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Evolution Equations & Control Theory

Open Access Articles

Mathematics of nonlinear acoustics
Barbara Kaltenbacher
2015, 4(4): 447-491 doi: 10.3934/eect.2015.4.447 +[Abstract](3963) +[PDF](724.1KB)
The aim of this paper is to highlight some recent developments and outcomes in the mathematical analysis of partial differential equations describing nonlinear sound propagation. Here the emphasis lies on well-posedness and decay results, first of all for the classical models of nonlinear acoustics, later on also for some higher order models. Besides quoting results, we also try to give an idea on their derivation by showning some of the crucial energy estimates. A section is devoted to optimization problems arising in the practical use of high intensity ultrasound.
    While this review puts a certain focus on results obtained in the context of the mentioned FWF project, we also provide some important additional references (although certainly not all of them) for interesting further reading.
Thomas Seidman, Samira El Yacoubi and Abdessamad Tridane
2015, 4(2): i-iv doi: 10.3934/eect.2015.4.2i +[Abstract](1783) +[PDF](141.4KB)
This special issue of the Journal of Evolution Equations and Control Theory (EECT) is dedicated to Professor Abdelhaq El Jai on the occasion of his retirement and in celebration of his significant achievements in the field of control and distributed parameter systems theory, both in his own research and in his leadership in the development of a Moroccan research community in DPS.

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Claudio Giorgi, Davide Guidetti and Maria Grazia Naso
2014, 3(3): i-ii doi: 10.3934/eect.2014.3.3i +[Abstract](1846) +[PDF](7910.0KB)
1.Mauro Fabrizio. This volume entitled ``Mathematical Models and Analytical Problems in Modern Continuum Thermomechanics" is dedicated to Mauro Fabrizio on the occasion of his retirement.

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Stephen W. Taylor and Roberto Triggiani
2013, 2(4): i-ii doi: 10.3934/eect.2013.2.4i +[Abstract](1733) +[PDF](305.1KB)
This volume is dedicated to Walter Littman on the occasion of his retirement.

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Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise
Kumarasamy Sakthivel and Sivaguru S. Sritharan
2012, 1(2): 355-392 doi: 10.3934/eect.2012.1.355 +[Abstract](3405) +[PDF](583.6KB)
In this paper, we establish the solvability of martingale solutions for the stochastic Navier-Stokes equations with Itô-Lévy noise in bounded and unbounded domains in $ \mathbb{R} ^d$,$d=2,3.$ The tightness criteria for the laws of a sequence of semimartingales is obtained from a theorem of Rebolledo as formulated by Metivier for the Lusin space valued processes. The existence of martingale solutions (in the sense of Stroock and Varadhan) relies on a generalization of Minty-Browder technique to stochastic case obtained from the local monotonicity of the drift term.
Alain Haraux and Irena Lasiecka
2012, 1(1): i-i doi: 10.3934/eect.2012.1.1i +[Abstract](2261) +[PDF](83.9KB)
The present Inaugural Volume is the first Issue of a new journal Evolution Equations and Control Theory [EECT], which is published within the AIMS Series. EECT is devoted to topics lying at the interface between Evolution Equations and Control Theory of Dynamics. Evolution equations are to be understood in a broad sense as Infinite Dimensional Dynamics which often arise in modeling physical systems as an infinite-dimensional process. This includes single PDE (Partial Differential Equations) or FDE (Functional Differential Equations) as well as coupled dynamics of different characteristics with an interface between them. Since modern control theory intrinsically depends on a good understanding of the qualitative theory of dynamics and evolution theory, the choice of these two topics appears synergistic and most natural. Past experience shows that new developments in control theory often depend on sufficient information related to the associated dynamical properties of the system. On the other hand, developments in evolution theory allow one to consider certain control theoretic formulations that alone would not appear treatable.

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