Global Semigroup of Conservative Solutions of the Nonlinear Variational Wave Equation

We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation u _tt − c(u)(c(u)u _x ) _x  = 0. We allow for initial data u| _{t = 0} and u _t | _t=0 that contain measures. We assume that 0 < k^-1 \leqq c(u) \leqq k{0 < \kappa^{-1} \leqq c(u) \leqq \kappa}. Solutions of this equation may experience concentration of the energy density (u_t²+c(u)²u_x²)dx{(u_t^2+c(u)^2u_x^2){\rm d}x} into sets of measure zero. The solution is constructed by introducing new variables related to the characteristics, whereby singularities in the energy density become manageable. Furthermore, we prove that the energy may focus only on a set of times of zero measure or at points where c′(u) vanishes. A new numerical method for constructing conservative solutions is provided and illustrated with examples.     

Time Periodic Solutions to the One-Dimensional Nonlinear Wave Equation

This paper is concerned with the time periodic solutions to the one-dimensional nonlinear wave equation with either variable or constant coefficients. By adjusting the basis of L ² function space, we can circumvent the difficulties caused by η _u  = 0 and obtain the existence of a weak periodic solution, which was posed as an open problem by Baubu and Pavel in (Trans Am Math Soc 349:2035–2048, 1997). Finally, an application to the forced Sine-Gordon equation is presented to illustrate the utility of this technique.     

Variational Methods in Semilinear Wave Equation with Nonlinear Boundary Conditions and Stability Questions

We present variational approach to the semilinear equation of the vibrating string \(x_{tt}(t,y)-{\Delta } x(t,y)+l(t,y,x(t,y))=0\) in bounded domain and certain type of nonlinearity on the boundary. To this effect we derive new dual variational method. Next the question of stability of solutions with respect to initial conditions is discussed.     

Breathers for a Relativistic Nonlinear Wave Equation

 A new class of one-dimensional relativistic nonlinear wave equations with a singular δ-type nonlinear term is considered. The sense of the equations is defined according to the least-action principle. The energy and momentum conservation is established. The main results are the existence of time-periodic finite-energy solutions, the existence of global solutions and soliton-type asymptotics for a class of finite-energy initial data. (Accepted May 28, 2002) Published online November 12, 2002 Communicated by G. FRIESECKE     

Conservative Solutions to a System of Variational Wave Equations of Nematic Liquid Crystals

In this paper, we solve a system of hyperbolic equations by modelling nematic liquid crystals.     

On Boundary Damping for a Weakly Nonlinear Wave Equation

In this paper an initial-boundary value problem for a weakly nonlinear string(or wave) equation with non-classical boundary conditions is considered. Oneend of the string is assumed to be fixed and the other end of the string isattached to a spring-mass-dashpot system, where the damping generated by thedashpot is assumed to be small. This problem can be regarded as a rather simple model describing oscillationsof flexible structures such as suspension bridges or overhead transmission lines in a windfield. A multiple-timescales perturbation method will be usedto construct formal asymptotic approximations of the solution. It will also beshown that all solutions tend to zero for a sufficiently large value of thedamping parameter. For smaller values of the damping parameter it will be shownhow the string-system eventually will oscillate.     

Weak Solutions for a Class of Nonlinear Systems of Viscoelasticity

The principal focus of the article is the construction of classical weak solutions of the initial value problem for a class of systems of viscoelasticity in arbitrary spatial dimension. The class of systems studied is large enough to incorporate certain requirements dictated by frame indifference and also has a structure which allows for a variational treatment of the time-discretized problem. Weak solutions for this system are constructed under certain monotonicity hypotheses and are shown to satisfy various a priori estimates, in particular giving improved regularity for the time derivative. Also measure-valued solutions are obtained under a uniform dissipation condition, which is much weaker than monotonicity. A special case of the viscoelastic system is the gradient flow of a non-convex potential, for which measure-valued solutions are here obtained, a new result in the vectorial case. Furthermore, in this setting it is possible to show that these measure-valued solutions satisfy a certain property which ensures they coincide with the classical weak solution when this exists, as for example in the convex case where existence and uniqueness are well known. Accepted July 1, 2000?Published online December 6, 2000     

Solutions of a Nonlinear Dirac Equation with External Fields

We study the stationary Dirac equation

Traveling Wave Solutions for a Class of One-Dimensional Nonlinear Shallow Water Wave Models

In this paper we consider a class of one-dimensional nonlinear shallow water wave models that support weak solutions. We construct new traveling wave solutions for these models. Moreover, we show that these new traveling wave solutions are stable.     

Asymptotic Expansion of Solutions of an Elliptic Equation Related to the Nonlinear Schrödinger Equation

We study the radially symmetric blow-up solutions of the nonlinear Schrödinger equation. We give a method for developing such a solution in a series which represents it asymptotically.     

On a Nonlinear Model for Tumor Growth: Global in Time Weak Solutions

We investigate the dynamics of a class of tumor growth models known as mixed models. The key characteristic of these type of tumor growth models is that the different populations of cells are continuously present everywhere in the tumor at all times. In this work we focus on the evolution of tumor growth in the presence of proliferating, quiescent and dead cells as well as a nutrient. The system is given by a multi-phase flow model and the tumor is described as a growing continuum Ω with boundary ?Ω both of which evolve in time. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion and viscosity in the weak formulation.     

Quasi-Periodic Solutions and Stability for a Weakly Damped Nonlinear Quasi-Periodic Mathieu Equation

Quasi-Periodic (QP) solutions are investigated for a weakly dampednonlinear QP Mathieu equation. A double parametric primary resonance(1:2, 1:2) is considered. To approximate QP solutions, a double multiple-scales method is applied to transform the original QP oscillator to anautonomous system performing two successive reductions. In a first step,the multiple-scales method is applied to the original equation to derive afirst reduced differential amplitude-phase system having periodiccomponents. The stability of stationary solutions of this reduced systemis analyzed. In a second step, the multiple-scales method is applied again tothe first reduced system (RS) to obtain a second autonomous differentialamplitude-phase RS. The problem for approximating QP solutions of theoriginal system is then transformed to the study of stationary regimesof the induced autonomous second RS. Explicit analytical approximationsto QP solutions are obtained and comparisons to numerical integrationare provided.     

Global Weak Solutions to a Generic Two-Fluid Model

This paper deals with mathematical properties of a generic two-fluid flow model commonly used in industrial applications. More precisely, we address the question of whether available mathematical results in the case of a single-fluid governed by the compressible barotropic Navier–Stokes equations may be extended to such a two-phase model. We focus on existence of global weak solutions, linear theory and determination of eigenvalues and invariant regions.     

Asymptotics of the Solutions of the Stochastic Lattice Wave Equation

We consider the long time limit for the solutions of a discrete wave equation with weak stochastic forcing. The multiplicative noise conserves energy, and in the unpinned case also conserves momentum. We obtain a time-inhomogeneous Ornstein-Uhlenbeck equation for the limit wave function that holds for both square integrable and statistically homogeneous initial data. The limit is understood in the point-wise sense in the former case, and in the weak sense in the latter. On the other hand, the weak limit for square integrable initial data is deterministic.     

Exact Explicit Peakon and Periodic Cusp Wave Solutions for Several Nonlinear Wave Equations

Using the method of dynamical systems for six nonlinear wave equations, the exact explicit parametric representations of the solitary cusp wave solutions and the periodic cusp wave solutions are given. These parametric representations follow that when travelling systems corresponding to these nonlinear wave equations has a singular straight line, under some parameter conditions, nonanalytic travelling wave solutions must appear. Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday     

Classification of Positive Solitary Solutions of the Nonlinear Choquard Equation

In this paper, we settle the longstanding open problem concerning the classification of all positive solutions to the nonlinear stationary Choquard equation $\Delta u-u+2u\left(\frac{1}{|x|}*|u|^2\right)=0, \quad u\in H^1(\mathbb{R}^3),$ which can be considered as a certain approximation of the Hartree–Fock theory for a one component plasma as explained in Lieb and Lieb–Simon’s papers starting from 1970s. We first prove that all the positive solutions of this equation must be radially symmetric and monotone decreasing about some fixed point. Interestingly, to use the new method of moving planes introduced by Chen–Li–Ou, we deduce the problem into an elliptic system. As a key step, we transform this differential system into a system of integral equations with the help of Riesz and Bessel potentials, and then use the method of a moving plane in an integral form. Next, using radial symmetry, we deduce the uniqueness result from Lieb’s work. Our argument can be adapted well to study the radial symmetry of positive solutions of the equation in the generalized form $u=K_1*F\left(u,K_2*u\right)$ .     

Lack of Uniqueness for Weak Solutions of the Incompressible Porous Media Equation

In this work we consider weak solutions of the incompressible two-dimensional porous media (IPM) equation. By using the approach of De Lellis–Székelyhidi, we prove non-uniqueness for solutions in L ^∞ in space and time.     

Asymptotic Variational Wave Equations

We investigate the equation (u _t +(f(u)) _x ) _x =f ^{′ ′}(u) (u _x )²/2 where f(u) is a given smooth function. Typically f(u)=u ²/2 or u ³/3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation u _tt − c(u) (c(u)u _x ) _x =0 which models some liquid crystals with a natural sinusoidal c. The equation itself is also the Euler–Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.We establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function f has a Lipschitz continuous second-order derivative. In the case where f is convex, the Cauchy problem is well posed also within the class of dissipative solutions. However, when f is not convex, we show that the dissipative solutions do not depend continuously on the initial data.