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Discrete & Continuous Dynamical Systems - S

Open Access Articles

On the geometry of the p-Laplacian operator
Bernd Kawohl and Jiří Horák
2017, 10(4): 799-813 doi: 10.3934/dcdss.2017040 +[Abstract](677) +[HTML](289) +[PDF](1487.0KB)

The \begin{document}$p$\end{document}-Laplacian operator \begin{document}$\Delta_pu={\rm div }\left(|\nabla u|^{p-2}\nabla u\right)$\end{document} is not uniformly elliptic for any \begin{document}$p\in(1,2)\cup(2,\infty)$\end{document} and degenerates even more when \begin{document}$p\to \infty$\end{document} or \begin{document}$p\to 1$\end{document}. In those two cases the Dirichlet and eigenvalue problems associated with the \begin{document}$p$\end{document}-Laplacian lead to intriguing geometric questions, because their limits for \begin{document}$p\to\infty$\end{document} or \begin{document}$p\to 1$\end{document} can be characterized by the geometry of \begin{document}$\Omega$\end{document}. In this little survey we recall some well-known results on eigenfunctions of the classical 2-Laplacian and elaborate on their extensions to general \begin{document}$p\in[1,\infty]$\end{document}. We report also on results concerning the normalized or game-theoretic \begin{document}$p$\end{document}-Laplacian

and its parabolic counterpart \begin{document}$u_t-\Delta_p^N u=0$\end{document}. These equations are homogeneous of degree 1 and \begin{document}$\Delta_p^N$\end{document} is uniformly elliptic for any \begin{document}$p\in (1,\infty)$\end{document}. In this respect it is more benign than the \begin{document}$p$\end{document}-Laplacian, but it is not of divergence type.

Hans-Christoph Kaiser, Dorothee Knees, Alexander Mielke, Joachim Rehberg, Elisabetta Rocca, Marita Thomas and Enrico Valdinoci
2017, 10(4): i-iv doi: 10.3934/dcdss.201704i +[Abstract](285) +[HTML](146) +[PDF](171.2KB)
Zhaosheng Feng
2017, 10(3): i-i doi: 10.3934/dcdss.201703i +[Abstract](287) +[HTML](154) +[PDF](71.7KB)
Preface: Diffusion on fractals and non-linear dynamics
Kurt Falk, Marc Kesseböhmer, Tobias Henrik Oertel-Jäger, Jens D. M. Rademacher and Tony Samuel
2017, 10(2): ⅰ-ⅳ doi: 10.3934/dcdss.201702i +[Abstract](346) +[HTML](174) +[PDF](140.8KB)
Zhouping Xin and Tong Yang
2016, 9(6): i-ii doi: 10.3934/dcdss.201606i +[Abstract](270) +[PDF](9733.8KB)
This special issue of DCDS is dedicated to Prof. Boling Guo on the occasion of his 80th birthday.

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Russell Johnson, Roberta Fabbri, Sylvia Novo, Carmen Núñez and Rafael Obaya
2016, 9(4): i-iii doi: 10.3934/dcdss.201604i +[Abstract](401) +[PDF](96.6KB)
Generally speaking, the term nonautonomous dynamics refers to the systematic use of dynamical tools to study the solutions of differential or difference equations with time-varying coefficients. The nature of the time variance may range from periodicity at one extreme, through Bohr almost periodicity, Birkhoff recurrence, Poisson recurrence etc. to stochasticity at the other extreme. The ``dynamical tools'' include almost everywhere Lyapunov exponents, exponential splittings, rotation numbers, and the theory of cocycles, but are by no means limited to these. Of course in practise one uses whatever ``works'' in the context of a given problem, so one usually finds dynamical methods used in conjunction with those of numerical analysis, spectral theory, the calculus of variations, and many other fields. The reader will find illustrations of this fact in all the papers of the present volume.

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Laurence Cherfils and Aziz Hamdouni
2016, 9(2): i-ii doi: 10.3934/dcdss.201602i +[Abstract](338) +[PDF](97.6KB)
This issue is specially dedicated to our colleague and friend Alain Cimetière.

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Ana Alonso Rodríguez, Luigi C. Berselli, Alessandro Morando and Paola Trebeschi
2016, 9(1): i-i doi: 10.3934/dcdss.2016.9.1i +[Abstract](253) +[PDF](87.2KB)
This special Issue of Discrete and Continuous Dynamical Systems - Series S entitled ``Fluid Dynamics and Electromagnetism: Theory and Numerical Approximation'' is in honor of the two leading Italian Mathematicians Paolo Secchi and Alberto Valli and this volume contains papers that engage a wide set of modern topics in the theory of linear and nonlinear partial differential equations and their applications.
    Paolo Secchi was born in 1954 in Trento (Italy) and graduated in Mathematics at the University of Trento in 1978, under the supervision of Prof. Hugo Beirão da Veiga. Since 1996 he is Full Professor in Mathematical Analysis at the University of Brescia. His research has ranged from Fluid Dynamics to the mathematical theory of Nonlinear Hyperbolic Equations and Systems. His current major interests focus on Free Boundary Problems in Fluid and Magneto-hydrodynamics and Initial-Boundary Value Problems for Hyperbolic Systems.
    Alberto Valli was born in 1953 in Castelnovo nè Monti (Reggio Emilia, Italy) and graduated in Mathematics at the University of Pisa in 1975, under the supervision of Prof. Giovanni Prodi. Since 1987 he is Full Professor in Mathematical Analysis at the University of Trento. His research has ranged from Fluid Dynamics to Numerical Methods for Partial Differential Equations, his main current interests being concerned with Finite Element Methods, Computational Electromagnetism, and Domain Decomposition Methods.
    Paolo Secchi and Alberto Valli are universally recognized as two of the leading experts in their respective research areas.
    In addition to this Volume, in order to celebrate their 60th birthdays an international conference was held at the C.I.R.M (International Center for Mathematical Research) in Levico Terme, Italy, June 3-6 2014. The Conference was aimed at bringing together leading scientists in the fields of Fluid Dynamics and Electromagnetism and to present high level contributions on recent developments in the theory and numerical analysis of partial differential equations related to these fields. We would like to warmly thank the C.I.R.M and its staff, which support and priceless assistance made possible the realization of the Conference, contributing to its success.
    As the guest editors, we are glad that {\it Discrete and Continuous Dynamical Systems - Series S} kindly agreed to publish this special Theme Issues in honor of Paolo Secchi and Alberto Valli. We are also very grateful to the Editor in Chief Prof. Miranville for his help, to the contributors of this Volume, as well as to the many reviewers, for their invaluable and essential support. On this occasion, we would like to express our deepest friendship to Paolo Secchi and Alberto Valli and to wish them all the best and many more productive years ahead.
Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain
Yoshihiro Shibata
2016, 9(1): 315-342 doi: 10.3934/dcdss.2016.9.315 +[Abstract](550) +[PDF](569.0KB)
In this paper, we prove the local well-posedness of the free boundary problems of Navier-Stokes equations in a general domain $\Omega\subset\mathbb{R}^N$ ($N \geq 2$). The velocity field is obtained in the maximal regularity class $W^{2,1}_{q,p}(\Omega\times(0, T)) = L_p((0, T), W^2_q(\Omega)^N) \cap W^1_p((0, T), L_q(\Omega)^N)$ ($2 < p < \infty$ and $N < q < \infty$) for any initial data satisfying certain compatibility conditions. The assumption of the domain $\Omega$ is the unique existence of solutions to the weak Dirichlet-Neumann problem as well as some uniformity of covering of the closure of $\Omega$. A bounded domain, a perturbed half space, and a perturbed layer satisfy the conditions for the domain, and therefore drop problems and ocean problems are treated in the uniform manner. Our method is based on the maximal $L_p$-$L_q$ regularity theorem of a linearized problem in a general domain.
Annihilation of two interfaces in a hybrid system
Shin-Ichiro Ei, Kei Nishi, Yasumasa Nishiura and Takashi Teramoto
2015, 8(5): 857-869 doi: 10.3934/dcdss.2015.8.857 +[Abstract](432) +[PDF](954.7KB)
We consider the mixed ODE-PDE system called a hybrid system, in which the two interfaces interact with each other through a continuous medium and their equations of motion are derived in a weak interaction framework. We study the bifurcation property of the resulting hybrid system and construct an unstable standing pulse solution, which plays the role of a separator for dynamic transition from standing breather to annihilation behavior between two interfaces.
Dietmar Hömberg and Guanghui Hu
2015, 8(3): i-i doi: 10.3934/dcdss.2015.8.3i +[Abstract](352) +[PDF](96.9KB)
The workshop Electromagnetics-Modelling, Simulation, Control and Industrial Applications was held at Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin during May 13-17, 2013. Organizers of this workshop were Dietmar Hömberg (WIAS), Ronald H. W. Hoppe (University of Augsburg/University of Houston), Olaf Klein (WIAS), Jürgen Sprekels (WIAS) and Fredi Tröltzsch (Technical University of Berlin). More than sixty researchers from mathematical, physical, engineering and industrial communities participated in this scientific meeting. This special issue of DCDS-S, which contains eleven research-level articles, is based on the talks presented during the workshop. Electromagnetism plays an important role in many modern high-technological applications. Our workshop brought together prominent worldwide experts from academia and industry to discuss recent achievements and future trends of modelling, computations and analysis in electromagnetics. The contributions to this volume cover the following topics: finite and boundary element discretization methods for the electromagnetic field equations in frequency and time domain, optimal control and model reduction for multi-physics problems involving electromagnetics, mathematical analysis of Maxwell's equations as well as direct and inverse scattering problems.
    We particularly emphasize that most articles are devoted to mathematically challenging issues in applied sciences and industrial applications. The optimal control and model reduction presented by S. Nicaise et al. arise from electromagnetic flow measurement in the real world. The contribution by G. Beck et al. focuses on a generalized telegrapher's model which describes the propagation of electromagnetic waves in non-homogeneous conductor cables with multi-wires. The numerical analysis of boundary integral formulations carried out by K. Schmidt et al. is motivated by asymptotic models for thin conducting sheets. The integral equation system established by B. Bugert et al. and the analysis and experiments performed by H. Gross et al. make new contributions to direct and inverse electromagnetic scattering from diffraction gratings, respectively. The locating and inversion schemes for detecting unknown configurations proposed by H. Ammari, G. Bao, J. Li and X. Liu et al. could be important and useful in radar and medical imaging, non-destructive testing and geophysical exploration. Last but not least, one can also find important mathematical applications regarding the estimate of the second Maxwell eigenvalues obtained by D. Pauly and the regularity of solutions to Maxwell's system at low frequencies due to P-E. Druet.
    We hope the presented papers will find a large audience and they may stimulate novel studies on electromagnetism. Finally we would like to express our gratitude to the Research Center MATHEON and the Weierstrass Institute, whose financial support made the workshop possible.
Zhaosheng Feng and Wei Feng
2014, 7(6): i-i doi: 10.3934/dcdss.2014.7.6i +[Abstract](343) +[PDF](71.8KB)
As we all know, many biological and physical systems, such as neuronal systems and disease systems, are featured by certain nonlinear and complex patterns in their elements and networks. These phenomena carry significant biological and physical information and regulate down-stream mechanism in many instances. This issue of Discrete and Continuous Dynamical Systems, Series S, comprises a collection of recent works in the general area of nonlinear differential equations and dynamical systems, and related applications in mathematical biology and engineering. The common themes of this issue include theoretical analysis, mathematical models, computational and statistical methods on dynamical systems and differential equations, as well as applications in fields of neurodynamics, biology, and engineering etc.
    Research articles contributed to this issue explore a large variety of topics and present many of the advances in the field of differential equations, dynamical systems and mathematical modeling, with emphasis on newly developed theory and techniques on analysis of nonlinear systems, as well as applications in natural science and engineering. These contributions not only present valuable new results, ideas and techniques in nonlinear systems, but also formulate a few open questions which may stimulate further study in this area. We would like to thank the authors for their excellent contributions, the referees for their tireless efforts in reviewing the manuscripts and making suggestions, and the chief editors of DCDS-S for making this issue possible. We hope that these works will help the readers and researchers to understand and make future progress in the field of nonlinear analysis and mathematical modeling.
Alain Miranville
2013, 6(6): i-i doi: 10.3934/dcdss.2013.6.6i +[Abstract](320) +[PDF](67.9KB)
The tenth edition of the French-Romanian Conference on Applied Mathematics took place in Poitiers, France, on august, 2010, and gathered around 130 scientists, mainly from France and Romania, but also from several other countries.
    This series of conferences was initiated in 1992 in Iasi. It was then decided to organize it every two years, alternatively in France and in Romania. The eleventh edition was organized in Bucarest, Romania, in august, 2012, and the twelfth one will be organized in Lyons, France, in 2014.
    This issue of DCDS S contains carefully refereed contributions from participants of the conference.
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](301) +[PDF](4231.6KB)
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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Elena Bonetti, Cecilia Cavaterra, Elisabetta Rocca and Riccarda Rossi
2013, 6(2): i-ii doi: 10.3934/dcdss.2013.6.2i +[Abstract](352) +[PDF](82.5KB)
This special volume of Discrete and Continuous Dynamical Systems - Series S is dedicated to Michel Frémond on the occasion of his 70th birthday, for his important contributions to several theoretical and applied problems in Mechanics, Thermodynamics and Engineering.

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A framework for the development of implicit solvers for incompressible flow problems
David J. Silvester, Alex Bespalov and Catherine E. Powell
2012, 5(6): 1195-1221 doi: 10.3934/dcdss.2012.5.1195 +[Abstract](473) +[PDF](1691.2KB)
This survey paper reviews some recent developments in the design of robust solution methods for the Navier--Stokes equations modelling incompressible fluid flow. There are two building blocks in our solution strategy. First, an implicit time integrator that uses a stabilized trapezoid rule with an explicit Adams--Bashforth method for error control, and second, a robust Krylov subspace solver for the spatially discretized system. Numerical experiments are presented that illustrate the effectiveness of our generic approach. It is further shown that the basic solution strategy can be readily extended to more complicated models, including unsteady flow problems with coupled physics and steady flow problems that are nondeterministic in the sense that they have uncertain input data.
Eduard Feireisl, Josef Málek and Mirko Rokyta
2012, 5(6): i-ii doi: 10.3934/dcdss.2012.5.6i +[Abstract](230) +[PDF](117.4KB)
This volume consists of five surveys that are focused on various aspects of theoretical fluid mechanics. In our opinion, mathematical analysis of fluid mechanics problems can lead to interesting and important results if the research shares at least two characteristics. First, it requires good understanding of how the models in consideration have been developed, what are their limitations, and in what situations can the models be applied. Second, the theoretical investigations must be driven by relevant and interesting problems in applications such as qualitative properties of the solution or the construction of numerical methods. Such a viewpoint has motivated the composition of the authors in this volume and resulted in five surveys that range from the constitutive theory of non-Newtonian fluids through the analysis of inhomogeneous incompressible fluid flows and the stability analysis of vortices up to the development of efficient computational methods for flows described by the incompressible Navier--Stokes equations and its generalizations.

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Dmitry Pelinovsky, Milena Stanislavova and Atanas Stefanov
2012, 5(5): i-iii doi: 10.3934/dcdss.2012.5.5i +[Abstract](385) +[PDF](125.6KB)
Partial differential equations viewed as dynamical systems on an infinite-dimensional space describe many important physical phenomena. Lately, an unprecedented expansion of this field of mathematics has found applications in areas as diverse as fluid dynamics, nonlinear optics and network communications, combustion and flame propagation, to mention just a few. In addition, there have been many recent advances in the mathematical analysis of differential difference equations with applications to the physics of Bose-Einstein condensates, DNA modeling, and other physical contexts. Many of these models support coherent structures such as solitary waves (traveling or standing), as well as periodic wave solutions. These coherent structures are very important objects when modeling physical processes and their stability is essential in practical applications. Stable states of the system attract dynamics from all nearby configurations, while the ability to control coherent structures is of practical importance as well. This special issue of Discrete and Continuous Dynamical Systems is devoted to the analysis of nonlinear equations of mathematical physics with a particular emphasis on existence and dynamics of localized modes. The unifying idea is to predict the long time behavior of these solutions. Three of the papers deal with continuous models, while the other three describe discrete lattice equations.

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Siegfried Carl, Salvatore A. Marano and Dumitru Motreanu
2012, 5(4): i-i doi: 10.3934/dcdss.2012.5.4i +[Abstract](347) +[PDF](88.0KB)
The present issue intends to provide an exposition of very recent topics and results in the qualitative study of nonlinear elliptic equations or systems such as, e.g., existence, multiplicity, and comparison principles. Emphasis is put on variational techniques, combined with topological arguments and sub-super-solution methods, in both a smooth and non-smooth framework.
    The collected papers investigate a wide range of questions. Let us mention for instance multiple solutions to elliptic equations and systems in bounded or unbounded domains, sub-super-solutions of elliptic problems whose relevant energy functionals can be non-differentiable, singular elliptic equations, asymptotically critical problems on higher dimensional spheres, local $C^1$-minimizers versus local $W^{1,p}$-minimizers.
    Each contribution is original and thoroughly reviewed.
Bernard Brighi
2012, 5(3): i-i doi: 10.3934/dcdss.2012.5.3i +[Abstract](257) +[PDF](90.4KB)
Michel Chipot was born in 1949 in Gérardmer (88), France.
    Michel started his mathematical research activities in the 70's, and defended first a "Thèse de 3ème cycle" at the university of Nancy I in 1975, where he was assistant since the end of 1971.

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