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Discrete & Continuous Dynamical Systems - S

2013 , Volume 6 , Issue 3

Issue on Evolution Equations and Mathematical Models in the Applied Sciences

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Preface
Silvia Romanelli, Anna Maria Candela, Marcello De Giosa, Rosa Maria Mininni and Alessandro Pugliese
2013, 6(3): i-v doi: 10.3934/dcdss.2013.6.3i +[Abstract](62) +[PDF](4231.6KB)
Abstract:
Nella settimana dal 29 giugno al 3 luglio 2009, presso l'Aula Magna della II Facoltà di Scienze Matematiche, Fisiche e Naturali dell'Università degli Studi di Bari ``Aldo Moro'', sede di Taranto, ed in collaborazione con il Dipartimento di Matematica della stessa università, si è svolto il Convegno Internazionale Evolution Equations and Mathematical Models in the Applied Sciences (EEMMAS), organizzato da Silvia Romanelli, Anna Maria Candela, Marcello De Giosa, Rosa Maria Mininni ed Alessandro Pugliese, e a cui hanno partecipato circa 60 matematici provenienti da università di diverse nazioni tra cui Algeria, Belgio, Colombia, Francia, Germania, Giappone, Israele, Italia, Lussemburgo, Romania, Stati Uniti.

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A constructive proof of Gibson's stability theorem
Fatiha Alabau-Boussouira and Piermarco Cannarsa
2013, 6(3): 611-617 doi: 10.3934/dcdss.2013.6.611 +[Abstract](99) +[PDF](300.8KB)
Abstract:
A useful stability result due to Gibson [SIAM J. Control Optim., 18 (1980), 311--316] ensures that, perturbing the generator of an exponentially stable semigroup by a compact operator, one obtains an exponentially stable semigroup again, provided the perturbed semigroup is strongly stable. In this paper we give a new proof of Gibson's theorem based on constructive reasoning, extend the analysis to Banach spaces, and relax the above compactness assumption. Moreover, we discuss some applications of such an abstract result to equations and systems of evolution.
Product structures and fractional integration along curves in the space
Valentina Casarino, Paolo Ciatti and Silvia Secco
2013, 6(3): 619-635 doi: 10.3934/dcdss.2013.6.619 +[Abstract](82) +[PDF](402.0KB)
Abstract:
In this paper we establish $L^p$ boundedness ($1 < p < \infty$) for a double analytic family of fractional integrals $S^{\gamma}_{z}$, $\gamma,z ∈\mathbb{C}$, when $\Re e z=0$. Our proof is based on product-type kernels arguments. More precisely, we prove that the convolution kernel of $S^{\gamma}_{z}$ is a product kernel on $\mathbb{R}^3$, adapted to the polynomial curve $x_1\mapsto (x_1^m,x_1^n)$ (here $m,n∈\mathbb{N},m ≥ 1, n > m $).
Schauder estimates for some perturbation of an infinite dimensional Ornstein--Uhlenbeck operator
Giuseppe Da Prato
2013, 6(3): 637-647 doi: 10.3934/dcdss.2013.6.637 +[Abstract](88) +[PDF](317.1KB)
Abstract:
We prove some Shauder estimates for an elliptic equation in Hilbert spaces.
On the essential self-adjointness of Ornstein-Uhlenbeck operators perturbed by inverse-square potentials
Tiziana Durante and Abdelaziz Rhandi
2013, 6(3): 649-655 doi: 10.3934/dcdss.2013.6.649 +[Abstract](89) +[PDF](333.6KB)
Abstract:
In this note we give sufficient conditions for the essential self-adjointness of some Kolmogorov operators perturbed by singular potentials. As an application we show that the space of test functions $C_c^∞(R^N \backslash \{0\})$ is a core for the operator $Au= Δu-Bx∇u+\frac{c}{|x|^2} u=:Lu+\frac{c}{|x|^2} u, u ∈ C_c^∞(R^N \backslash \{0\}),$ in $L^2(R^N,\mu)$ provided that $c\le \frac{(N-2)^2}{4}-1$. Here $B$ is a positive definite $N\times N$ hermitian matrix and $\mu$ is the unique invariant measure for the Ornstein-Uhlenbeck operator $L$.
Arithmetic progressions -- an operator theoretic view
Tanja Eisner and Rainer Nagel
2013, 6(3): 657-667 doi: 10.3934/dcdss.2013.6.657 +[Abstract](77) +[PDF](363.0KB)
Abstract:
Motivated by the recent Green--Tao theorem on arithmetic progressions in the primes, we discuss some of the basic operator theoretic techniques used in its proof. In particular, we obtain a complete proof of Szemerédi's theorem for arithmetic progressions of length $3$ (Roth's theorem) and the Furstenberg--Sárközy theorem.
Semiclassical limit of Husimi function
Hassan Emamirad and Philippe Rogeon
2013, 6(3): 669-676 doi: 10.3934/dcdss.2013.6.669 +[Abstract](80) +[PDF](356.3KB)
Abstract:
We will show that Liouville and quantum Liouville operators $L$ and $L_\hbar$ generate two $C_0$-groups $e^{tL}$ and $e^{tL_h}$ of isometries in $L^2(\mathbb{R}^{2n})$ and $e^{tL_h}$ converges ultraweakly to $e^{tL}$. As a consequence we show that the Gaussian mollifier of the Wigner function, called Husimi function, converges in $L^1(\mathbb{R}^{2n})$ to the solution of the Liouville equation.
Simulation of lava flows with power-law rheology
Marilena Filippucci, Andrea Tallarico and Michele Dragoni
2013, 6(3): 677-685 doi: 10.3934/dcdss.2013.6.677 +[Abstract](121) +[PDF](434.3KB)
Abstract:
In this work we studied the effect of a power-law rheology on a gravity driven lava flow. Assuming a viscous fluid with constant temperature and constant density and assuming a steady flow in an inclined rectangular channel, the equation of the motion is solved by the finite volume method and a classical iterative solutor. Comparisons with observed channeled lava flows indicate that the assumption of the power-law rheology causes relevant differences in average velocity and volume flow rate with respect to the Newtonian rheology.
Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates
Genni Fragnelli
2013, 6(3): 687-701 doi: 10.3934/dcdss.2013.6.687 +[Abstract](115) +[PDF](433.6KB)
Abstract:
We prove null controllability results for the one dimensional degenerate heat equation in non divergence form with a drift term and Neumann boundary conditions. To this aim we prove Carleman estimates for the associated adjoint problem. Some linear extensions are considered.
On an Allen-Cahn type integrodifferential equation
Gianni Gilardi
2013, 6(3): 703-709 doi: 10.3934/dcdss.2013.6.703 +[Abstract](75) +[PDF](307.8KB)
Abstract:
An Allen-Cahn type system si transformed into an integraodifferential equation. Results on well-posedness and long time behavior are presented.
Convergence to a stationary state of solutions to inverse problems of parabolic type
Davide Guidetti
2013, 6(3): 711-722 doi: 10.3934/dcdss.2013.6.711 +[Abstract](84) +[PDF](342.2KB)
Abstract:
We illustrate some results of existence and uniqueness of solutions to inverse parabolic problems of partial recostruction of the forcing term. In particular, we look for conditions assuring that the solution and the unknown part of the forcing term converge to a stationary state.
Dynamic behaviour of a periodic competitive system with pulses
Benedetta Lisena
2013, 6(3): 723-729 doi: 10.3934/dcdss.2013.6.723 +[Abstract](78) +[PDF](284.1KB)
Abstract:
In this article we consider an $n$-dimensional competitive Lotka-Volterra system with periodic coefficients and impulses. We provide sufficient conditions for the existence and global attractivity of a positive periodic solution.
Nonautonomous Kolmogorov equations in the whole space: A survey on recent results
Luca Lorenzi
2013, 6(3): 731-760 doi: 10.3934/dcdss.2013.6.731 +[Abstract](71) +[PDF](554.6KB)
Abstract:
In this paper we survey some recent results concerned with nonautonomous Kolmogorov elliptic operators. Particular attention is paid to the case of the nonautonomous Ornstein-Uhlenbeck operator
Non-hamiltonian Schrödinger systems
Sandra Lucente and Eugenio Montefusco
2013, 6(3): 761-770 doi: 10.3934/dcdss.2013.6.761 +[Abstract](69) +[PDF](340.6KB)
Abstract:
In this paper we study local and global in time existence for the Cauchy Problem of some semilinear Schrödinger systems. In particular we do not assume that the nonlinear term guarantees conservation of charge or energy.
Schrödinger type evolution equations with monotone nonlinearity of non-power type
Yoshiki Maeda and Noboru Okazawa
2013, 6(3): 771-781 doi: 10.3934/dcdss.2013.6.771 +[Abstract](83) +[PDF](352.8KB)
Abstract:
Existence of unique strong solutions is established for Schrödinger type evolution equations with monotone nonlinearity. The proof is based on a perturbation theorem for $m$-accretive operators in a complex Hilbert space.
Dispersive waves with multiple tunnel effect on a star-shaped network
F. Ali Mehmeti, R. Haller-Dintelmann and V. Régnier
2013, 6(3): 783-791 doi: 10.3934/dcdss.2013.6.783 +[Abstract](71) +[PDF](359.9KB)
Abstract:
We consider the Klein-Gordon equation on a star-shaped network composed of $n$ half-axes connected at their origins. We add a potential which is constant but different on each branch. The corresponding spatial operator is self-adjoint and we state explicit expressions for its resolvent and its resolution of the identity in terms of generalized eigenfunctions. This leads to a generalized Fourier type inversion formula in terms of an expansion in generalized eigenfunctions. This paper is a survey of a longer article, nevertheless the proof of the central formula is indicated.
The historical changes of borders separating pure mathematics from applied mathematics
Jean-Paul Pier
2013, 6(3): 793-801 doi: 10.3934/dcdss.2013.6.793 +[Abstract](118) +[PDF](198.0KB)
Abstract:
Tracing the pioneering phenomenon of Greek mathematics one may not foresee any later systematic barriers between pure and applied mathematics.
On backward stochastic differential equations in infinite dimensions
Jan A. Van Casteren
2013, 6(3): 803-824 doi: 10.3934/dcdss.2013.6.803 +[Abstract](93) +[PDF](444.2KB)
Abstract:
In the present paper we present a result in which probabilistic methods are used to prove existence and uniqueness of a backward partial differential equation in a Hilbert space. This equation is of the form (7) in Theorem 1.1 below. In particular semi-linear conditions on the coefficient $f$ are imposed.
Growth order and blow up points for the parabolic Burgers' equation under dynamical boundary conditions
Joachim von Below, Gaëlle Pincet Mailly and Jean-François Rault
2013, 6(3): 825-836 doi: 10.3934/dcdss.2013.6.825 +[Abstract](139) +[PDF](647.9KB)
Abstract:
We investigate the blow up points of the one--dimensional parabolic Burgers' equation $$\partial_t u=\partial_x^2 u-u\partial_xu+u^p $$ under a dissipative dynamical boundary condition $\sigma \partial_t u+\partial_\nu u=0$ for one bump initial data. A numerical example of a solution pertaining exactly two bumps stemming from its initial data is presented. Moreover, we discuss the growth order of the $L^\infty$--norm of the solutions when approaching the blow up time.

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