Finite-Time Blow-Up of Solutions to Semilinear Wave Equations

This work studies the finite-time blow-up of solutions to the equation u_tt−Δu=F(u) in Minkowski space. We develop a new technique which simplifies some of the existing arguments. The approach we use is a modification of the so-called method of conformal compactification. In this we are motivated by the work of Christodoulou, and Baez, Segal, and Zhou on nonlinear wave equations, as well as the recent developments in the rigorous theory of nonlinear quantum fields.     

Hyperbolic mean curvature flow

In this paper we introduce the hyperbolic mean curvature flow and prove that the corresponding system of partial differential equations is strictly hyperbolic, and based on this, we show that this flow admits a unique short-time smooth solution and possesses the nonlinear stability defined on the Euclidean space with dimension larger than 4. We derive nonlinear wave equations satisfied by some geometric quantities related to the hyperbolic mean curvature flow. Moreover, we also discuss the relation between the equations for hyperbolic mean curvature flow and the equations for extremal surfaces in the Minkowski space-time.     

Thermodynamic formalization of the fluid dynamics equations for a charged dielectric in an electromagnetic field

Minkowski??s classical work underlying modern electrodynamics is described. Primary attention is given to the mathematical refinements that are required if the parameters ? and ?? depend on the properties of the dielectric fluid, i.e., the medium carrying charges in the field under study. It is shown that the motion of the medium and the accompanying evolution of the electromagnetic field are described by differential equations that are symmetric and hyperbolic in the sense of Friedrichs. This property guarantees their well-posedness. Note that this class of equations was not known in Minkowski??s time. At present, it plays an important role in the mathematical simulation of nonstationary processes and in the design of numerical algorithms. The author??s view of the mathematical foundations of Minkowski??s work is presented, which relates the latter to present-day insights into the theory of differential equations. This paper can possibly be of interest to physicists.     

Minkowski Equations in Nonlinear Magnetizable and Polarizable Moving Media

On the basis of the covariance of the Maxwell equations under Lorentz transformations, we generalize the Minkowski equations by deducing the material relations for nonlinear magnetizable and polarizable moving media.     

Nonlinear semigroups and the yang-mills equations with the metallic boundary conditions

A generalization of Segal's theorem on nonlinear perturbations of semigroups is proven. The Yang-Mills equations in a spatially bounded subset of the Minkowski space are studied under the assumption of temporal gauge. It is shown that the Cauchy problem for these equations is uniquely solvable (locally in time) if the metallic boundary conditions are imposed. An a'priori estimate of the second Sobolev norm of a 1-form is given.     

The inverse Penrose transform of a solution to the Maxwell-Dirac-Weyl field equations

An explicit family of solutions to the nonlinear coupled Maxwell-Dirac-Weyl equations in Minkowski space is presented. The abstract results of Henkin and Manin (Phys. Lett. B, 95 (1980), 405–408) show that these solutions are equivalent by the Penrose transform to a coupled system of cohomology classes and a complex line bundle on ambitwistor space, the space of null lines in Minkowski space. The explicit inverse Penrose transform of this family of solutions is computed giving explicit expressions for the line bundle (transform of the vector potential), the obstruction to extension (transform of the charge), and the two cohomology classes (transform of the Dirac-Weyl coupled spinor fields).     

Nonlinear partial differential equations associated with binormal motions of constant torsion curves in Minkowski 3-space

We give three nonlinear partial differential equations which are associated with binormal motions of constant torsion curves in Minkowski 3-space. We also give B?cklund transformations for these equations, as well as for surfaces swept out by related moving curves. As applications, from some trivial binormal motions we construct some new binormal motions.     

Symmetry of translating solutions to mean curvature flows

First, we review the authors' recent results on translating solutions to mean curvature flows in Euclidean space as well as in Minkowski space, emphasizing on the asymptotic expansion of rotationally symmetric solutions. Then we study the sufficient condition for which the translating solution is rotationally symmetric. We will use a moving plane method to show that this condition is optimal for the symmetry of solutions to fully nonlinear elliptic equations without ground state condition.     

Local existence in retarded time under a weak decay on complete null cones

In the previous paper(see Li and Zhu(2014)), for a characteristic problem with not necessarily small initial data given on a complete null cone decaying like that in the work of the stability of Minkowski spacetime by Christodoulou and Klainerman(1993), we proved the local existence in retarded time, which means the solution to the vacuum Einstein equations exists in a uniform future neighborhood, while the global existence in retarded time is the weak cosmic censorship conjecture. In this paper, we prove that the local existence in retarded time still holds when the data is assumed to decay slower, like that in Bieri's work(2007)on the extension to the stability of Minkowski spacetime. Such decay guarantees the existence of the limit of the Hawking mass on the initial null cone, when approaching to infinity, in an optimal way.     

Algorithms for computing Minkowski operators and their application in differential games

The Minkowski operators are considered, which extend the concepts of the Minkowski sum and difference to the case where one of the summands depends on an element of the other term. The properties of these operators are examined. Convolution methods of computer geometry and algorithms for computing the values of the Minkowski operators are developed. These algorithms are used to construct epsilon-optimal control strategies in a nonlinear differential game with a nonconvex target set. The errors of the proposed algorithms are estimated in detail. Numerical results for the conflicting control of a nonlinear pendulum are presented.     

Aesthetic fields: a two-dimensional lattice solution satisfying integrability

In our previous paper we obtained examples of lattice solutions to the aesthetic field equations. A drawback with these solutions was that the integrability equations were not satisfied. In this paper we find a solution to the aesthetic field equations which describes a two-dimensional lattice structure. The integrability equations are satisfied in this case. We work within a six-dimensional framework in this paper. Three of the coordinates are real and three of the coordinates are pure imaginary. It is a generalization of the Minkowski version of aesthetic field theory.     

A priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary

We study the relativistic Euler equations on the Minkowski spacetime background. We make assumptions on the equation of state and the initial data that are relativistic analogs of the well-known physical vacuum boundary condition, which has played an important role in prior work on the non-relativistic compressible Euler equations. Our main result is the derivation, relative to Lagrangian (also known as co-moving) coordinates, of local-in-time a priori estimates for the solution. The solution features a fluid-vacuum boundary, transported by the fluid four-velocity, along which the hyperbolicity of the equations degenerates. In this context, the relativistic Euler equations are equivalent to a degenerate quasilinear hyperbolic wave-map-like system that cannot be treated using standard energy methods.     

Hierarchies of Huygens' Operators and Hadamard's Conjecture

We develop a new unified approach to the problem of constructing linear hyperbolic partial differential operators that satisfy Huygens' principle in the sense of J. Hadamard. The underlying method is essentially algebraic and based on a certain nonlinear extension of similarity (gauge) transformations in the ring of analytic differential operators.The paper provides a systematic and self-consistent review of classical and recent results on Huygens' principle in Minkowski spaces. Most of these results are carried over to more general pseudo-Riemannian spaces with the metric of a plane gravitational wave.A particular attention is given to various connections of Huygens' principle with integrable systems and the soliton theory. We discuss the link to nonlinear KdV-type evolution equations, Darboux–Bäcklund transformations and the bispectral problem in the sense of Duistermaat, Grünbaum and Wilson.     

Cauchy-Riemann equations in Minkowski plane

The properties of the symmetry and ordering of Minkowski plane are discussed by using hyperbolic imaginary unit and elliptic imaginary unit of Clifford algebra, and the representations of Cauchy-Riemann equations are given in Minkowski plane.     

Pointwise Estimates and Regularity in Geometric Optics and Other Generated Jacobian Equations

The study of reflector surfaces in geometric optics necessitates the analysis of certain nonlinear equations of Monge‐Ampère type known as generated Jacobian equations. This class of equations, whose general existence theory has been recently developed by Trudinger, goes beyond the framework of optimal transport. We obtain pointwise estimates for weak solutions of such equations under minimal structural and regularity assumptions, covering situations analogous to those of costs satisfying the A3‐weak condition introduced by Ma, Trudinger, and Wang in optimal transport. These estimates are used to develop a C^1,α regularity theory for weak solutions of Aleksandrov type. The results are new even for all known near‐field reflector/refractor models, including the point source and parallel beam reflectors, and are applicable to problems in other areas of geometry, such as the generalized Minkowski problem.© 2017 Wiley Periodicals, Inc.     

Uniqueness results for the Minkowski problem extended to hedgehogs

The classical Minkowski problem has a natural extension to hedgehogs, that is to Minkowski differences of closed convex hypersurfaces. This extended Minkowski problem is much more difficult since it essentially boils down to the question of solutions of certain Monge-Ampère equations of mixed type on the unit sphere \mathbbSⁿ \mathbb{S}^n of ℝ ⁿ⁺¹. In this paper, we mainly consider the uniqueness question and give first results.     

Solution globale des equations de Maxwell-Dirac-Klein-Gordon

We prove the global existence on Minkowski space time of a solution of the Cauchy problem for the non linear system of coupled Maxwell, Dirac and Klein-Gordon equations, for small data with appropriate decay at space-like infinity. The method uses the conformal mapping of Minkowski space time onto a bounded open set of the Einstein cylinder.     

Applications of the extended tanh method for coupled nonlinear evolution equations

In this work, we establish exact solutions for coupled nonlinear evolution equations. The extended tanh method is used to construct solitary and soliton solutions of nonlinear evolution equations. The extended tanh method presents a wider applicability for handling nonlinear wave equations.     

Application of Exp-function method to wave solutions of the Sine-Gordon and Ostrovsky equations

In this work, the Exp-function method is employed to find new wave solutions for the Sine-Gordon and Ostrovsky equation. The equations are simplified to the nonlinear partial differential equations and then different types of exact solutions are extracted by this method. It is shown that the Exp-function method is a powerful analytical method for solving other nonlinear equations occurring in nonlinear physical phenomena. Results are presented in contour plots that show the different values of effective parameters on the velocity profiles.     

Classification of Local Conservation Laws of Maxwell's Equations

A complete and explicit classification of all independent local conservation laws of Maxwell's equations in four dimensional Minkowski space is given. Besides the elementary linear conservation laws, and the well-known quadratic conservation laws associated to the conserved stress-energy and zilch tensors, there are also chiral quadratic conservation laws which are associated to a new conserved tensor. The chiral conservation laws possess odd parity under the electric–magnetic duality transformation of Maxwell's equations, in contrast to the even parity of the stress-energy and zilch conservation laws. The main result of the classification establishes that every local conservation law of Maxwell's equations is equivalent to a linear combination of the elementary conservation laws, the stress-energy and zilch conservation laws, the chiral conservation laws, and their higher order extensions obtained by replacing the electromagnetic field tensor by its repeated Lie derivatives with respect to the conformal Killing vectors on Minkowski space. The classification is based on spinorial methods and provides a direct, unified characterization of the conservation laws in terms of Killing spinors.