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• 1.
Gifu University, Japan.
Eindhoven University of Technology.
On uniqueness of a weak solution of one-dimensional concrete carbonation problem2011In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 29, no 4, p. 1345-1365Article in journal (Refereed)

In our previous works we studied a one-dimensional free-boundary model related to the aggressive penetration of gaseous carbon dioxide in unsaturated concrete. Essentially, global existence and uniqueness of weak solutions to the model were obtained when the initial functions are bounded on the domain. In this paper we investigate the well-posedness of the problem for the case when the initial functions belong to a $\displaystyle{{L}}^{{2}}-$ class. Specifically, the uniqueness of weak solutions is proved by applying the dual equation method.

• 2.
Moscow MV Lomonosov State Univ, Fac Computat Math & Cybernet, Moscow 119991, Russia.
Linnaeus University, Faculty of Technology, Department of Mathematics. Linnaeus University, Faculty of Technology, Department of Mathematics.
Ergodicity criteria for non-expanding transformations of 2-adic spheres2014In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 34, no 2, p. 367-377Article in journal (Refereed)

In the paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of discrete dynamical systems < f; S2-r (a)> on 2-adic spheres S2-r (a) of radius 2(-r), r >= 1, centered at some point a from the ultrametric space of 2-adic integers Z(2). The map f: Z(2) -> Z(2) is assumed to be non-expanding and measure-preserving; that is, f satisfies a Lipschitz condition with a constant 1 with respect to the 2-adic metric, and f preserves a natural probability measure on Z(2), the Haar measure mu(2) on Z(2) which is normalized so that mu(2)(Z(2)) = 1.

• 3.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Keldysh Institute of Applied Mathematics, Russian Academy of Science.
Symmetries of evolution equations with non-local operators and applications to the Boltzmann equation2009In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 24, no 1, p. 35-57Article in journal (Refereed)

In this paper we consider Lie group symmetries of evolutionequations with non-local operators in context of applications tononlinear kinetic equations. As an illustration we consider theBoltzmann equation and calculate the admitted group of pointtransformations.

• 4.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Dipartimento di Matematica e Informatica Universita di Perugia.
On nonlocal symmetries generated by recursion operators : second-order evolution equations2017In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 37, no 8, p. 4239-4247Article in journal (Refereed)

We introduce a new type of recursion operator to generate a class of nonlocal symmetries for second-order evolution equations in 1+1 dimensions, namely those evolution equations which allow the complete integration of their stationary equations. We show that this class of evolution equations is C-integrable (linearizable by a point transformation). We also discuss some applications.

• 5.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
Fraunhofer Chalmers Res Ctr Ind Math, SE-41288 Gothenburg, Sweden..
Spectral Properties Of Renormalization For Area-Preserving Maps2016In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 36, no 7, p. 3651-3675Article in journal (Refereed)

Area-preserving maps have been observed to undergo a universal period-doubling cascade, analogous to the famous Feigenbaum-Coullet-Tresser period doubling cascade in one-dimensional dynamics. A renormalization approach has been used by Eckmann, Koch and Wittwer in a computer-assisted proof of existence of a conservative renormalization fixed point. Furthermore, it has been shown by Gaidashev, Johnson and Martens that infinitely renormalizable maps in a neighborhood of this fixed point admit invariant Cantor sets with vanishing Lyapunov exponents on which dynamics for any two maps is smoothly conjugate. This rigidity is a consequence of an interplay between the decay of geometry and the convergence rate of renormalization towards the fixed point. In this paper we prove a result which is crucial for a demonstration of rigidity: that an upper bound on this convergence rate of renormalizations of infinitely renormalizable maps is sufficiently small.

• 6. Geyer, A.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Traveling wave solutions of a highly nonlinear shallow water equation2018In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 38, no 3, p. 1567-1604Article in journal (Refereed)

Motivated by the question whether higher-order nonlinear model equations, which go beyond the Camassa-Holm regime of moderate amplitude waves, could point us to new types of waves profiles, we study the traveling wave solutions of a quasilinear evolution equation which models the propagation of shallow water waves of large amplitude. The aim of this paper is a complete classification of its traveling wave solutions. Apart from symmetric smooth, peaked and cusped solitary and periodic traveling waves, whose existence is well-known for moderate amplitude equations like Camassa-Holm, we obtain entirely new types of singular traveling waves: periodic waves which exhibit singularities on both crests and troughs simultaneously, waves with asymmetric peaks, as well as multi-crested smooth and multi-peaked waves with decay. Our approach uses qualitative tools for dynamical systems and methods for integrable planar systems.

• 7.
Delft Univ Technol, Fac EEMCS, Delft Inst Appl Math, Mekelweg 4, NL-2628 CD Delft, Netherlands..
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
SHALLOW WATER MODELS FOR STRATIFIED EQUATORIAL FLOWS2019In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 39, no 8, p. 4533-4545Article in journal (Refereed)

Our aim is to study the effect of a continuous prescribed density variation on the propagation of ocean waves. More precisely, we derive KdV-type shallow water model equations for unidirectional flows along the Equator from the full governing equations by taking into account a prescribed but arbitrary depth-dependent density distribution. In contrast to the case of constant density, we obtain for each fixed water depth a different model equation for the horizontal component of the velocity field. We derive explicit formulas for traveling wave solutions of these model equations and perform a detailed analysis of the effect of a given density distribution on the depth-structure of the corresponding traveling waves.

• 8. Kaloshin, V.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits2006In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 15, no 2, p. 611-640Article in journal (Refereed)

Let M be a compact manifold of dimension three with a nondegenerate volume form Omega and Diff(Omega)(r) (M) be the space of C-r-smooth (Omega-) volume-preserving difffeomorphisms of M with 2 <= r <= infinity. In this paper we prove two results. One of them provides the existence of a Newhouse domain N in Diff(Omega)(r)(M). The proof is based on the theory of normal forms [13], construction of certain renormalization limits, and results from [23, 26, 28, 32]. To formulate the second one, associate to each diffeomorphism a sequence P-n(f) which gives for each n the number of isolated periodic points of f of period n. The main result of this paper states that for a Baire generic diffeomorphism f in N, the number of periodic points P-n(f) grows with n faster than any prescribed sequence of numbers {a(n)} (n is an element of Z+) along a subsequence, i.e., P-ni (f) > ani for some n(i) -> with infinity i -> infinity. The strategy of the proof is similar to the one of the corresponding 2-dimensional non volume-preserving result [16]. The latter one is, in its turn, based on the Gonchenko-Shilnikov-Turaev Theorem [8, 9].

• 9.
Seoul Natl Univ, Dept Math Sci, Seoul 08826, South Korea..
Seoul Natl Univ, Dept Math Sci, Seoul 08826, South Korea.;Korea Inst Adv Study, Seoul 02455, South Korea.. KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
HOMOGENIZATION OF THE BOUNDARY VALUE FOR THE DIRICHLET PROBLEM2019In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 39, no 12, p. 6843-6864Article in journal (Refereed)

In this paper, we give a mathematically rigorous proof of the averaging behavior of oscillatory surface integrals. Based on ergodic theory, we find a general geometric condition which we call irrational direction dense condition, abbreviated as IDDC, under which the averaging takes place. It should be stressed that IDDC does not imply any control on the curvature of the given surface. As an application, we prove homogenization for elliptic systems with Dirichlet boundary data, in C-1-domains.

• 10.
University of California, United States .
Weak geodesic flow and global solutions of the Hunter-saxton equation2007In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 18, no 4, p. 643-656Article in journal (Refereed)

We show how global weak solutions of the Hunter-Saxton equation can be naturally constructed using the geometric interpretation of the equation as the Euler equation for the geodesic flow on an L2-sphere. The approach involves forming a weak extension of the geodesic flow and relating it to a corresponding weak formulation of the Hunter-Saxton equation.

• 11.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Norwegian Univ Sci & Technol, Dept Math Sci, N-7491 Trondheim, Norway.
Infinity-harmonic potentials and their streamlines2019In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 39, no 8, p. 4731-4746Article in journal (Refereed)

We consider certain solutions of the Infinity-Laplace Equation in planar convex rings. Their ascending streamlines are unique while the descending ones may bifurcate. We prove that bifurcation occurs in the generic situation and as a consequence, the solutions cannot have Lipschitz continuous gradients.

• 12. Lukkassen, Dag
Narvik University College, 8505 Narvik, Norway. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Multiscale homogenization of monotone operators2008In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 22, no 3, p. 711-727Article in journal (Refereed)

In this paper we prove a generalization of the iterated homogenization theorem for monotone operators, proved by Lions et al. in [ ] and [ ]. Our results enable us to homogenize more realistic models of multiscale structures.

• 13.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Entropy range problems and actions of locally normal groups2009In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 25, no 3, p. 981-989Article in journal (Refereed)

This paper deals with the problem of finding the range of entropy values resulting from actions of discrete amenable groups by automorphisms of compact abelian groups. When the acting group G is locally normal, we obtain an entropy formula and show that the full range of entropy values [0, infinity] occurs for actions of G. We consider related entropy range problems, give sufficient conditions for zero entropy and, as a consequence, verify the known relationship between completely positive entropy and mixing for these actions.

• 14.
Department of Mathematical Sciences, Chalmers University of Technology, Sweden.
Department of Mathematics, University of Bergen, Norway.
Integrability of Nonholonomically Coupled Oscillators2014In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 34, no 3, p. 1121-1130Article in journal (Refereed)

We study a family of nonholonomic mechanical systems. These systems consist of harmonic oscillators coupled through nonholonomic constraints. The family includes the contact oscillator, which has been used as a test problem for numerical methods for nonholonomic mechanics. The systems under study constitute simple models for continuously variable transmission gearboxes.     The main result is that each system in the family is integrable reversible with respect to the canonical reversibility map on the cotangent bundle. By using reversible Kolmogorov--Arnold--Moser theory, we then establish preservation of invariant tori for reversible perturbations. This result explains previous numerical observations, that some discretisations of the contact oscillator have favourable structure preserving properties.

• 15.
University West, Department of Engineering Science, Division of Natural Sciences and Electrical and Surveying Engineering.
University of Copenhagen, Department of Mathematical Sciences.
Simple skew category algebras associated with minimal partially defined dynamical systems2013In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 33, no 9, p. 4157-4171Article in journal (Refereed)

In this article, we continue our study of category dynamical systems, that is functors s from a category G to Topop, and their corresponding skew category algebras. Suppose that the spaces s(e), for e ∈ ob(G), are compact Hausdorff. We show that if (i) the skew category algebra is simple, then (ii) G is inverse connected, (iii) s is minimal and (iv) s is faithful. We also show that if G is a locally abelian groupoid, then (i) is equivalent to (ii), (iii) and (iv). Thereby, we generalize results by Öinert for skew group algebras to a large class of skew category algebras.

• 16.
CMUP and Dep. de Matemática Pura, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal.
Centro de Matemática da Universidade do Porto (CMUP) and Dep. de Matemática Pur, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal. School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, United Kingdom.
Generating functions for Hopf bifurcation with Sn-symmetry2009In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 25, no 3, p. 823-842Article in journal (Refereed)

Hopf bifurcation in the presence of the symmetric group $\displaystyle{S}_{{n}}$ (acting naturally by permutation of coordinates) is a problem with relevance to coupled oscillatory systems. To study this bifurcation it is important to construct the Taylor expansion of the equivariant vector field in normal form. We derive generating functions for the numbers of linearly independent invariants and equivariants of any degree, and obtain recurrence relations for these functions. This enables us to determine the number of invariants and equivariants for all $\displaystyle{n}$, and show that this number is independent of $\displaystyle{n}$ for sufficiently large $\displaystyle{n}$. We also explicitly construct the equivariants of degree three and degree five, which are valid for arbitrary $\displaystyle{n}$.

• 17.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences, Engineering and Mathematics.
Noncommutative AKNS systems and multisoliton solutions to the matrix sine-Gordon equation2009In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, no Suppl, p. 678-690Article in journal (Refereed)

The main result is a very general solution formula for the noncommutative AKNS system, extending work by Bauhardt and P¨oppe. As anapplication, we construct for the matrix sine-Gordon equation N-soliton solutions analogous to the multisoliton solutions for the KdV equation due toGoncharenko.

• 18.
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
Infinitely many solutions for some singular elliptic problems2013In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 33, no 1, p. 321-333Article in journal (Refereed)

We prove the existence of an unbounded sequence of critical points of the functional J(lambda) (u) = 1/p integral(RN) parallel to x vertical bar(-a)del(k)u vertical bar(p) - lambda h(x)parallel to x vertical bar(-(a+k))u vertical bar(p) - 1/q integral(RN) Q(x)parallel to x vertical bar(-b) u vertical bar(q) related to the Caffarelli-Kohn-Nirenberg inequality and its higher order variant by Lin. We assume Q <= 0 at 0 and infinity and consider two essentially different cases: h equivalent to 1 and h in a certain weighted Lebesgue space.

• 19.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Almost every interval translation map of three intervals is finite type2014In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 34, no 5, p. 2307-2314Article in journal (Refereed)

Interval translation maps (ITMs) are a non-invertible generalization of interval exchange transformations (IETs). The dynamics of finite type ITMs is similar to IETs, while infinite type ITMs are known to exhibit new interesting effects. In this paper, we prove the finiteness conjecture for the ITMs of three intervals. Namely, the subset of ITMs of finite type contains an open, dense, and full Lebesgue measure subset of the space of ITMs of three intervals. For this, we show that any ITM of three intervals can be reduced either to a rotation or to a double rotation.

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