This paper studies the Cauchy problem for the nonlinear fractional power dissipative equation ut + (−△) u = F (u) for initial data in the Lebesgue space L(R) with r ≥ rd , nb/(2α− d) or the homogeneous Besov space Ḃ p,∞(R ) with σ = (2α − d)/b − n/p and 1 ≤ p ≤ ∞, where α > 0, F (u) = f(u) or Q(D)f(u) with Q(D) being a homogeneous pseudo-differential operator of order d ∈ [0, 2α) and f(u) is a ...

An explicit asymptotic model for transient Love waves is derived from the exact equations of anti-plane elasticity. The perturbation procedure relies upon the slow decay of low frequency Love waves to approximate the displacement field in the substrate by a power series in the depth coordinate. When appropriate decay conditions are imposed on the series, one obtains a model equation governing t...

In this paper we give a complete description of the Fredholm spectrum of the linearized 3D Euler equation in terms of the dynamical spectrum of the cocycle governing the evolution of shortwave perturbations. The argument is based on the pseudo differential representation of the Euler group and an application of an abstract isomorphism theorem for operator C∗-algebras. We further show that for a...

Abstract An extended Harry Dym hierarchy is constructed by using eigenfunctions and adjoint eigenfunctions of the spectral problems of the Harry Dym hierarchy associated with the pseudo-differential operator L = u∂ +u0 +u1∂−1 + · · ·. The corresponding Lax presentation possesses a self-consistent source involving squared eigenfunctions. The resulting extended Harry Dym hierarchy is reduced to t...

Abstract. We prove Strichartz estimates for the absolutely continuous evolution of a Schrödinger operator H = (i∇ + A) + V in R, n ≥ 3. Both the magnetic and electric potentials are time-independent and satisfy pointwise polynomial decay bounds. The vector potential A(x) is assumed to be continuous but need not possess any Sobolev regularity. This work is a refinement of previous methods, which...

We develop an algorithm for the computation of general Fourier integral operators associated with canonical graphs. The algorithm is based on dyadic parabolic decomposition using wave packets and enables the discrete approximate evaluation of the action of such operators on data in the presence of caustics. The procedure consists in the construction of a universal operator representation throug...

We consider a model of cellular detonations in gases. It consists in conservation laws with a non-local pseudo-differential operator whose symbol is asymptotically |ξ|, where 0 < λ ≤ 2; it can be decomposed as the λ/2 fractional power of the Laplacian plus a convolution term. After defining the notion of entropy solution, we prove the wellposedness in the L∞ framework. In the case where 1 < λ ≤...

We present a novel approach to the Kadomtsev-Petviashvili (KP) hierarchy and its modified counterpart, the mKP hierarchy based on factorizations of formal pseudo-differential operators and a matrix-valued Lax operator for the mKP hierarchy. As a result of this framework we obtain new Bäcklund transformations for the KP hierarchy and the possibility of transferring classes of KP solutions into t...

We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semi-finite von Neumann algebra. Our results have several applications. We deduce a formula for the Chern character of an odd L-summable Breuer-Fredholm module in terms of a Hochschild 1-cycle. We explain how to derive a Wodzicki res...

‎In the present paper‎, ‎by making use of the differential subordination and‎ ‎superordination results of Miller and Mocanu‎, ‎  ‎certain classes of admissible functions ‎are determined so that subordination as well as superordination implications‎ ‎of functions associated with an integral operator hold‎. ‎Additionally‎, ‎differential sandwich-type result is obtained‎.