# American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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## Discrete & Continuous Dynamical Systems - S

2016 , Volume 9 , Issue 1

Issue on fluid dynamics and electromagnetism: Theory and numerical approximation

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2016, 9(1): iii-ix doi: 10.3934/dcdss.2016.9.1iii +[Abstract](471) +[PDF](280.3KB)
Abstract:
The research of Professor Paolo Secchi concerns the theory of partial differential equations, especially from fluid dynamics.

2016, 9(1): i-i doi: 10.3934/dcdss.2016.9.1i +[Abstract](338) +[PDF](87.2KB)
Abstract:
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.
2016, 9(1): xi-xvii doi: 10.3934/dcdss.2016.9.1xi +[Abstract](482) +[PDF](240.2KB)
Abstract:
The scientific activity of Professor Alberto Valli has been mainly devoted to three different subjects: theoretical analysis of partial differential equations in fluid dynamics; domain decomposition methods; numerical approximation of problems arising in low-frequency electromagnetism.

2016, 9(1): 1-13 doi: 10.3934/dcdss.2016.9.1 +[Abstract](478) +[PDF](389.3KB)
Abstract:
In this paper we prove the global existence of large amplitude finite energy solutions for a system describing Quantum Fluids with nonlinear nonlocal interaction terms. The system may also (but not necessarily) include dissipation terms which do not provide any help to get the global existence. The method is based on the polar factorization of the wave function (which somehow generalizes the WKB method), the construction of approximate solutions via a fractional step argument and the deduction of Strichartz type estimates for the approximate solutions. Finally local smoothing and a compactness argument of Lions Aubin type allow to show the convergence.
2016, 9(1): 15-32 doi: 10.3934/dcdss.2016.9.15 +[Abstract](453) +[PDF](447.2KB)
Abstract:
We are concerned with the problem of the global (in time) existence of weak solutions to hyperbolic systems of conservation laws, in one spatial dimension. First, we provide a survey of the different facets of a technique that has been used in several papers in the last years: the path decomposition. Then, we report on two very recent results that have been achieved by means of suitable applications of this technique. The first one concerns a system of three equations arising in the dynamic modeling of phase transitions, the second one is the famous Euler system for nonisentropic fluid flow. In both cases, the results concern classes of initial data with possibly large total variation.
2016, 9(1): 33-42 doi: 10.3934/dcdss.2016.9.33 +[Abstract](460) +[PDF](561.7KB)
Abstract:
Isogeometric collocation mixed methods for spatial rods are presented and studied. A theoretical analysis of stability and convergence is available. The proposed schemes are locking-free, irrespective of the selected approximation spaces.
2016, 9(1): 43-52 doi: 10.3934/dcdss.2016.9.43 +[Abstract](459) +[PDF](369.5KB)
Abstract:
In these notes we consider second order linear elliptic boundary value problems in the framework of different spaces on continuous functions. We appeal to a general formulation which contains some interesting particular cases as, for instance, a new class of functional spaces, called here Hölog spaces and denoted by the symbol $\, C^{0,\,\lambda}_\alpha(\overline{\Omega}) \,,$ $\,0 \leq\,\lambda<\,1\,,$ and $\,\alpha \in\,\mathbb{R}\,.$ One has the following inclusions $$C^{0,\,\lambda+\,\epsilon}(\overline{\Omega})\subset \,C^{0,\,\lambda}_\alpha(\overline{\Omega})\subset \,C^{0,\,\lambda}(\overline{\Omega}) \subset \,C^{0,\,\lambda,}_{-\alpha}(\overline{\Omega}) \subset\,C^{0,\,\lambda-\,\epsilon}(\overline{\Omega})\,,$$ for $\,\alpha>\,0\,$ ($\epsilon >\,0\,$ arbitrarily small). Roughly speaking, for each fixed $\,\lambda\,,$ the family $\, C^{0,\,\lambda}_\alpha(\overline{\Omega}) \,$ is a refinement of the single Hölder classical space $\, C^{0,\,\lambda}(\overline{\Omega})=\,C^{0,\,\lambda}_0(\overline{\Omega})\,.$ On the other hand, for $\,\lambda=\,0\,$ and $\,\alpha>\,0\,,$ $\,C^{0,\,0}_\alpha(\overline{\Omega})=\,\, D^{0,\,\alpha}(\overline{\Omega})\,$ is a Log space. The more interesting feature is that, as for classical Hölder (and Sobolev) spaces, full regularity occurs. namely, for each $\,\lambda>\,0\,$ and arbitrary real $\,\alpha\,,$ $\,\nabla^2\,u$ and $\,f\,$ enjoy the same $\, C^{0,\,\lambda}_\alpha(\overline{\Omega}) \,$ regularity. All the above setup is presented as part of a more general picture.
2016, 9(1): 53-71 doi: 10.3934/dcdss.2016.9.53 +[Abstract](432) +[PDF](427.3KB)
Abstract:
We consider a class of nonlinear elliptic systems and we prove regularity up to the boundary for second order derivatives. In the proof we trace carefully the dependence on the various parameters of the problem, in order to establish, in a further work, results for more general systems.
2016, 9(1): 73-88 doi: 10.3934/dcdss.2016.9.73 +[Abstract](513) +[PDF](404.1KB)
Abstract:
We prove that the entropy for an $L^\infty$-solution to a scalar conservation laws with continuous initial data is concentrated on a countably $1$-rectifiable set. To prove this result we introduce the notion of Lagrangian representation of the solution and give regularity estimates on the solution.
2016, 9(1): 89-107 doi: 10.3934/dcdss.2016.9.89 +[Abstract](625) +[PDF](1081.7KB)
Abstract:
The aim of this paper is to provide a survey of the state of the art in the finite element approach to the Immersed Boundary Method (FE-IBM) which has been investigated by the authors during the last decade. In a unified setting, we present the different formulation proposed in our research and highlight the advantages of the one based on a distributed Lagrange multiplier (DLM-IBM) over the original FE-IBM.
2016, 9(1): 109-123 doi: 10.3934/dcdss.2016.9.109 +[Abstract](412) +[PDF](735.4KB)
Abstract:
We propose a parallel algorithm for the numerical solution of the eikonal-diffusion equation, by means of a dynamic domain decomposition technique. The new method is an extension of the patchy domain decomposition method presented in [5] for first order Hamilton-Jacobi-Bellman equations. Using the connection with stochastic optimal control theory, the semi-Lagrangian scheme underlying the original method is modified in order to deal with (possibly degenerate) diffusion. We show that under suitable relations between the discretization parameters and the diffusion coefficient, the parallel computation on the proposed dynamic decomposition can be faster than that on a static decomposition. Some numerical tests in dimension two are presented, in order to show the features of the proposed method.
2016, 9(1): 125-137 doi: 10.3934/dcdss.2016.9.125 +[Abstract](361) +[PDF](434.9KB)
Abstract:
We report some recent results on weak and semi-strong solutions to a coupled hyperbolic-elliptic system of Von Karman type on ${IR}^{2m}$, $m \in {IN}_{\geq 2}$.
2016, 9(1): 139-155 doi: 10.3934/dcdss.2016.9.139 +[Abstract](376) +[PDF](287.5KB)
Abstract:
Motivated by the formation of brine channels, this paper is devoted to a continuum model for salt separation and phase transition in saline water. The mass density and the concentrations of salt and ice are the pertinent variables describing saline water. Hence the balance of mass is considered for the single constituents (salt, water, ice). To keep the model as simple as possible, the balance of momentum and energy are considered for the mixture as a whole. However, due to the internal structure of the mixture, an extra-energy flux is allowed to occur in addition to the heat flux. Also, the mixture is allowed to be viscous. The constitutive equations involve the dependence on the temperature, the mass density of the mixture, the salt concentration and the ice concentration, in addition to the stretching tensor, and the gradient of temperature and concentrations. The balance of mass for the single constituents eventually result in the evolution equations for the concentrations. A whole set of constitutive equations compatible with thermodynamics are established. A free energy function is given which allows for capturing the main feature which occurs during the freezing of the salted water. That is, the salt entrapment in small regions (brine channels) where the cryoscopic effect forbids complete ice formation.
2016, 9(1): 157-172 doi: 10.3934/dcdss.2016.9.157 +[Abstract](515) +[PDF](427.1KB)
Abstract:
We consider weak solutions of the instationary Navier-Stokes system in general unbounded smooth domains $\Omega\subset \mathbb{R}^3$ and discuss several criteria to prove that the weak solution is locally or globally in time a strong solution in the sense of Serrin. Since the usual Stokes operator cannot be defined on all types of unbounded domains we have to replace the space $L^q(\Omega)$, $q>2$, by $\tilde L^q(\Omega) = L^q(\Omega) \cap L^2(\Omega)$ and Serrin's class $L^r(0,T;L^q(\Omega))$ by $L^r(0,T;\tilde L^q(\Omega))$ where $2< r <\infty$, $3< q <\infty$ and $\frac{2}{r} + \frac{3}{q} =1$.
2016, 9(1): 173-183 doi: 10.3934/dcdss.2016.9.173 +[Abstract](481) +[PDF](362.9KB)
Abstract:
We consider the Euler-Cahn-Hilliard system proposed by Lowengrub and Truskinovsky describing the motion of a binary mixture of compressible fluids. We show that the associated initial-value problem possesses infinitely many global-in-time weak solutions for any finite energy initial data. A modification of the method of convex integration is used to prove the result.
2016, 9(1): 185-214 doi: 10.3934/dcdss.2016.9.185 +[Abstract](614) +[PDF](7513.4KB)
Abstract:
We deal with the discretization of generalized transient advection problems for differential forms on bounded spatial domains. We pursue an Eulerian method of lines approach with explicit timestepping. Concerning spatial discretization we extend the jump stabilized Galerkin discretization proposed in $[$ H. HEUMANN and R.HIPTMAIR, Stabilized Galerkin methods for magnetic advection, Math. Modelling Numer. Analysis, 47 (2013), pp.1713--1732$]$ to forms of any degree and, in particular, advection velocities that may have discontinuities resolved by the mesh. A rigorous a priori convergence theory is established for Lipschitz continuous velocities, conforming meshes and standard finite element spaces of discrete differential forms. However, numerical experiments furnish evidence of the good performance of the new method also in the presence of jumps of the advection velocity.
2016, 9(1): 215-234 doi: 10.3934/dcdss.2016.9.215 +[Abstract](381) +[PDF](419.8KB)
Abstract:
The Giroux correspondence and the notion of a near force-free magnetic field are used to topologically characterize near force-free magnetic fields which describe a variety of physical processes, including plasma equilibrium. As a byproduct, the topological characterization of force-free magnetic fields associated with current-carrying links, as conjectured by Crager and Kotiuga, is shown to be necessary and conditions for sufficiency are given. Along the way a paradox is exposed: The seemingly unintuitive mathematical tools, often associated to higher dimensional topology, have their origins in three dimensional contexts but in the hands of late-onset visually impaired topologists. This paradox was previously exposed in the context of algorithms for the visualization of three-dimensional magnetic fields. For this reason, the paper concludes by developing connections between mathematics and cognitive science in this specific context.
2016, 9(1): 235-253 doi: 10.3934/dcdss.2016.9.235 +[Abstract](450) +[PDF](1502.2KB)
Abstract:
A numerical scheme based on Nédélec finite elements has been recently introduced to solve the eigenvalue problem for the curl operator in simply connected domains. This topological assumption is not just a technicality, since the eigenvalue problem is ill-posed on multiply connected domains, in the sense that its spectrum is the whole complex plane. However, additional constraints can be added to the eigenvalue problem in order to recover a well-posed problem with a discrete spectrum. Vanishing circulations on each non-bounding cycle in the complement of the domain have been chosen as additional constraints in this paper. A mixed weak formulation including a Lagrange multiplier (that turns out to vanish) is introduced and shown to be well-posed. This formulation is discretized by Nédélec elements, while standard finite elements are used for the Lagrange multiplier. Spectral convergence is proved as well as a priori error estimates. It is also shown how to implement this finite element discretization taking care of these additional constraints. Finally, a numerical test to assess the performance of the proposed methods is reported.
2016, 9(1): 255-267 doi: 10.3934/dcdss.2016.9.255 +[Abstract](405) +[PDF](420.7KB)
Abstract:
We study existence and uniqueness of regular solutions to the Navier-Stokes initial boundary value problem in bounded or exterior domains $\Omega$ ($\partial\Omega$ sufficiently smooth) under the assumption $v_\circ$ in $L^n(\Omega)$, sufficiently small, and we prove global in time existence. The results are known in literature (see Remark 3), however the proof proposed here seems shorter, and we give a result concerning the behavior in time of the $L^q$-norm ($q\in[n,\infty]$) of the solutions and of the $L^n$-norm of the time derivative, with a sort of continuous dependence on the data, which, as far as we know, are new, and are close to the ones of the solution to the Stokes problem. Moreover, the constant for the $L^q$-estimate is independent of $q$.
2016, 9(1): 269-287 doi: 10.3934/dcdss.2016.9.269 +[Abstract](581) +[PDF](285.8KB)
Abstract:
We aim to provide a finite element analysis for the elastoacoustic vibration problem. We use a dual-mixed variational formulation for the elasticity system and combine the lowest order Lagrange finite element in the fluid domain with the reduced symmetry element known as PEERS and introduced for linear elasticity in [1]. We show that the resulting global nonconforming scheme provides a correct spectral approximation and we prove quasi-optimal error estimates. Finally, we confirm the asymptotic rates of convergence by numerical experiments.
2016, 9(1): 289-313 doi: 10.3934/dcdss.2016.9.289 +[Abstract](572) +[PDF](570.6KB)
Abstract:
We present a recent result [23] for the free boundary problem for contact discontinuities in ideal compressible magnetohydrodynamics (MHD). They are characteristic discontinuities with no flow across the discontinuity for which the pressure, the magnetic field and the velocity are continuous whereas the density and the entropy may have a jump. Under the Rayleigh-Taylor sign condition $[\partial p/\partial N]<0$ on the jump of the normal derivative of the pressure satisfied at each point of the unperturbed contact discontinuity, we prove the well-posedness in Sobolev spaces of the linearized problem for 2D planar MHD flows. This is a necessary step to prove a local-in-time existence theorem [24] for the original nonlinear free boundary problem provided that the Rayleigh-Taylor sign condition is satisfied at each point of the initial discontinuity. The uniqueness of a solution to this problem follows already from the basic a priori estimate deduced for the linearized problem.
2016, 9(1): 315-342 doi: 10.3934/dcdss.2016.9.315 +[Abstract](674) +[PDF](569.0KB)
Abstract:
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.
2016, 9(1): 343-362 doi: 10.3934/dcdss.2016.9.343 +[Abstract](662) +[PDF](523.2KB)
Abstract:
We study an atherosclerosis model described by a reaction-diffusion system of three equations, in one dimension, with homogeneous Neumann boundary conditions. The method of upper and lower solutions and its associated monotone iteration (the monotone iterative method) are used to establish existence, uniqueness and boundedness of global solutions for the problem. Upper and lower solutions are derived for the corresponding steady-state problem. Moreover, solutions of Cauchy problems defined for time-dependent system are presented as alternatives upper and lower solutions. The stability of constant steady-state solutions and the asymptotic behavior of the time-dependent solutions are studied.

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