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It is known since the works of Zariski that the essential difficulty in the local uniformization problem is met already in the case of valuations of height one. In this paper we prove that local uniformization of schemes and non-archimedean analytic spaces rigorously follows from the case of valuations of height one. For non-archimedean spaces this result reduces the problem to studying local structure of smooth Berkovich spaces. [ABSTRACT FROM AUTHOR]

In this paper, we obtain global small solutions and decay estimates for the two-dimensional MHD boundary layer in Gevrey space without any structural assumptions, generalizing the results of [20] in analytic space. The proof method is mainly inspired by [14] and [26] , using new auxiliary functions and finer structural analysis to overcome the difficulty of the loss of derivatives and then we obtain the global well-posedness of the MHD boundary layer in the Gevrey 3 2 space. [ABSTRACT FROM AUTHOR]

In this note, we provide a corrected proof of Lemma 3.2 in [4]. We also point out some improvements to Remark 5.5 in the same paper. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the global well-posedness of solutions to magnetohydrodynamics (MHD) boundary layer equations in analytic function spaces. When the initial data is a small perturbation around a selected profile, and such a profile is governed by an one dimensional heat equation with a source term, we establish the global in time existence and uniqueness of analytic solutions to the two dimensional (2D) MHD boundary layer equations. It is noted that the far-field state of velocity is not required to be small initially, but decays to zero as time tends to infinity with suitable decay rates. The whole analysis is divided into two parts: When the initial far-field states are not small, we reformulate the original problem into a small perturbation problem by extracting a suitable background profile which is governed by a heat equation, then we prove a long-time existence of solutions, and the lower bound of lifespan of solutions is given in term of the small parameter of initial perturbation; After establishing the long-time existence, combining the decay properties of far-field state and the properties of solutions to the heat equation, it is shown that both the solution obtained in the first part and the background profile indeed satisfy some smallness requirements when time goes to the lower bound of lifespan of solutions. Then we can show the global existence of solutions after this moment. [ABSTRACT FROM AUTHOR]

In this paper we extend vessel theory, or equivalently, the theory of overdetermined 2 D systems to the Pontryagin space setting. We focus on realization theorems of the various characteristic functions associated to such vessels. In particular, we develop an indefinite version of de Branges-Rovnyak theory over real compact Riemann surfaces. To do so, we use the theory of contractions in Pontryagin spaces and the theory of analytic kernels with a finite number of negative squares. Finally, we utilize the indefinite de Branges-Rovnyak theory on compact Riemann surfaces in order to prove a Beurling type theorem on indefinite Hardy spaces on finite bordered Riemann surfaces. [ABSTRACT FROM AUTHOR]

In this paper we introduce the family of spaces R M (p , q) , 1 ≤ p , q ≤ + ∞. They are spaces of holomorphic functions in the unit disc with average radial integrability. This family contains the classical Hardy spaces (when p = ∞) and Bergman spaces (when p = q). We characterize the inclusion between R M (p 1 , q 1) and R M (p 2 , q 2) depending on the parameters. For 1 < p , q < ∞ , our main result provides a characterization of the dual spaces of R M (p , q) by means of the boundedness of the Bergman projection. We show that R M (p , q) is separable if and only if q < + ∞. In fact, we provide a method to build isomorphic copies of ℓ ∞ in R M (p , ∞). [ABSTRACT FROM AUTHOR]