Asymptotic Stability of High-dimensional Zakharov–Kuznetsov Solitons

We prove that solitons (or solitary waves) of the Zakharov–Kuznetsov (ZK) equation, a physically relevant high dimensional generalization of the Korteweg–de Vries (KdV) equation appearing in Plasma Physics, and having mixed KdV and nonlinear Schrödinger (NLS) dynamics, are strongly asymptotically stable in the energy space. We also prove that the sum of well-arranged solitons is stable in the same space. Orbital stability of ZK solitons is well-known since the work of de Bouard [Proc R Soc Edinburgh 126:89–112, 1996]. Our proofs follow the ideas of Martel [SIAM J Math Anal 157:759–781, 2006] and Martel and Merle [Math Ann 341:391–427, 2008], applied for generalized KdV equations in one dimension. In particular, we extend to the high dimensional case several monotonicity properties for suitable half-portions of mass and energy; we also prove a new Liouville type property that characterizes ZK solitons, and a key Virial identity for the linear and nonlinear part of the ZK dynamics, obtained independently of the mixed KdV–NLS dynamics. This last Virial identity relies on a simple sign condition which is numerically tested for the two and three dimensional cases with no additional spectral assumptions required. Possible extensions to higher dimensions and different nonlinearities could be obtained after a suitable local well-posedness theory in the energy space, and the verification of a corresponding sign condition.     

Strichartz Estimates and Moment Bounds for the Relativistic Vlasov–Maxwell System

We consider the relativistic Vlasov–Maxwell system with data of unrestricted size and without compact support in momentum space. In the two-dimensional and the two-and-a-half-dimensional cases, Glassey–Schaeffer proved (Commun Math Phys 185:257–284, 1997; Arch Ration Mech Anal 141:331–354, 1998; Arch Ration Mech Anal. 141:355–374, 1998) that for regular initial data with compact momentum support this system has unique global in time classical solutions. In this work we do not assume compact momentum support for the initial data and instead require only that the data have polynomial decay in momentum space. In the two-dimensional and the two-and-a-half-dimensional cases, we prove the global existence, uniqueness and regularity for solutions arising from this class of initial data. To this end we use Strichartz estimates and prove that suitable moments of the solution remain bounded. Moreover, we obtain a slight improvement of the temporal growth of the \({L^\infty_x}\) norms of the electromagnetic fields compared to Glassey and Schaeffer (Commun Math Phys 185:257–284, 1997; Arch Ration Mech Anal 141:355–374, 1998). In the three-dimensional case, we apply Strichartz estimates and moment bounds to show that a regular solution can be extended as long as \({{\|p_0^{\theta} f \|_{L^{q}_{x}L^1_{p}}}}\) remains bounded for \({\theta > \frac{2}{q}}\), \({2 < q \leqq \infty}\). This improves previous results of Pallard (Indiana Univ Math J 54(5):1395–1409, 2005; Commun Math Sci 13(2):347–354, 2015).     

On the Regularity Set and Angular Integrability for the Navier–Stokes Equation

We investigate the size of the regular set for suitable weak solutions of the Navier–Stokes equation, in the sense of Caffarelli–Kohn–Nirenberg (Commun Pure Appl Math 35:771–831, 1982). We consider initial data in weighted Lebesgue spaces with mixed radial-angular integrability, and we prove that the regular set increases if the data have higher angular integrability, invading the whole half space \({\{t > 0\}}\) in an appropriate limit. In particular, we obtain that if the \({L^{2}}\) norm with weight \({|x|^{-\frac12}}\) of the data tends to 0, the regular set invades \({\{t > 0\}}\); this result improves Theorem D of Caffarelli et al. (Commun Pure Appl Math 35:771–831, 1982).     

Beltrami Fields with a Nonconstant Proportionality Factor are Rare

We consider the existence of Beltrami fields with a nonconstant proportionality factor f in an open subset U of \({\mathbb{R}^3}\). By reformulating this problem as a constrained evolution equation on a surface, we find an explicit differential equation that f must satisfy whenever there is a nontrivial Beltrami field with this factor. This ensures that there are no nontrivial regular solutions for an open and dense set of factors f in the C^k topology, \({k\geqq 7}\). In particular, there are no nontrivial Beltrami fields whenever f has a regular level set diffeomorphic to the sphere. This provides an explanation of the helical flow paradox of Morgulis et al. (Commun Pure Appl Math 48:571–582, 1995).     

Long-time Stability in Systems of Conservation Laws,Using Relative Entropy/Energy

We study the long-time stability of shock-free solutions of hyperbolic systems of conservation laws, under an arbitrarily large initial disturbance in L ²∩ L ^∞. We use the relative entropy method, a robust tool which allows us to consider rough and large disturbances. We display practical examples in several space dimensions, for scalar equations as well as isentropic gas dynamics. For full gas dynamics, we use a trick from Chen [1], in which the estimate is made in terms of the relative mechanical energy instead of the relative mathematical entropy.     

On Lie symmetries and soliton solutions of $$$$-dimensional Bogoyavlenskii equations

The present article is devoted to find some invariant solutions of the \((2+1)\)-dimensional Bogoyavlenskii equations using similarity transformations method. The system describes \((2+1)\)-dimensional interaction of a Riemann wave propagating along y-axis with a long wave along x-axis. All possible vector fields, commutative relations and symmetry reductions are obtained by using invariance property of Lie group. Meanwhile, the method reduces the number of independent variables by one, which leads to the reduction of Bogoyavlenskii equations into a system of ordinary differential equations. The system so obtained is solved under some parametric restrictions and provides invariant solutions. The derived solutions are much efficient to explain the several physical properties depending upon various existing arbitrary constants and functions. Moreover, some of them are more general than previously established results (Peng and Shen in Pramana 67:449–456, 2006; Malik et al. in Comput Math Appl 64:2850–2859, 2012; Zahran and Khater in Appl Math Model 40:1769–1775, 2016; Zayed and Al-Nowehy in Opt Quant Electron 49(359):1–23, 2017). In order to provide rich physical structures, the solutions are supplemented by numerical simulation, which yield some positons, negatons, kinks, wavefront, multisoliton and asymptotic nature.     

Global Weak Solutions for Kolmogorov–Vicsek Type Equations with Orientational Interactions

We study the global existence and uniqueness of weak solutions to kinetic Kolmogorov–Vicsek models that can be considered as non-local, non-linear, Fokker–Planck type equations describing the dynamics of individuals with orientational interactions. This model is derived from the discrete Couzin–Vicsek algorithm as mean-field limit (Bolley et al., Appl Math Lett, 25:339–343, 2012; Degond et al., Math Models Methods Appl Sci 18:1193–1215, 2008), which governs the interactions of stochastic agents moving with a velocity of constant magnitude, that is, the corresponding velocity space for these types of Kolmogorov–Vicsek models is the unit sphere. Our analysis for L^p estimates and compactness properties take advantage of the orientational interaction property, meaning that the velocity space is a compact manifold.     

On Reducibility of 1d Wave Equation with Quasiperiodic in Time Potentials

In this paper we use a KAM theorem of Grébert and Thomann (Commun Math Phys 307:383–427, 2011) to prove the reducibility of the 1d wave equation with Dirichlet boundery conditions on \([0,\pi ]\) with a quasi-periodic in time potential under some symmetry assumptions. From Mathieu–Hill operator’s known results (Eastham in The spectral theory of periodic differential operators, Hafner, New York, 1974; Magnus and Winkler in Hill’s equation, Wiley-Interscience, London, 1969) and Bourgain’s techniques (Commun Math Phys 204:207–247, 1999), we prove that for any \(\epsilon \) small enough, there exist a \(0<m_{\epsilon }\le 1\) and one solution \(u_{\epsilon }(t,x)\) with

$$\begin{aligned} \Vert u_{\epsilon }(t_n,x)\Vert _{H^1({\mathbb {T}})}\rightarrow \infty , \qquad |t_n|\rightarrow \infty , \end{aligned}$$

where \(u_{\epsilon }(t,x)\) satisfies 1d wave equation

$$\begin{aligned} u_{tt}-u_{xx}+m_{\epsilon }u-\epsilon \cos 2t u=0, \end{aligned}$$

with Dirichlet boundery conditions on \([0,\pi ]\).

     

Multiple Periodic Solutions of an Equation with State-Dependent Delay

Given \({N \in \mathbb N}\) we prove the existence, for parameter values in a certain range, of N distinct periodic solutions of a state-dependent delay equation studied by Walther (Differ Integral Equ 15:923–944, 2002).     

Anisotropic Regularity Conditions for the Suitable Weak Solutions to the 3D Navier–Stokes Equations

We are concerned with the problem, originated from Seregin (159–200, 2007), Seregin (J. Math. Sci. 143: 2961–2968, 2007), Seregin (Russ. Math. Surv. 62:149–168, 2007), what are minimal sufficiently conditions for the regularity of suitable weak solutions to the 3D Navier–Stokes equations. We prove some interior regularity criteria, in terms of either one component of the velocity with sufficiently small local scaled norm and the rest part with bounded local scaled norm, or horizontal part of the vorticity with sufficiently small local scaled norm and the vertical part with bounded local scaled norm. It is also shown that only the smallness on the local scaled L ² norm of horizontal gradient without any other condition on the vertical gradient can still ensure the regularity of suitable weak solutions. All these conclusions improve pervious results on the local scaled norm type regularity conditions.     

Bounded Correctors in Almost Periodic Homogenization

We show that certain linear elliptic equations (and systems) in divergence form with almost periodic coefficients have bounded, almost periodic correctors. This is proved under a new condition we introduce which quantifies the almost periodic assumption and includes (but is not restricted to) the class of smooth, quasiperiodic coefficient fields which satisfy a Diophantine-type condition previously considered by Kozlov (Mat Sb (N.S), 107(149):199–217, 1978). The proof is based on a quantitative ergodic theorem for almost periodic functions combined with the new regularity theory recently introduced by Armstrong and Shen (Pure Appl Math, 2016) for equations with almost periodic coefficients. This yields control on spatial averages of the gradient of the corrector, which is converted into estimates on the size of the corrector itself via a multiscale Poincaré-type inequality.     

Topological and Measure-Theoretical Entropies of Nonautonomous Dynamical Systems

We consider the dynamics of a nonautonomous dynamical system determined by a sequence of continuous self-maps \(f_n:X \rightarrow X,\) where \( n \in {\mathbb {N}},\) defined on a compact metric space X. Applying the theory of the Carathéodory structures, elaborated by Pesin (Dimension Theory in Dynamical Systems. Chicago Lectures in Mathematics. The University of Chicago Press, Chicago, 1997), we construct a Carathéodory structure whose capacity coincides with the topological entropy of the considered system. Generalizing the notion of local measure entropy, introduced by Brin and Katok (in: Palis (ed) Geometric Dynamics, Lecture Notes in Mathematics. Springer, Berlin 1983) for a single map, to a nonautonomous dynamical system we provide a lower and upper estimations of the topological entropy by local measure entropies. The theorems of the paper generalize results of Kawan (Nonautonomous Stoch Dyn Syst 1:26–52, 2013) and of Feng and Huang (J Funct Anal 263:2228–2254, 2012). Also, we construct a new entropy-like invariant such the entropy of a sequence \(\{f_n:X \rightarrow X\}_{n=1}^{\infty }\) of Lipschitz continuous maps with the same Lipschitz constant \(L >1,\) restricted to a subset \(Y\subset X,\) is upper bounded by Hausdorff dimension of Y multiplied by the logarithm of the Lipschitz constant L. This gives a generalizations of results of Dai et al. (Sci China Ser A 41:1068–1075, 1998) and Misiurewicz (Discret Contin Dyn Syst 10:827–833, 2004).     

The magnetic field of a permanent hollow cylindrical magnet

Based on the rational version of MAXWELL’s equations according to TRUESDELL and TOUPIN or KOVETZ, cf. (Kovetz in Electromagnetic theory, Oxford University Press, Oxford, 2000; Truesdell and Toupin in Handbuch der Physik, Bd. III/1, Springer, Berlin, pp 226–793; appendix, pp 794–858, 2000), we present, for stationary processes, a closed-form solution for the magnetic flux density of a hollow cylindrical magnet. Its magnetization is constant in axial direction. We consider MAXWELL’s equations in regular and singular points that are obtained by rational electrodynamics, adapted to stationary processes. The magnetic flux density is calculated analytically by means of a vector potential. We obtain a solution in terms of complete elliptic integrals. Therefore, numerical evaluation can be performed in a computationally efficient manner. The solution is written in dimensionless form and can easily be applied to cylinders of arbitrary shape. The relation between the magnetic flux density and the magnetic field is linear, and an explicit relation for the field is presented. With a slight modification the result can be used to obtain the field of a solid cylindrical magnet. The mathematical structure of the solution and, in particular, singularities are discussed.     

Dynamic Transitions of Generalized Burgers Equation

In this article, we study the dynamic transition for the one dimensional generalized Burgers equation with periodic boundary condition. The types of transition are dictated by the sign of an explicitly given parameter b, which is derived using the dynamic transition theory developed by Ma and Wang (Phase transition dynamics. Springer, New York, 2014). The rigorous result demonstrates clearly the types of dynamics transition in terms of length scale l, dispersive parameter δ and viscosity ν.     

Complete Global and Bifurcation Analysis of a Stoichiometric Predator–Prey Model

Loladze et al. (Bull Math Biol 62:1137–1162, 2000) proposed a highly cited stoichiometric predator–prey system, which is nonsmooth, and thus it is extremely difficult to analyze its global dynamics. The main challenge comes from the phase plane fragmentation and parameter space partitioning in order to perform a detailed and complete global stability and bifurcation analysis. Li et al. (J Math Biol 63:901–932, 2011) firstly discussed its global dynamical behavior with Holling type I functional response and found that the system has no limit cycles, and the internal equilibrium is globally asymptotically stable if it exists. Secondly, for the system with Holling type II functional response, Li et al. (2011) fixed all parameters (with realistic values) except K to perform the bifurcation analysis and obtained some interesting phenomena, for instance, the appearance of bistability and many bifurcation types. The aim of this paper is to provide a complete global analysis for the system with Holling type II functional response without fixing any parameter. Our analysis shows that the model has far richer dynamics than those found in the previous paper (Li et al. 2011), for example, four types of bistability appear: besides the bistability between an internal equilibrium and a limit cycle as shown in Li et al. (2011), the other three bistabilities occur between an internal equilibrium and a boundary equilibrium, between two internal equilibria, or between a boundary equilibrium and a limit cycle. In addition, this paper rigorously provides all possible bifurcation passways of this stoichiometric model with Holling type II functional response.     

Magnetostriction of a sphere: stress development during magnetization and residual stresses due to the remanent field

Based on the principles of rational continuum mechanics and electrodynamics (see Truesdell and Toupin in Handbuch der Physik, Springer, Berlin, 1960 or Kovetz in Electromagnetic theory, Oxford University Press, Oxford, 2000), we present closed-form solutions for the mechanical displacements and stresses of two different magnets. Both magnets are initially of spherical shape. The first (hard) magnet is uniformly magnetized and deforms due to the field induced by the magnetization. In the second problem of a (soft) linear-magnetic sphere, the deformation is caused by an applied external field, giving rise to magnetization. Both problems can be used for modeling parts of general magnetization processes. We will address the similarities between both settings in context with the solutions for the stresses and displacements. In both problems, the volumetric Lorentz force density vanishes. However, a Lorentz surface traction is present. This traction is determined from the magnetic flux density. Since the obtained displacements and stresses are small in magnitude, we may use Hooke’s law with a small-strain approximation, resulting in the Lamé-Navier equations of linear elasticity theory. If gravity is neglected and azimuthal symmetry is assumed, these equations can be solved in terms of a series. This has been done by Hiramatsu and Oka (Int J Rock Mech Min Sci Geomech Abstr 3(2):89–90, 1966) before. We make use of their series solution for the displacements and the stresses and expand the Lorentz tractions of the analyzed problems suitably in order to find the expansion coefficients. The resulting algebraic system yields finite numbers of nonvanishing coefficients. Finally, the resulting stresses, displacements, principal strains and the Lorentz tractions are illustrated and discussed.     

Two-Scale <Emphasis Type="Italic">Γ</Emphasis>-Convergence of Integral Functionals and its Application to Homogenisation of Nonlinear High-Contrast Periodic Composites

An analytical framework is developed for passing to the homogenisation limit in (not necessarily convex) variational problems for composites whose material properties oscillate with a small period ε and that exhibit high contrast of order \({\varepsilon^{-1}}\) between the constitutive, “stress-strain”, response on different parts of the period cell. The approach of this article is based on the concept of “two-scale Γ-convergence”, which is a kind of “hybrid” of the classical Γ-convergence (De Giorgi and Franzoni in Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur (8)58:842–850, 1975) and the more recent two-scale convergence (Nguetseng in SIAM J Math Anal 20:608–623, 1989). The present study focuses on a basic high-contrast model, where “soft” inclusions are embedded in a “stiff” matrix. It is shown that the standard Γ-convergence in the L ^p -space fails to yield the correct limit problem as \({\varepsilon \to 0,}\) due to the underlying lack of L ^p -compactness for minimising sequences. Using an appropriate two-scale compactness statement as an alternative starting point, the two-scale Γ-limit of the original family of functionals is determined via a combination of techniques from classical homogenisation, the theory of quasiconvex functions and multiscale analysis. The related result can be thought of as a “non-classical” two-scale extension of the well-known theorem by Müller (Arch Rational Mech Anal 99:189–212, 1987).     

A Rigidity Result for a Reduced Model of a Cubic-to-Orthorhombic Phase Transition in the Geometrically Linear Theory of Elasticity

In this article we study a simplified two-dimensional model for a cubic-to-orthorhombic phase transition occurring in certain shape-memory-alloys. In the low temperature regime the linear theory of elasticity predicts various possible patterns of martensite arrangements: Apart from the well known laminates this phase transition displays additional structures involving four martensitic variants—so called crossing twins.Introducing a variational model including surface energy, we show that these structures are rigid under small energy perturbations. Combined with an upper bound construction this gives the optimal scaling behavior of incompatible microstructures. These results are related to papers by Capella and Otto (Commun. Pure Appl. Math. 62(12):1632–1669, 2009; Proc. R. Soc. Edinb., Sect. A, Math. 142:273–327, 2012) as well as to a paper by Dolzmann and Müller (Meccanica 30:527–539, 1995).     

On the interplay between fast reaction and slow diffusion in the concrete carbonation process: a matched-asymptotics approach

A matched-asymptotics approach is proposed to show the occurrence of two distinct characteristic length scales in the carbonation process. The separation of these scales arises due to the strong competition between reaction and diffusion effects. We show that for sufficiently large times τ the width of the carbonated region is proportional to \(\sqrt{\tau}\), while the width of the reaction front is proportional to \(\tau^{\frac{p-1}{2(p+1)}}\) for carbonation-reaction rates with a power law structure like k[CO₂] ^p [Ca(OH)₂] ^q , where k>0 and p,q>1 and identify the proportionality coefficient asymptotically. We emphasize the occurrence of a water barrier in the reaction zone which may hinder the penetration of CO₂ by locally filling with water air parts of the pores. This non-linear effect may be one of the causes why a purely linear extrapolation of accelerated carbonation test results to natural carbonation settings is (even theoretically) not reasonable. Finally, we compare our asymptotic penetration law against measured penetration depths from Bune (Zum Karbonatisierungsbedingten Verlust der Dauerhaftigkeit von Außenbauteilen aus Stahlbeton, 1994). The novelty consists in the fact that the factor multiplying \(\sqrt{\tau}\) is now identified asymptotically by solving a non-linear system of ordinary differential equations, and hence, fitting arguments are not necessary to estimate its size. We offer an alternative to the (asymptotic) \(\sqrt{\tau}\) expression of the carbonation-front position obtained in Papadakis et al. (AIChE J. 35:1639, 1989).     

Micromorphic approach for gradient-extended thermo-elastic–plastic solids in the logarithmic strain space

The coupled thermo-mechanical strain gradient plasticity theory that accounts for microstructure-based size effects is outlined within this work. It extends the recent work of Miehe et al. (Comput Methods Appl Mech Eng 268:704–734, 2014) to account for thermal effects at finite strains. From the computational viewpoint, the finite element design of the coupled problem is not straightforward and requires additional strategies due to the difficulties near the elastic–plastic boundaries. To simplify the finite element formulation, we extend it toward the micromorphic approach to gradient thermo-plasticity model in the logarithmic strain space. The key point is the introduction of dual local–global field variables via a penalty method, where only the global fields are restricted by boundary conditions. Hence, the problem of restricting the gradient variable to the plastic domain is relaxed, which makes the formulation very attractive for finite element implementation as discussed in Forest (J Eng Mech 135:117–131, 2009) and Miehe et al. (Philos Trans R Soc A Math Phys Eng Sci 374:20150170, 2016).