Showing papers in "Annali Dell'universita' Di Ferrara in 2006"
TL;DR: In this paper, the authors study the system of equations describing a stationary thermoconvective flow of a non-Newtonian fluid and prove the existence of a weak solution under general assumptions and the uniqueness under smallness conditions.
Abstract: We study the system of equations describing a stationary thermoconvective flow of a non-Newtonian fluid. We assume that the stress tensor S has the form $\displaystyle \mathbf{S}=-P\mathbf{I}+\left( \mu (\theta )+\tau (\theta ){|\mathbf{D(u)}|}^{p(\theta )-2}\right) {\mathbf{D(u)}}, $
where u is the vector velocity, P is the pressure, θ is the temperature and μ ,p and τ are the given coefficients depending on the temperature. D and I are respectively the rate of strain tensor and the unit tensor. We prove the existence of a weak solution under general assumptions and the uniqueness under smallness conditions. Keywords: Non-Newtonian fluids, Nonlinear thermal diffusion equations, Heat and mass transfer Mathematics Subject Classification (2000): 76A05, 76D07, 76E30, 35G15
325 citations
TL;DR: In this paper, the authors proved the solvability of the Stokes problem in bounded or exterior domain under the assumption that the initial value of the velocity vector field v0 belongs to C.............. s (Ω), $s\in [0, 2]$ fixme (in particular, it can be only continuous).
Abstract: We prove the solvability of the evolution Stokes problem in bounded or exterior domain Ω∈ R
n
under the assumption that the initial value of the velocity vector field v0 belongs to C
s
(Ω), $s\in [0,2)$
(in particular, it can be only continuous). The solution is obtained in some weighted Holder spaces. This result makes it possible to prove the local solvability of a nonlinear problem under the same assumption concerning v0. Keywords: Stokes equations, Weighted norms Mathematics Subject Classification (2000): 35Q30, 76D03
25 citations
TL;DR: In this article, it was shown that SG-pseudodifferential operators of nonnegative order possess complex powers, and that the powers are again classical and derive an explicit formula for all homogeneous components.
Abstract: Under a suitable ellipticity condition, we show that classical SG-pseudodifferential operators of nonnegative order possess complex powers. We show that the powers are again classical and derive an explicit formula for all homogeneous components. Keywords: Complex power, Weighted symbols, Noncompact manifolds
24 citations
TL;DR: In this article, a class of vector-valued deformations of the classical harmonic oscillator, called noncommutative harmonic oscillators, is described, with special emphasis on the Poisson relation and clustering properties of the eigenvalues.
Abstract: Some spectral properties of certain 2×2 globally elliptic systems of ordinary differential operators, a class of vector-valued deformations of the classical harmonic oscillator here called noncommutative harmonic oscillators, will be described, with special emphasis on the Poisson relation and clustering properties of the eigenvalues. Keywords: Clustering theorems, Periodic trajectories, Poisson relations, Noncommutative harmonic oscillators
20 citations
TL;DR: In this article, the authors examined the spectrum of the Lax-Phillips semigroup Z(t) for trapping obstacles having at least one trapped ray in a neighborhood of an obstacle.
Abstract: We examine the cut-off resolvent R
χ
(λ) = χ (–Δ
D
– λ2)–1χ, where Δ
D
is the Laplacian with Dirichlet boundary condition and $\chi \in C_0^{\infty}(\mathbb{R}^n)$
equal to 1 in a neighborhood of the obstacle K. We show that if R
χ
(λ) has no poles for $\Im \lambda \geq -\delta,\: \delta > 0$
, then $\|R_{\chi}(\lambda)\|_{L^2 \to L^2} \leq C|\lambda|^{n-2},\: \lambda \in \mathbb{R},\: |\lambda| \geq C_0.$
This estimate implies a local energy decay. We study the spectrum of the Lax-Phillips semigroup Z(t) for trapping obstacles having at least one trapped ray. Keywords: Trapping obstacles, Resonances, Local energy decay, Cut-off resolvent
20 citations
TL;DR: In this paper, the analytical properties of multilane traffic flow models based on hyperbolic balance laws are analyzed and compared with the operator splitting method for traffic flow model with a fixed number of vehicles.
Abstract: We give rigorous results on the analytical properties of multilane traffic flow models based on hyperbolic balance laws. Keywords: Traffic flows, Hyperbolic conservatin laws, Operator splitting method
19 citations
TL;DR: In this article, asymptotic estimates for a class of Fourier integrals which are defined on surfaces with singular points were studied and applied to obtain decay estimates for solutions of Cauchy problems of constant coefficient hyperbolic equations.
Abstract: We study asymptotic estimates for a class of Fourier integrals which are defined on surfaces with singular points and show how these estimates can be applied to obtain decay estimates for solutions of Cauchy problems of constant coefficient hyperbolic equations. Integrals of the type which we consider appear e.g. in the study of solutions of the system of elasticity for cubic crystals. Related estimates for smooth surfaces or for more general oscillatory integrals in a fully C∞-setting have a long tradition in analysis and have proved useful in many situations.
12 citations
TL;DR: In this article, it was shown that there exists a solution which is not asymptotically free provided that the coefficient tends slowly to some constant, where the coefficient is of Lipschitz class and has some stability condition.
Abstract: We shall find asymptotic profiles for strictly hyperbolic equations with time-dependent coefficients which are of Lipschitz class and have some stability condition. More precisely, it will be shown that there exists a solution which is not asymptotically free provided that the coefficient tends slowly to some constant. Keywords: Wave equation, Asymptotic profiles, Asymptotic integrations
9 citations
TL;DR: In this paper, the authors studied relations between modulus of continuity of the coefficients and loss of derivatives in the Cauchy problem for evolution operators with real characteristics in the Petrovsky sense.
Abstract: We study relations between modulus of continuity of the coefficients and loss of derivatives in the Cauchy problem for evolution operators with real characteristics in the Petrovsky sense. We also provide counterexamples to show that the obtained classification is sharp. Keywords: Cauchy problem, Evolution equations, Loss of regularity of the solution
8 citations
TL;DR: In this article, the existence of a critical exponent for large data smooth solutions of the Cauchy Problem associated with the semilinear weakly hyperbolic equations is studied.
Abstract: In this paper, we deal with some global existence results for the large data smooth solutions of the Cauchy Problem associated with the semilinear weakly hyperbolic equations $$ u_{tt}-a_{\lambda_1}(t)\Delta_x u=-a_{\lambda_2}(t)|u|^{p-1}u. $$
Here u=u(x,t), $x\in \mathbb{R}^n$
and for λ≥ 0, a
λ
≥ 0 is a continuous function that behaves as |t–t0|
λ
close to some t0>0. We conjecture the existence of a critical exponent p
c
(λ1,λ2,n) such that for p≤ p
c
(λ1,λ2,n) a global existence theorem holds. For suitable λ1,λ2,n, we recall some known results and add new ones. Keywords: Critical exponents for semilinear equations, Weak hyperbolicity
5 citations
TL;DR: In this paper, the authors considered a non-characteristic boundary value problem for equations of eikonal type and showed that, near the boundary, the viscosity solution inherits the regularity of the data.
Abstract: We consider a non-characteristic boundary value problem for equations of eikonal type and we show that, near the boundary, the viscosity solution inherits the regularity of the data. As a consequence, we slightly improve the results in [1] on the structure of the cut-locus of a class of distance functions. Keywords: Viscosity solutions, Eikonal equation, Cut-locus, Analytic regularity
TL;DR: In this article, the problem of global analytic and Gevrey hypoellipticity and solvability for a class of partial differential operators on a torus is considered.
Abstract: In this paper we consider the problem of global analytic and Gevrey hypoellipticity and solvability for a class of partial differential operators on a torus. We prove that global analytic and Gevrey hypoellipticity and solvability on the torus is equivalent to certain Diophantine approximation properties. Keywords: Global hypoellipticity, Global solvability, Gevrey classes, Diophantine approximation property Mathematics Subject Classification (2000): 35D05, 46E10, 46F05, 58J99
TL;DR: Inverse Map Theorem for a map f from the Heisenberg group into itself, provided the Pansu differential of f is continuous, non-singular and satisfies some growth conditions at infinity.
Abstract: We prove a global Inverse Map Theorem for a map f from the Heisenberg group into itself, provided the Pansu differential of f is continuous, non singular and satisfies some growth conditions at infinity. An estimate for the Lipschitz constant (with respect to the Carnot–Caratheodory distance in $\mathbb{H}$
) of a continuously Pansu differentiable map is included. This gives a characterization of (continuously Pansu differentiable) globally biLipscitz deformations of $\mathbb{H}$
in term of a pointwise estimate of their differential. Keywords: Inverse problem, Heisenberg group
TL;DR: In this article, the main results of this paper give two criteria for certain infinite series of rational numbers to be Liouville numbers, and some examples of such numbers are also included.
Abstract: The main results of this paper give two criteria for certain infinite series of rational numbers to be Liouville. Some examples are also included. Keywords: Liouville numbers, Infinite series Mathematics Subject Classification (2000): 11J82
TL;DR: In this article, the existence and uniqueness of a viscosity solution of the Dirich-let problem related to the prescribed Levi mean curvature equation, under suitable assumptions on the boundary data and on the Levi curvature of the domain, were proved.
Abstract: We prove existence and uniqueness of a viscosity solution of the Dirich- let problem related to the prescribed Levi mean curvature equation, under suitable assumptions on the boundary data and on the Levi curvature of the domain. We also show that such a solution is Lipschitz continuous by proving that it is the uni- form limit of a sequence of classical solutions of elliptic problems and by building Lipschitz continuous barriers.
TL;DR: In this paper, the well posedness of the Cauchy problem for the operator P =D¯¯¯¯ t�Ω(n)€ 2€ 2 Ω (n + 2 n)€ 0.
Abstract: The well posedness of the Cauchy problem for the operator P=D
t
2–D
x
a(t,x)
n
D
x
, $t,x \in \mathbb{R}$
with data on t=0 is studied assuming a ∈ C
N
(
$[0,T];\gamma^{(s_0)}$
(R)), s0>1 and sufficiently close to 1, a(t,x)≥ 0. Well posedness is proved in Gevrey classes γ(s), for $1\leq s< \frac{Nn^2}{2(N+2n)}$
, n≥ n0. Keywords: Partial differential equations, Cauchy problem, Well posedness
TL;DR: In this article, a Waring type problem for partially symmetric tensors is addressed and an explicit answer in lower dimensional cases is given in terms of the Noether-Fano inequality.
Abstract: Here we address a Waring type problem for partially symmetric tensors, extending previous work by Massimiliano Mella in the totally symmetric case of forms. In particular, we provide an explicit answer in lower dimensional cases. Keywords: Waring problem, Partially symmetric tensor, Segre-Veronese embedding, Noether-Fano inequality, Weakly defective variety Mathematics Subject Classification (2000): 14N05
TL;DR: In this article, the Littlewood-Paley decomposition technique was used to obtain a C∞-well-posedness result for weakly hyperbolic equations with a finite order of degeneration.
Abstract: We use the Littlewood-Paley decomposition technique to obtain a C∞-well-posedness result for a weakly hyperbolic equation with a finite order of degeneration. Keywords: Littlewood-Paley decomposition, Hyperbolic equations, C∞-well-posedness, Approximate energy method
TL;DR: In this paper, the authors considered the Cauchy problem for a second order equation of hyperbolic type, and they gave an appropriate Levi condition on the lower order terms in order to get C∞ well posedness of the problem.
Abstract: We consider the Cauchy problem for a second order equation of hyperbolic type. This equation degenerates in two different ways. On one hand, the coefficients have a bad behavior with respect to time: there is a blow-up phenomenon in the first time derivative of the principal part’s coefficients, that is the derivative vanishes at the time t=0. On the other hand, the equation is weakly hyperbolic and the multiplicity of the roots is not constant, but zeroes are of finite order. Here we overcome the blow-up problem and, moreover, the finitely degeneration of the Cauchy problem allows us to give an appropriate Levi condition on the lower order terms in order to get C∞ well posedness of the Cauchy problem. Keywords: Cauchy problem, Hyperbolic equations, Levi conditions
TL;DR: In this paper, a uniqueness result for the Cauchy problem associated to a particular type of ODE was presented, under the only assumption of continuity of the right hand side at the initial point.
Abstract: We present a uniqueness result for the Cauchy problem associated to a particular type of ordinary differential equation (ODE), under the only assumption of continuity of the right hand side at the initial point. Keywords: Polar coordinates, Tangent vector, Inner product Mathematics Subject Classification (2000): 34A12
TL;DR: In this article, a new property of the quasi-symmetrizer which allows to generalize the result in (6) to semi-linear systems with coefficients depending also on the space variables is discussed.
Abstract: The technique of quasi-symmetrizer has been applied to the well-po- sedness of the Cauchy problem for scalar operators (10), (13) and linear sys- tems (5), (15), (4), and to the propagation of analitycity for solutions to semi-linear systems (6). In all these works, it is assumed that the principal symbol depends only on the time variable. In this note we illustrate, in some special cases, a new property of the quasi- symmetrizer which allows us to generalize the result in (6) to semi-linear systems with coefficients depending also on the space variables (21). Such a property is closely connected with some interesting inequalities on the eigenvalues of a hy- perbolic matrix. We expect that this technique applies also to other problems.
TL;DR: In this paper, the higher codimensional version of Theorem 1.1 was proved for a class of PDO's with a symplectic characteristic manifold, and a complete proof of Fefferman's SAK Principle was given.
Abstract: We prove in details the higher codimensional version of Theorem 1.1 [11]. This provides a complete proof of Fefferman’s SAK Principle for a class of PDO’s with symplectic characteristic manifold. Keywords: A priori estimates, General theory of PDO’s
TL;DR: In this article, the authors studied the propagation of singularities and the microlocal behaviour at infinity for the solution of the Cauchy problem associated to an SG-hyperbolic operator with one characteristic of constant multiplicity.
Abstract: We study the propagation of singularities and the microlocal behaviour at infinity for the solution of the Cauchy problem associated to an SG-hyperbolic operator with one characteristic of constant multiplicity. We perform our analysis in the framework of tempered ultradistributions, cf. Introduction, using an appropriate notion of wave front set. Keywords: Wave front set at infinity, Tempered ultradistributions, Hyperbolic equations
TL;DR: In this paper, the authors solved locally in time the solutions to the Cauchy problem for first order quasilinear hyperbolic systems of which coefficients of principal part and of lower order terms are μ- Holder and $\mu'$ -Holder continuous in time variable respectively and in Gevrey class of index s with respect to space variables.
Abstract: In this paper we shall solve locally in time the solutions to the Cauchy problem for first order quasilinear hyperbolic systems of which coefficients of principal part and of lower order terms are μ- Holder and $\mu'$
- Holder continuous in time variable respectively and in Gevrey class of index s with respect to space variables under the assumption $1\le s <\min\{1 + \frac\mu
u,1+\frac{1-\mu+\mu'}{
u}\}, 0<\mu\le1$
, where ν denotes the maximal muliplicity of characteristics of systems. Keywords: Nonlinear hyperbolic systems, Cauchy problem, Gevrey classes
TL;DR: In this article, a necessary and sufficient condition for the Cauchy problem for first order hyperbolic systems with constant coefficient principal part is given, without any assumptions on the rank.
Abstract: In this article we shall introduce the results obtained in [16], ie, we shall give a necessary and sufficient condition that the Cauchy problem for first order hyperbolic systems with constant coefficient principal part is C∞ well-posed under the maximal rank condition (see the condition (R) below) We shall also give a simple sufficient condition without any assumptions on the rank Keywords: Hyperbolic system, Cauchy problem, Constant coefficient principal part
TL;DR: In this article, the existence of weak periodic solutions for a parabolic-elliptic system proposed as a model for a time dependent thermistor with degenerate thermal conductivity was proved.
Abstract: This paper deals with the existence of weak periodic solutions for a parabolic-elliptic system proposed as a model for a time dependent thermistor with degenerate thermal conductivity. Applying the maximal monotone mappings theory, we prove an existence result for weak periodic solutions. Keywords: Nonlinear parabolic-elliptic system of degenerate type, Periodic solutions, Thermistor problem Mathematics Subject Classification (2000): 35B10, 35J60, 35K65
TL;DR: In this article, the authors characterize the finite groups in which the intersection of the maximal non-nilpotent subgroups is nilpotent, but different from Φ(G).
Abstract: The authors characterize the finite groups in which $\,\mathcal{H} (G)$
, the intersection of the maximal non-nilpotent subgroups of G, is nilpotent, but different from Φ(G). Further, if $\,\mathcal{F} \,$
is a saturated formation and if $\,\mathcal{F}(G)\,$
is the intersection of all maximal subgroups of G not belonging to $\,\mathcal{F}$
, a necessary and sufficient condition is given for $\,\mathcal{F}(G)\,$
to be nilpotent different from Φ(G). Keywords: Frattini subgroup, Maximal subgroups, Saturated formation Mathematics Subject Classification (2000): 20B05, 20D10, 20D25,20E28
TL;DR: In this paper, the authors considered the system of equations of viscous gas motion whose pressure is related to the density by the law $p = h \varrho^\gamma$ with 1 <γ <2, in a domain defined by two levels of geopotential.
Abstract: We consider the system of equations of viscous gas motion whose pressure is related to the density by the law $p = h \varrho^\gamma$
with 1<γ <2, in a domain defined by two levels of geopotential. Under the force due to geopotential and the Coriolis force, we prove the stability of the equilibrium state in a suitable Sobolev space. Keywords: Viscous barotropic gas, Equilibrium state, Coriolis force Mathematics Subject Classification (2000): 35Q35, 76N15
TL;DR: Using the decomposition of an antisymmetric 2-tensor as a sum of two orthogonal bivectors, the various canonical forms of the electromagnetic tensor field are analyzed, recovering known results as discussed by the authors.
Abstract: Using the decomposition of an antisymmetric 2-tensor as a sum of two orthogonal bivectors, the various canonical forms of the electromagnetic tensor field are analyzed, recovering known results. However, introducing 1+3 spacetime splitting techniques, the canonical forms are associated to special frames and observers and this helps to clarify the role played by “measurable” quantities (electric and magnetic fields, Poynting vector) in the classification problem itself. Keywords: Electromagnetic field, Canonical form Mathematics Subject Classification (2000): 83A05, 78A25
TL;DR: In this paper, a generalization of the identity proved by Worpitzky was proposed, by expressing each power x ≥ n as a linear combination of the images of β ≥ n under the powers of the shift operator E (here $\beta_m(x):=\frac{x^{\underline{m}}}{m!}$cffff ).
Abstract: We give a generalization of the identity proved by J. Worpitzky in [4], by expressing each power x
n
as a linear combination of the images of β
m
under the powers of the shift operator E (here $\beta_m(x):=\frac{x^{\underline{m}}}{m!}$
). We encode the coefficients of these linear combinations in a 3-dimensional array - the Eulerian octant - and we find recurrences formulae, explicit expressions and generating functions for its entries. Keywords: Recursive matrices, Eulerian numbers Mathematics Subject Classification (2000): 05A19