Global Solution of 2D Hyperbolic Liquid Crystal System for Small Initial Data

Global strong solution to the semi-linear Keller–Segel system of parabolic–parabolic type with small data in scale invariant spaces

The first target of this article is the local well-posedness question for 1D quasilinear Schrödinger equations with cubic nonlinearities. The study of this class of problems, in all dimensions, was initiated in pioneering work of Kenig-Ponce-Vega for localized initial data, and then continued by Marzuola-Metcalfe-Tataru for initial data in Sobolev spaces. Our objective here is to fully redevelop the study of this problem in the 1D case, and to prove a sharp local well-posedness result. The second goal of this article is to consider the long-time/global existence of solutions for the same problem. This is motivated by a broad conjecture formulated by the authors in earlier work, which reads as follows: “Cubic defocusing dispersive one dimensional flows with small initial data have global dispersive solutions”; the conjecture was initially proved for a class of semilinear Schrödinger type models. Our work here establishes the above conjecture for 1D quasilinear Schrödinger flows. Precisely, we show that if the problem has phase rotation symmetry and is conservative and defocusing, then small data in Sobolev spaces yields global, scattering solutions. This is the first result of this type for 1D quasilinear dispersive flows where no localization condition is imposed on the data. Furthermore, we prove the global well-posedness at the minimal Sobolev regularity as in our local well-posedness result. The defocusing condition is essential in our global result. Without it, the authors have conjectured that small, $\epsilon $ ϵ size data yields long-time solutions on the $\epsilon ^{-8}$ ϵ − 8 time-scale. A third goal of this paper is to also prove this second conjecture for 1D quasilinear Schrödinger flows, also at minimal regularity.

We study scattering theory for the semilinear wave equation Utt Au = JujP-lu in two space dimensions. We show that if p > p0 = (3+x/ii)/2, the scattering operator exists for smooth and small data. The lower bound p0 of p is considered to be optimal (see Glassey [6, 7], Schaeffer [ 18]). Our result is an extension of the results by Strauss [19], Klainerman [10], and Mochizuki and Motai [14, 15]. The construction of the scattering operator for small data does not follow directly from the proofs in [7, 13, 20 and 22] concerning the global existence of solutions for the Cauchy problem of the above equation with small initial data given at t = 0 in two space dimensions, because we have to consider the integral equation with unbounded integral region associated to the above equation: u(x, t) = u(x, t)? I[ [(II )(,s dy ds, ? 27r IJ Jix-yl<t-s /(tS)2 Ix y12 for t e R, where u-(x, t) is a solution of utt Au = 0 which u(x, t) approaches asymptotically as t -+ -oo. The proof of the basic estimate for the above integral equation is more difficult and complicated than that for the Cauchy problem of utt Au = Iu Pu in two space dimensions.

Global Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces: High dimensions

We study the (n + 1)-dimensional generalization of the dispersionless Kadomtsev–Petviashvili (dKP) equation, a universal equation describing the propagation of weakly nonlinear, quasi-one-dimensional waves in n + 1 dimensions, and arising in several physical contexts, such as acoustics, plasma physics and hydrodynamics. For n = 2, this equation is integrable, and has been recently shown to be a prototype model equation in the description of the two-dimensional wave breaking of localized initial data. We construct an exact solution of the (n + 1)-dimensional model containing an arbitrary function of one variable, corresponding to its parabolic invariance, describing waves, constant on their paraboloidal wave front, breaking simultaneously in all points of it. Then, we use such a solution to build a uniform approximation of the solution of the Cauchy problem, for small and localized initial data, showing that such a small and localized initial data evolving according to the (n + 1)-dimensional dKP equation break, in the long time regime, if and only if 1 ⩽ n ⩽ 3, i.e., in physical space. Such a wave breaking takes place, generically, in a point of the paraboloidal wave front, and the analytic aspects of it are given explicitly in terms of the small initial data.

We prove existence of global solutions to a semilinear massless Dirac system with small initial data. We study solutions in generalised Sobolev spaces suggested by S. Klainerman. Our approach is based on using conservation law of charge together with Sobolev type weighted estimates for the spinor field. Our result seems to be sharp in a view of blowing-up results obtained by F. John (see [7]). We also study decay properties of the spinor field. With similar methods we prove global existence for a nonlinear wave equation in three space dimension. The same equation was studied by T. Sideris [14] and H. Takamura [15]. They proved global existence for spherically symmetrical initial data. In this work we remove this condition on the initial data.

Under the internal dissipative condition, the Cauchy problem for inhomogeneous quasilinear hyperbolic systems with small initial data admits a unique global C1 solution, which exponentially decays to zero as t → +∞, while if the coefficient matrix Θ of boundary conditions satisfies the boundary dissipative condition, the mixed initial-boundary value problem with small initial data for quasilinear hyperbolic systems with nonlinear terms of at least second order admits a unique global C1 solution, which also exponentially decays to zero as t → +∞. In this paper, under more general conditions, the authors investigate the combined effect of the internal dissipative condition and the boundary dissipative condition, and prove the global existence and exponential decay of the C1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems with small initial data. This stability result is applied to a kind of models, and an example is given to show the possible exponential instability if the corresponding conditions are not satisfied.

Global solutions to semilinear parabolic systems for small data

For a class of special three-dimensional quasl\ilinear wqve equations, we study the blowup mechanism of classical soulutions . More precisely,under the nondegenerate conditions, any radially symmetric soul\tion with small initial data is shown to develop singualritiws in the second ordeer deriva tives while hte irst order derivatioves and itself remain continuous , moreover the vlowup of solution is of "cusp type".

Small Data Blowup for Systems of Semilinear Wave Equations with Different Propagation Speeds in Three Space Dimensions

In this paper and the companion work [J. Funct. Anal. 281 (2021)], we prove that the Schrödinger map flows from \mathbb{R}^d with d\ge 2 to compact Kähler manifolds with small initial data in critical Sobolev spaces are global. The main difficulty compared with the constant sectional curvature case is that the gauged equation now is not self-contained due to the curvature part. Our main idea is to use a novel bootstrap-iteration scheme to reduce the gauged equation to an approximate constant curvature system in finite times of iteration. This paper together with the companion work solves the open problem raised by Tataru.

We study scattering theory for the semilinear wave equation utt -Au = \u\p~lu in two space dimensions.We show that if p > po = (3+VT7)/2, the scattering operator exists for smooth and small data.The lower bound po of p is considered to be optimal (see Glassey [6,7], Schaeffer [18]).Our result is an extension of the results by Strauss [19], Klainerman [10], and Mochizuki and Motai [14,15].The construction of the scattering operator for small data does not follow directly from the proofs in [7, 13, 20 and 22] concerning the global existence of solutions for the Cauchy problem of the above equation with small initial data given at t = 0 in two space dimensions, because we have to consider the integral equation with unbounded integral region associated to the above equation:for t e R, where u^(x, t) is a solution of uu -Au = 0 which u(x, t) approaches asymptotically as t - -oo .The proof of the basic estimate for the above integral equation is more difficult and complicated than that for the Cauchy problem of Uu -Au = \u\p~iu in two space dimensions.

Global existence of weak solutions to quasilinear degenerate Keller–Segel systems of parabolic–parabolic type with small data

This paper is concerned with the time decay rates of the N−species Vlasov–Poisson system with small data in the whole space. The global existence and large time behaviors are obtained in and more higher dimensional space. For the proof, the classical (for , n ≥ 4) and the modified (for ) vector field method and the bootstrap argument are mainly employed. Compared to the unipolar case, there are some crucial new ideas introduced to handle the multi‐species case, such as a new bootstrap assumption with some necessary parameters and the multipolar version of vector field method with new coefficients corresponding to different species charged particles, respectively.

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