Equivariant Wave Maps Research Articles

On Gibbs measures and topological solitons of exterior equivariant wave maps

We consider k -equivariant wave maps from the exterior spatial domain \mathbb{R}^{3} \setminus B(0,1) into the target \mathbb{S}^{3} . This model has infinitely many topological solitons Q_{n,k} , which are indexed by their topological degree n\in \mathbb{Z} . For each n\in \mathbb{Z} and k\geq 1 , we prove the existence and invariance of a Gibbs measure supported on the homotopy class of Q_{n,k} . As a corollary, we obtain that soliton resolution fails for random initial data. Since soliton resolution is known for initial data in the energy space, this reveals a sharp contrast between deterministic and probabilistic perspectives.

Large deviations for (1 + 1)-dimensional stochastic geometric wave equation

We consider stochastic wave map equation on real line with solutions taking values in a d-dimensional compact Riemannian manifold. We show first that this equation has unique, global, strong in PDE sense, solution in local Sobolev spaces. The main result of the paper is a proof of the Large Deviations Principle for solutions in the case of vanishing noise.

Soliton Resolution for Critical Co-rotational Wave Maps and Radial Cubic Wave Equation

In this paper we prove the soliton resolution conjecture for all times, for all solutions in the energy space, of the co-rotational wave map equation. To our knowledge this is the first such result for all initial data in the energy space for a wave-type equation. We also prove the corresponding results for radial solutions, which remain bounded in the energy norm, of the cubic (energy-critical) nonlinear wave equation in space dimension 4.

Continuous time soliton resolution for two-bubble equivariant wave maps

Continuous time soliton resolution for two-bubble equivariant wave maps

Asymptotic stability of large energy harmonic maps under the wave map from 2D hyperbolic spaces to 2D hyperbolic spaces

In this paper, we prove that the large energy harmonic maps from H2 to H2 are asymptotically stable under the wave map equation. This work is the last paper in a series of works [46,45,44], which together prove the asymptotic stability for all “admissible” harmonic maps of arbitrary size without any symmetry assumptions.

Development of singularities of the Skyrme model

The Skyrme model is a geometric field theory and a quasilinear modification of the Nonlinear Sigma Model (Wave Maps). In this paper, we study the development of singularities for the equivariant Skyrme Model, in the strong-field limit, where the restoration of scale invariance allows us to look for self-similar blow-up behavior. After introducing the Skyrme Model and reviewing what’s known about formation of singularities in equivariant Wave Maps, we prove the existence of smooth self-similar solutions to the [Formula: see text]-dimensional Skyrme Model in the strong-field limit, and use that to conclude that the solution to the corresponding Cauchy problem blows-up in finite time, starting from a particular class of everywhere smooth initial data.

Global well-posedness for the three dimensional simplified inertial Ericksen-Leslie systems near equilibrium

We study a simplified inertial Ericksen-Leslie system for the nematic liquid crystal flow, which can be viewed as a system coupling Navier-Stokes equations and wave map equations. We prove the global existence of classical solution with initial data near equilibrium.

Biharmonic wave maps into spheres

A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed. The equation is reformulated as a conservation law and solved by a suitable Ginzburg-Landau-type approximation.

Vortex structures for some geometric flows from pseudo-Euclidean spaces

For some geometric flows (such as wave map equations, Schrödinger flows) from pseudo-Euclidean spaces to a unit sphere contained in a three-dimensional Euclidean space, we construct solutions with various vortex structures (vortex pairs, elliptic/hyperbolic vortex circles, and also elliptic vortex helices). The approaches base on the transformations associated with the symmetries of the nonlinear problems, which will lead to two-dimensional elliptic problems with resolution theory given by the finite-dimensional Lyapunov-Schmidt reduction method in nonlinear analysis.

Construction of two-bubble solutions for energy-critical wave equations

We construct pure two-bubbles for some energy-critical wave equations, that is solutions which in one time direction approach a superposition of two stationary states both centered at the origin, but asymptotically decoupled in scale. Our solution exists globally, with one bubble at a fixed scale and the other concentrating in infinite time, with an error tending to $0$ in the energy space. We treat the cases of the power nonlinearity in space dimension $6$, the radial Yang-Mills equation and the equivariant wave map equation with equivariance class $k\geq 3$. The concentration speed of the second bubble is exponential for the first two models and a power function in the last case.

Universality of Blow up Profile for Small Blow up Solutions to the Energy Critical Wave Map Equation

In this paper we introduce the channel of energy argument to the study of energy critical wave maps into the sphere. More precisely, we prove a channel of energy type inequality for small energy wave maps, and as an application we show that for a wave map that has energy just above the degree one harmonic maps and that blows up in finite time, the solution asymptotically de-couples into a regular part plus a traveling wave with small momentum, in the energy space. In particular, the only possible form of energy concentration is through the concentration of traveling waves. This is often called quantization of energy at blow up. We also give a brief review of important background results in the subcritical and critical regularity theory for the two dimensional wave maps.

Threshold for blowup for equivariant wave maps in higher dimensions

We consider equivariant wave maps from to in supercritical dimensions . Using mixed numerical and analytic methods, we show that the threshold of blowup is given by the codimension-one stable manifold of a self-similar solution with one instability. To probe self-similar blowup, we develop a novel numerical method, based on an adaptive rescaling of coordinates, which may be of independent interest in numerical studies of singularity formation.

On the full bosonic string from Minkowski space to Riemannian manifolds

We study the action of the full bosonic string for the domain being two-dimensional Minkowski space and the target a Riemannian manifold. Its critical points couple the wave map equation to a scalar and a two-form potential. Besides investigating their basic features we establish existence results for the latter.

Stability of stationary equivariant wave maps from the hyperbolic plane

In this paper we initiate the study of equivariant wave maps from $2d$ hyperbolic space, ${\Bbb H}^2$, into rotationally symmetric surfaces. This problem exhibits markedly different phenomena than its Euclidean counterpart due to the exponential volume growth of concentric geodesic spheres on the domain. In particular, when the target is ${\Bbb S}^2$, we find a family of equivariant harmonic maps ${\Bbb H}^2\to{\Bbb S}^2$, indexed by a parameter that measures how far the image of each harmonic map wraps around the sphere. These maps have energies taking all values between zero and the energy of the unique co-rotational Euclidean harmonic map, $Q_{{\rm euc}}$, from ${\Bbb R}^2$ to ${\Bbb S}^2$, given by stereographic projection. We prove that the harmonic maps are asymptotically stable for values of the parameter smaller than a threshold that is large enough to allow for maps that wrap more than halfway around the sphere. Indeed, we prove Strichartz estimates for the operator obtained by linearizing around such a harmonic map. However, for harmonic maps with energies approaching the Euclidean energy of $Q_{{\rm euc}}$, asymptotic stability via a perturbative argument based on Strichartz estimates is precluded by the existence of gap eigenvalues in the spectrum of the linearized operator. When the target is ${\Bbb H}^2$, we find a continuous family of asymptotically stable equivariant harmonic maps ${ \Bbb H}^2\to{\Bbb H}^2$ with arbitrarily small and arbitrarily large energies. This stands in sharp contrast to the corresponding problem on Euclidean space, where all finite energy solutions scatter to zero as time tends to infinity.

Equivariant wave maps on the hyperbolic plane with large energy

In this paper we continue the analysis of equivariant wave maps from 2-dimensional hyperbolic space into surfaces of revolution that was initiated in [13, 14]. When the target is the hyperbolic plane we proved in [13] the existence and asymptotic stability of a 1-parameter family of finite energy harmonic maps indexed by how far each map wraps around the target. Here we conjecture that each of these harmonic maps is globally asymptotically stable, meaning that the evolution of any arbitrarily large finite energy perturbation of a harmonic map asymptotically resolves into the harmonic map itself plus free radiation. Since such initial data exhaust the energy space, this is the soliton resolution conjecture for this equation. The main result is a verification of this conjecture for a nonperturbative subset of the harmonic maps

Asymptotic decomposition for semilinear Wave and equivariant wave map equations

In this paper we give a unified proof to the soliton resolution conjecture along a sequence of times, for the semilinear focusing energy critical wave equations in the radial case and two dimensional equivariant wave map equations, including the four dimensional radial Yang Mills equation, without using outer energy type inequalities. Such inequalities have played a crucial role in previous works with similar results. Roughly speaking, we prove that along a sequence of times $t_n\to T_+$ (the maximal time of existence), the solution decouples to a sum of rescaled solitons and a term vanishing in the energy space, plus a free radiation term in the global case or a regular part in the finite time blow up case. The main difficulty is that in general (for instance for the radial four dimensional Yang Mills case and the radial six dimensional semilinear wave case) we do not have a favorable outer energy inequality for the associated linear wave equations. Our main new input is the simultaneous use of two virial identities.

A class of large global solutions for the wave-map equation

In this paper we consider the equation for equivariant wave maps from R3+1 to S-3 and we prove global in forward time existence of certain C-infinity-smooth solutions which have infinite critical Sobolev norm (H) overdot(3/4) (R-3) x (H) overdot(1/2) (R-3). Our construction provides solutions which can moreover satisfy the additional size condition parallel to u(0, .)parallel to L-infinity(vertical bar-vertical bar >= 1) > M for arbitrarily chosen M > 0. These solutions are also stable under suitable perturbations. Our method, strongly inspired by work of Krieger and Schlag, is based on a perturbative approach around suitably constructed approximate self similar solutions.

Stability of stationary wave maps from a curved background to a sphere

We study time and space equivariant wave maps from $M\times\mathbb{R}\rightarrow S^2,$ where $M$ is diffeomorphic to a two dimensional sphere and admits an action of $SO(2)$ by isometries. We assume that metric on $M$ can be written as $dr^2+f^2(r)d\theta^2$ away from the two fixed points of the action, where the curvature is positive, and prove that stationary (time equivariant) rotationally symmetric (of any rotation number) smooth wave maps exist and are stable in the energy topology. The main new ingredient in the construction, compared with the case where $M$ is isometric to the standard sphere (considered by Shatah and Tahvildar-Zadeh [34]), is the the use of triangle comparison theorems to obtain pointwise bounds on the fundamental solution on a curved background.

MultiSymplectic Discretization of Wave Map Equations

We present a new multi-symplectic formulation of constrained Hamiltonian partial differential equations, and we study the associated local conservation laws. A multi-symplectic discretisation based on this new formulation is exemplified by means of the Euler box scheme. When applied to the wave map equation, this numerical scheme is explicit, preserves the constraint and can be seen as a generalisation of the Shake algorithm for constrained mechanical systems. Furthermore, numerical experiments show excellent conservation properties of the numerical solutions.

On the Global Stability of the Wave-map Equation in Kerr Spaces with Small Angular Momentum

This paper is motivated by the problem of the nonlinear stability of the Kerr solution for axially symmetric perturbations. We consider a model problem concerning the axially symmetric perturbations of a wave map $$\Phi $$ defined from a fixed Kerr solution $${\mathcal K}(M,a)$$ , $$0\le a \le M $$ , with values in the two dimensional hyperbolic space $${\mathbb H}^2$$ . A particular such wave map is given by the complex Ernst potential associated to the axial Killing vector-field $$\mathbf{Z}$$ of $${\mathcal K}(M,a)$$ . We conjecture that this stationary solution is stable, under small axially symmetric perturbations, in the domain of outer communication (DOC) of $${\mathcal K}(M,a)$$ , for all $$0\le a<M$$ and we provide preliminary support for its validity, by deriving convincing stability estimates for the linearized system.