相关论文: Global Regularity of Wave Maps from R^{3+1} to H^2

We prove the existence of Sinai-Ruelle-Bowen measures for a class of $C^2$ self-mappings of a rectangle with unbounded derivatives. The results can be regarded as a generalization of a well-known one dimensional Folklore Theorem on the…

We define and compute hyperbolic coordinates and associated foliations which provide a new way to describe the geometry of the standard map. We also identify a uniformly hyperbolic region and a complementary 'critical' region containing a…

We consider the space $\mathcal{D}'^r_L(M;E)$ of distributional sections of the smooth complex vector bundle $E\rightarrow M$ whose Sobolev wave front set of order $r\in\mathbb{R}$ lies in the closed conic subset $L$ of $T^*M\backslash0$.…

This paper is concerned with two-dimensional, steady, periodic water waves propagating at the free surface of water in a flow of constant vorticity over an impermeable flat bed. The motion of these waves is assumed to be governed both by…

We prove there is a class of maps $\gamma:\mathbb{T}^{2n}\rightarrow\mathbb{S}^1$ such that a conservative dynamically coherent partially hyperbolic skew-product on $\mathbb{T}^{2n}\times\mathbb{S}^1$ with fixed hyperbolic dynamics on the…

This article is concerned with the infinite depth water wave equation in two space dimensions. We consider this problem expressed in position-velocity potential holomorphic coordinates,and prove that small localized data leads to global…

The goal for this paper is twofold. Our first main objective is to develop Bahouri-Gerard type profile decompositions for waves on hyperbolic space. Recently, such profile decompositions have proved to be a versatile tool in the study of…

We propose global surjectivity theorems of differentiable maps based on second order conditions. Using the homotopy continuation method, we demonstrate that, for a $C^2$ differentiable map from a Hilbert space to a finite-dimensional…

We prove that any simple planar travelling wave solution to the membrane equation in spatial dimension $d \geq 3$ with bounded spatial extent is globally nonlinearly stable under sufficiently small compactly-supported perturbations, where…

In this paper we initiate the study of equivariant wave maps from 2d hyperbolic space into rotationally symmetric surfaces. This problem exhibits markedly different phenomena than its Euclidean counterpart due to the exponential volume…

In this paper, we prove the global existence of smooth solutions to the three-dimensional incompressible magneto-hydrodynamical system with initial data close enough to the equilibrium state, $(e_3,0).$ Compared with the the previous works…

We consider second-order hyperbolic equations with rough time-independent coefficients. Our main result is that such equations are well posed on the Hardy spaces $\mathcal{H}^{s,1}_{FIO}(\mathbb{R}^{n})$ and…

In this paper we construct smooth Riemannian metrics on the sphere which admit smooth Zoll families of minimal hypersurfaces. This generalizes a theorem of Guillemin for the case of geodesics. The proof uses the Nash-Moser Inverse Function…

We consider 1-equivariant wave maps from 1+2 dimensions to the 2-sphere. For wave maps of topological degree zero we prove global existence and scattering for energies below twice the energy of harmonic map, Q, given by stereographic…

We study the Cauchy problem for a quasilinear wave equation with low-regularity data. A space-time $L^2$ estimate for the variable coefficient wave equation plays a central role for this purpose. Assuming radial symmetry, we establish the…

We consider the full irrotational water waves system with surface tension and no gravity in dimension two (the capillary waves system), and prove global regularity and modified scattering for suitably small and localized perturbations of a…

This paper aims to establish the local and global well-posedness theory in $L^1$, inspired by the approach of Keel and Tao [Internat. Math. Res. Notices, 1998], for the forced wave map equation in the ``external'' formalism. In this…

We prove global well-posedness for a cubic, non-local Schr\"odinger equation with radially-symmetric initial data in the critical space $L^2(\R^2)$, using the framework of Kenig-Merle and Killip-Tao-Visan. As a consequence, we obtain a…

We show that on compact Riemann surfaces of negative curvature, the generalized periods, i.e. the $\nu$-th order Fourier coefficient of eigenfunctions $e_\lambda$ over a period geodesic $\gamma$ goes to 0 at the rate of…

A harmonic map from a Riemannian manifold into a Grassmannian manifold is characterized by a vector bundle, a space of sections of this bundle and a Laplace operator. We apply our main theorem, itself a generalization of a Theorem of…