechanism, which arises in several
applications such as the collective motion of cells and traffic flows. It is
proved that the solution regularity of this class of conservation laws
will persist at least for a short time. This persistency may continue as long
as the solution gradient remains bounded. Based on this result, we further
identify sub-thresholds for finite time shock formation in traffic flow models
with Arrhenius look-ahead dynamics
Global regularity, and wave breaking phenomena in a class of nonlocal dispersive equations
This paper is concerned with a class of nonlocal dispersive models -- the
-equation proposed by H. Liu [ On discreteness of the Hopf equation,
{\it Acta Math. Appl. Sin.} Engl. Ser. {\bf 24}(3)(2008)423--440]: including integrable equations such as the
Camassa-Holm equation, , and the Degasperis-Procesi equation,
, as special models. We investigate both global regularity of
solutions and wave breaking phenomena for . It is shown
that as increases regularity of solutions improves: (i) , the solution will blow up when the momentum of initial data satisfies
certain sign conditions; (ii) , the solution will blow
up when the slope of initial data is negative at one point; (iii) and , global existence
of strong solutions is ensured. Moreover, if the momentum of initial data has a
definite sign, then for any global smoothness of the
corresponding solution is proved. Proofs are either based on the use of some
global invariants or based on exploration of favorable sign conditions of
quantities involving solution derivatives. Existence and uniqueness results of
global weak solutions for any are also presented. For
some restricted range of parameters results here are equivalent to those known
for the equations [e.g. J. Escher and Z. Yin, Well-posedness, blow-up
phenomena, and global solutions for the b-equation, {\it J. reine angew.
Math.}, {\bf 624} (2008)51--80.]Comment: 21 page
Recovery of high frequency wave fields from phase space based measurements
Computation of high frequency solutions to wave equations is important in
many applications, and notoriously difficult in resolving wave oscillations.
Gaussian beams are asymptotically valid high frequency solutions concentrated
on a single curve through the physical domain, and superposition of Gaussian
beams provides a powerful tool to generate more general high frequency
solutions to PDEs. An alternative way to compute Gaussian beam components such
as phase, amplitude and Hessian of the phase, is to capture them in phase space
by solving Liouville type equations on uniform grids. In this work we review
and extend recent constructions of asymptotic high frequency wave fields from
computations in phase space. We give a new level set method of computing the
Hessian and higher derivatives of the phase. Moreover, we prove that the
order phase space based Gaussian beam superposition converges to the
original wave field in at the rate of \ep^{\frac{k}{2}-\frac{n}{4}} in
dimension .Comment: a revision of introductio
The Cauchy-Dirichlet problem for the FENE dumbbell model of polymeric fluids
The FENE dumbbell model consists of the incompressible Navier-Stokes equation
and the Fokker-Planck equation for the polymer distribution. In such a model,
the polymer elongation cannot exceed a limit , yielding all
interesting features near the boundary. In this paper we establish the local
well-posedness for the FENE dumbbell model under a class of Dirichlet-type
boundary conditions dictated by the parameter . As a result, for each
we identify a sharp boundary requirement for the underlying density
distribution, while the sharpness follows from the existence result for each
specification of the boundary behavior. It is shown that the probability
density governed by the Fokker-Planck equation approaches zero near boundary,
necessarily faster than the distance function for , faster than for , and as fast as for . Moreover, the sharp
boundary requirement for is also sufficient for the distribution to
remain a probability density.Comment: 32 page
Error Estimates of the Bloch Band-Based Gaussian Beam Superposition for the Schr\"odinger Equation
This work is concerned with asymptotic approximations of the semi-classical
Schr\"odinger equation in periodic media using Gaussian beams. For the
underlying equation, subject to a highly oscillatory initial data, a hybrid of
the Gaussian beam approximation and homogenization leads to the Bloch
eigenvalue problem and associated evolution equations for Gaussian beam
components in each Bloch band. We formulate a superposition of Bloch-band based
Gaussian beams to generate high frequency approximate solutions to the original
wave field. For initial data of a sum of finite number of band eigen-functions,
we prove that the first-order Gaussian beam superposition converges to the
original wave field at a rate of , with the
semiclassically scaled constant, as long as the initial data for Gaussian beam
components in each band are prepared with same order of error or smaller. For a
natural choice of initial approximation, a rate of of initial
error is verified.Comment: 27 page
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