54,132 research outputs found
A non-local vector calculus,non-local volume-constrained problems,and non-local balance laws
A vector calculus for nonlocal operators is developed, including the definition of nonlocal divergence, gradient, and curl operators and the derivation of the corresponding adjoints operators. Nonlocal analogs of several theorems and identities of the vector calculus for differential operators are also presented. Relationships between the nonlocal operators and their differential counterparts are established, first in a distributional sense and then in a weak sense by considering weighted integrals of the nonlocal adjoint operators. The nonlocal calculus gives rise to volume-constrained problems that are analogous to elliptic boundary-value problems for differential operators; this is demonstrated via some examples. Another application is posing abstract nonlocal balance laws and deriving the corresponding nonlocal field equations
Maximal -regularity for stochastic evolution equations
We prove maximal -regularity for the stochastic evolution equation
\{{aligned} dU(t) + A U(t)\, dt& = F(t,U(t))\,dt + B(t,U(t))\,dW_H(t),
\qquad t\in [0,T],
U(0) & = u_0, {aligned}. under the assumption that is a sectorial
operator with a bounded -calculus of angle less than on
a space . The driving process is a cylindrical
Brownian motion in an abstract Hilbert space . For and
and initial conditions in the real interpolation space
\XAp we prove existence of unique strong solution with trajectories in
L^p(0,T;\Dom(A))\cap C([0,T];\XAp), provided the non-linearities
F:[0,T]\times \Dom(A)\to L^q(\mathcal{O},\mu) and B:[0,T]\times \Dom(A) \to
\g(H,\Dom(A^{\frac12})) are of linear growth and Lipschitz continuous in their
second variables with small enough Lipschitz constants. Extensions to the case
where is an adapted operator-valued process are considered as well.
Various applications to stochastic partial differential equations are worked
out in detail. These include higher-order and time-dependent parabolic
equations and the Navier-Stokes equation on a smooth bounded domain
\OO\subseteq \R^d with . For the latter, the existence of a unique
strong local solution with values in (H^{1,q}(\OO))^d is shown.Comment: Accepted for publication in SIAM Journal on Mathematical Analysi
Fractional Calculus in Wave Propagation Problems
Fractional calculus, in allowing integrals and derivatives of any positive
order (the term "fractional" kept only for historical reasons), can be
considered a branch of mathematical physics which mainly deals with
integro-differential equations, where integrals are of convolution form with
weakly singular kernels of power law type. In recent decades fractional
calculus has won more and more interest in applications in several fields of
applied sciences. In this lecture we devote our attention to wave propagation
problems in linear viscoelastic media. Our purpose is to outline the role of
fractional calculus in providing simplest evolution processes which are
intermediate between diffusion and wave propagation. The present treatment
mainly reflects the research activity and style of the author in the related
scientific areas during the last decades.Comment: 33 pages, 9 figures. arXiv admin note: substantial text overlap with
arXiv:1008.134
Bounded -calculus for cone differential operators
We prove that parameter-elliptic extensions of cone differential operators
have a bounded -calculus. Applications concern the Laplacian and the
porous medium equation on manifolds with warped conical singularities
On the fractional Poisson process and the discretized stable subordinator
The fractional Poisson process and the Wright process (as discretization of
the stable subordinator) along with their diffusion limits play eminent roles
in theory and simulation of fractional diffusion processes. Here we have
analyzed these two processes, concretely the corresponding counting number and
Erlang processes, the latter being the processes inverse to the former.
Furthermore we have obtained the diffusion limits of all these processes by
well-scaled refinement of waiting times and jumpsComment: 30 pages, 4 figures. A preliminary version of this paper was an
invited talk given by R. Gorenflo at the Conference ICMS2011, held at the
International Centre of Mathematical Sciences, Pala-Kerala (India) 3-5
January 2011, devoted to Prof Mathai on the occasion of his 75 birthda
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