802 research outputs found
A Mixed Discrete-Continuous Fragmentation Model
Motivated by the occurrence of "shattering" mass-loss observed in purely
continuous fragmentation models, this work concerns the development and the
mathematical analysis of a new class of hybrid discrete--continuous
fragmentation models. Once established, the model, which takes the form of an
integro-differential equation coupled with a system of ordinary differential
equations, is subjected to a rigorous mathematical analysis, using the theory
and methods of operator semigroups and their generators. Most notably, by
applying the theory relating to the Kato--Voigt perturbation theorem, honest
substochastic semigroups and operator matrices, the existence of a unique,
differentiable solution to the model is established. This solution is also
shown to preserve nonnegativity and conserve mass
Reflections on Dubinskii's nonlinear compact embedding theorem
We present an overview of a result by Ju. A. Dubinskii [Mat. Sb. 67 (109)
(1965); translated in Amer. Math. Soc. Transl. (2) 67 (1968)], concerning the
compact embedding of a seminormed set in , where
is a Banach space and ; we establish a
variant of Dubinskii's theorem, where a seminormed nonnegative cone is used
instead of a seminormed set; and we explore the connections of these results
with a nonlinear compact embedding theorem due to E. Maitre [Int. J. Math.
Math. Sci. 27 (2003)].Comment: 17 pages, 1 figur
Discontinuous Galerkin finite element methods for time-dependent Hamilton--Jacobi--Bellman equations with Cordes coefficients
We propose and analyse a fully-discrete discontinuous Galerkin time-stepping
method for parabolic Hamilton--Jacobi--Bellman equations with Cordes
coefficients. The method is consistent and unconditionally stable on rather
general unstructured meshes and time-partitions. Error bounds are obtained for
both rough and regular solutions, and it is shown that for sufficiently smooth
solutions, the method is arbitrarily high-order with optimal convergence rates
with respect to the mesh size, time-interval length and temporal polynomial
degree, and possibly suboptimal by an order and a half in the spatial
polynomial degree. Numerical experiments on problems with strongly anisotropic
diffusion coefficients and early-time singularities demonstrate the accuracy
and computational efficiency of the method, with exponential convergence rates
under combined - and -refinement.Comment: 40 pages, 3 figures, submitted; extended version with supporting
appendi
Existence of global weak solutions to compressible isentropic finitely extensible nonlinear bead-spring chain models for dilute polymers
We prove the existence of global-in-time weak solutions to a general class of
models that arise from the kinetic theory of dilute solutions of nonhomogeneous
polymeric liquids, where the polymer molecules are idealized as bead-spring
chains with finitely extensible nonlinear elastic (FENE) type spring
potentials. The class of models under consideration involves the unsteady,
compressible, isentropic, isothermal Navier-Stokes system in a bounded domain
in , or , for the density, the velocity and
the pressure of the fluid. The right-hand side of the Navier-Stokes momentum
equation includes an elastic extra-stress tensor, which is the sum of the
classical Kramers expression and a quadratic interaction term. The elastic
extra-stress tensor stems from the random movement of the polymer chains and is
defined through the associated probability density function that satisfies a
Fokker-Planck-type parabolic equation, a crucial feature of which is the
presence of a centre-of-mass diffusion term. We require no structural
assumptions on the drag term in the Fokker-Planck equation; in particular, the
drag term need not be corotational. With a nonnegative initial density for the
continuity equation; a square-integrable initial velocity datum for the
Navier-Stokes momentum equation; and a nonnegative initial probability density
function for the Fokker-Planck equation, which has finite relative entropy with
respect to the Maxwellian associated with the spring potential in the model, we
prove, via a limiting procedure on certain discretization and regularization
parameters, the existence of a global-in-time bounded-energy weak solution to
the coupled Navier-Stokes-Fokker-Planck system, satisfying the prescribed
initial condition.Comment: 83 pages, 1 figure. arXiv admin note: text overlap with
arXiv:1112.4781, arXiv:1004.143
A partial Fourier transform method for a class of hypoelliptic Kolmogorov equations
We consider hypoelliptic Kolmogorov equations in spatial dimensions,
with , where the differential operator in the first spatial
variables featuring in the equation is second-order elliptic, and with respect
to the st spatial variable the equation contains a pure transport term
only and is therefore first-order hyperbolic. If the two differential
operators, in the first and in the st co-ordinate directions, do not
commute, we benefit from hypoelliptic regularization in time, and the solution
for is smooth even for a Dirac initial datum prescribed at . We
study specifically the case where the coefficients depend only on the first
variables. In that case, a Fourier transform in the last variable and standard
central finite difference approximation in the other variables can be applied
for the numerical solution. We prove second-order convergence in the spatial
mesh size for the model hypoelliptic equation subject to
the initial condition , with and , proposed by Kolmogorov, and for an
extension with . We also demonstrate exponential convergence of an
approximation of the inverse Fourier transform based on the trapezium rule.
Lastly, we apply the method to a PDE arising in mathematical finance, which
models the distribution of the hedging error under a mis-specified derivative
pricing model
Finite element approximation of an incompressible chemically reacting non-Newtonian fluid
We consider a system of nonlinear partial differential equations modelling
the steady motion of an incompressible non-Newtonian fluid, which is chemically
reacting. The governing system consists of a steady convection-diffusion
equation for the concentration and the generalized steady Navier-Stokes
equations, where the viscosity coefficient is a power-law type function of the
shear-rate, and the coupling between the equations results from the
concentration-dependence of the power-law index. This system of nonlinear
partial differential equations arises in mathematical models of the synovial
fluid found in the cavities of moving joints. We construct a finite element
approximation of the model and perform the mathematical analysis of the
numerical method in the case of two space dimensions. Key technical tools
include discrete counterparts of the Bogovski\u{\i} operator, De Giorgi's
regularity theorem in two dimensions, and the Acerbi-Fusco Lipschitz truncation
of Sobolev functions, in function spaces with variable integrability exponents.Comment: 40 page
Optimal order finite difference approximation of generalized solutions to the biharmonic equation in a cube
We prove an optimal order error bound in the discrete norm for
finite difference approximations of the first boundary-value problem for the
biharmonic equation in space dimensions, with , whose
generalized solution belongs to the Sobolev space , for , where . The result extends the range of the Sobolev index in the best
convergence results currently available in the literature to the maximal range
admitted by the Sobolev embedding of into in
space dimensions
Regularity and approximation of strong solutions to rate-independent systems
Rate-independent systems arise in a number of applications. Usually, weak
solutions to such problems with potentially very low regularity are considered,
requiring mathematical techniques capable of handling nonsmooth functions. In
this work we prove the existence of H\"older-regular strong solutions for a
class of rate-independent systems. We also establish additional higher
regularity results that guarantee the uniqueness of strong solutions. The proof
proceeds via a time-discrete Rothe approximation and careful elliptic
regularity estimates depending in a quantitative way on the (local) convexity
of the potential featuring in the model. In the second part of the paper we
show that our strong solutions may be approximated by a fully discrete
numerical scheme based on a spatial finite element discretization, whose rate
of convergence is consistent with the regularity of strong solutions whose
existence and uniqueness are established.Comment: 32 page
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