We prove a dimension-free Lp(Ω)×Lq(Ω)×Lr(Ω)→L1(Ω×(0,∞)) embedding for triples of elliptic operators in divergence form with complex coefficients and subject to mixed boundary conditions on Ω, and for ...triples of exponents p,q,r∈(1,∞) mutually related by the identity 1/p+1/q+1/r=1. Here Ω is allowed to be an arbitrary open subset of Rd. Our assumptions involving the exponents and coefficient matrices are expressed in terms of a condition known as p-ellipticity. The proof utilizes the method of Bellman functions and heat flows. As a corollary, we give applications to (i) paraproducts and (ii) square functions associated with the corresponding operator semigroups, moreover, we prove (iii) inequalities of Kato–Ponce type for elliptic operators with complex coefficients. All the above results are the first of their kind for elliptic divergence-form operators with complex coefficients on arbitrary open sets. Furthermore, the approach to (ii),(iii) through trilinear embeddings seems to be new.
With initial data that are analytic in a strip, solutions to the 2D generalized Zakharov-Kuznetsov equation continue to be analytic in a strip the width of which will decrease as time goes. We obtain ...algebraic (instead of exponential) lower bounds on the decreasing rate of the uniform radius of analyticity of the solutions.
We prove almost everywhere convergence of continuous-time quadratic averages with respect to two commuting
$\mathbb {R}$
-actions, coming from a single jointly measurable measure-preserving
$\mathbb ...{R}^2$
-action on a probability space. The key ingredient of the proof comes from recent work on multilinear singular integrals; more specifically, from the study of a curved model for the triangular Hilbert transform.
In the present paper, we consider the Cauchy problem of nonlinear Schrödinger equations with a derivative nonlinearity which depends only on ū and its first derivatives. The well-posedness of the ...equation at the scaling subcritical regularity was proved by A. Grünrock (2000). We prove the well-posedness of the equation and the scattering for the solution at the scaling critical regularity by using U2 space and V2 space which are applied to prove the well-posedness and the scattering for KP-II equation at the scaling critical regularity by Hadac, Herr and Koch (2009).
The initial value problem for the generalized Korteweg-deVries equation of order three on the line is shown to be globally well posed for rough data. Our proof is based on the multilinear estimate ...and the
I
-method introduced by Colliander, Keel, Staffilani, Takaoka, and Tao.
We study the local well-posedness of the nonlinear Schrödinger equation associated to the Grushin operator with random initial data. To the best of our knowledge, no well-posedness result is known in ...the Sobolev spaces Hk when k⩽32. In this article, we prove that there exists a large family of initial data such that, with respect to a suitable randomization in Hk, k∈(1,32, almost-sure local well-posedness holds. The proof relies on bilinear and trilinear estimates.
In this paper, we show that the Fornberg–Whitham equation is Well-posed in Sobolev spaces Hs, for s>3/2, and in the periodic case. We then show that the Well-posedness is sharp in the sense that the ...continuity of the data-to-solution map is not better than continuous by using the method of approximate solutions. However, we also show that the solution map is Hölder continuous in a weaker topology. These results are based on the Well-posedness result, as well as the solution size and lifespan estimates.
This article studies the global well-posedness (GWP) for a class of defocusing, generalized sixth-order Boussinesq equations, extending a previous result obtained by Wang-Esfahani (Wang and Esfahani, ...2014) for the case when the nonlinear term is cubic.
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