Bilinear Estimates Research Articles

White noise driven Ostrovsky equation

The current paper is devoted to the Cauchy problem for the white noise driven Ostrovsky equation with positive dispersion. We obtain the local well-posedness for the initial data u0(⋅,ω)∈Hs(R) (a.e. ω∈Ω) which is F0-measurable with s>−34 and Φ∈L20,s∩L20(L2(R),H˙s,−12+ϵ(R)), and the globally well-posedness for the initial data u0(⋅,ω)∈L2(R) (a.e. ω∈Ω) which is F0-measurable and Φ∈L20,0∩L20(L2(R),H˙0,−12+ϵ(R)). The key ingredients that we used in this paper are some bilinear estimates in Xs,b-type spaces given in subsection 2.1, Itô formula, the BDG inequality and stopping time techniques as well as frequency truncated technique. Our method can be applied to the case γ=0, which is just the KdV equation, thus, our result improves the local well-posedness result of A. de Bouard, A. Debussche and Y. Tsutsumi (1999) [3].

Growth-in-time of higher Sobolev norms of solutions to the 1D Dirac–Klein–Gordon system

We study the growth-in-time of higher order Sobolev norms of solutions to the Dirac–Klein–Gordon (DKG) equations in one space dimension. We show that these norms grow at most polynomially-in-time. The main ingredients in the proof are the upside-down [Formula: see text]-method which was introduced by Colliander, Keel, Staffilani, Takaoka and Tao, and bilinear null-form estimates.

The Cauchy problem for higher-order modified Camassa–Holm equations on the circle

In this paper, we investigate the Cauchy problem for the shallow water type equation ut+∂x2n+1u+12∂x(u2)+∂x(1−∂x2)−1u2+12ux2=0with low regularity data in the periodic settings. Firstly, we proved that the bilinear estimate related to the nonlinear term of the equation in space Ws (defined in page 5) is invalid with s<−n2+1. Then, the locally well-posed of the Cauchy problem for the periodic shallow water-type equation is obtained in Hs(T) with s>−n+32,n≥2 for arbitrary initial data. Thus, our result improves the result of Himonas and Misiolek (Commun. Partial Differ. Equ, 23(1998), 123–139.), where they have proved that the problem is locally well-posed for small initial data in Hs(T) with s≥−n2+1,n∈N+ with the aid of the standard Fourier restriction norm method.

Multi-scale bilinear restriction estimates for general phases

We prove (adjoint) bilinear restriction estimates for general phases at different scales in the full non-endpoint mixed norm range, and give bounds with a sharp and explicit dependence on the phases. These estimates have applications to high-low frequency interactions for solutions to partial differential equations, as well as to the linear restriction problem for surfaces with degenerate curvature. As a consequence, we obtain new bilinear restriction estimates for elliptic phases and wave/Klein–Gordon interactions in the full bilinear range, and give a refined Strichartz inequality for the Klein–Gordon equation. In addition, we extend these bilinear estimates to hold in adapted function spaces by using a transference type principle which holds for vector valued waves.

The Korteweg-de Vries equation on an interval

The initial-boundary value problem (IBVP) for the Korteweg-de Vries (KdV) equation on an interval is studied by extending a novel approach recently developed for the well-posedness of the KdV on the half-line, which is based on the solution formula produced via Fokas’ unified transform method for the associated forced linear IBVP. Replacing in this formula the forcing by the nonlinearity and using data in Sobolev spaces suggested by the space-time regularity of the Cauchy problem of the linear KdV gives an iteration map for the IBVP which is shown to be a contraction in an appropriately chosen solution space. The proof relies on key linear estimates and a bilinear estimate similar to the one used for the KdV Cauchy problem by Kenig, Ponce, and Vega.

A Hierarchical Approach for Joint Parameter and State Estimation of a Bilinear System with Autoregressive Noise

This paper is concerned with the joint state and parameter estimation methods for a bilinear system in the state space form, which is disturbed by additive noise. In order to overcome the difficulty that the model contains the product term of the system input and states, we make use of the hierarchical identification principle to present new methods for estimating the system parameters and states interactively. The unknown states are first estimated via a bilinear state estimator on the basis of the Kalman filtering algorithm. Then, a state estimator-based recursive generalized least squares (RGLS) algorithm is formulated according to the least squares principle. To improve the parameter estimation accuracy, we introduce the data filtering technique to derive a data filtering-based two-stage RGLS algorithm. The simulation example indicates the efficiency of the proposed algorithms.

Well-posedness of the “good” Boussinesq equation in analytic Gevrey spaces and time regularity

The Cauchy problem for the “good” Boussinesq equation with data in analytic Gevrey spaces on the line and the circle is considered and its local well-posedness in these spaces is proved. The proof is based on bilinear estimates in Bourgain type spaces incorporating the symbol of the linear part of the equation and an exponential weight expressing the analytic Gevrey regularity of the solution in the spatial variable. Also, Gevrey regularity of the solution in time variable is provided.

Remarks on Chern-Simons gauged O(3) sigma model in one space dimension

We find the stationary solutions of the Chern-Simons gauged O(3) sigma model in one space dimension and study an initial value problem under the gauge condition A1 = 0. We also study low regularity local well-posedness using bilinear estimates for wave-Sobolev space H1, θ. As a by-product, we construct a finite energy global solution by considering the conservation of energy.

Scattering results for Dirac Hartree-type equations with small initial data

We consider the Dirac equations with cubic Hartree-type nonlinearity which are derived by uncoupling the Dirac-Klein-Gordon systems. We prove small data global well-posedness and scattering results in the full scaling subcritical regularity regime. The strategy of the proof relies on the localized Strichartz estimates and bilinear estimates in \begin{document}$ V^2 $\end{document} spaces, together with the use of the null structure that the nonlinear term exhibits. This result is shown to be almost optimal in the sense that the iteration method based on Duhamel's formula fails over the supercritical range.

Variational estimates for martingale paraproducts

We show that bilinear variational estimates of Do, Muscalu, and Thiele (arXiv:1009.5187) remain valid for a pair of general martingales with respect to the same filtration. Our result can also be viewed as an off-diagonal generalization of the Burkholder--Davis--Gundy inequality for martingale rough paths by Chevyrev and Friz (arXiv:1704.08053).

Sharp bilinear estimates and its application to a system of quadratic derivative nonlinear Schrödinger equations

In the present paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations for the spatial dimension d=2 and 3. This system was introduced by Colin and Colin (2004). The first author obtained some well-posedness results in the Sobolev space Hs. But under some condition for the coefficient of Laplacian, this result is not optimal. We improve the bilinear estimate by using a nonlinear version of the classical Loomis–Whitney inequality, and prove well-posedness in Hs for s≥1∕2 if d=2, and s>1∕2 if d=3.

Critical well-posedness and scattering results for fractional Hartree-type equations

Scattering for the mass-critical fractional Schrödinger equation with a cubic Hartree-type nonlinearity for initial data in a small ball in the scale-invariant space of three-dimensional radial and square-integrable initial data is established. For this, we prove a bilinear estimate for free solutions and extend it to perturbations of bounded quadratic variation. This result is shown to be sharp by proving the discontinuity of the flow map in the super-critical range.

$$L^p$$ L p Bilinear Quasimode Estimates

In this paper, we investigate the $$L^p$$ bilinear quasimode estimates on compact Riemannian manifolds. We obtain results in the full range $$p\ge 2$$ on all n-dimensional manifolds with $$n\ge 2$$ . This in particular implies the $$L^p$$ bilinear eigenfunction estimates. We further show that all of these estimates are sharp by constructing various quasimodes and eigenfunctions that saturate our estimates.

The Cauchy problem for nonlinear quadratic interactions of the Schrödinger type in one dimensional space

In this work, we study the well-posedness of the Cauchy problem associated with the coupled Schrödinger equations with quadratic nonlinearities, which appears modeling problems in nonlinear optics. We obtain the local well-posedness for data in Sobolev spaces with low regularity. To obtain the local theory, we prove new bilinear estimates for the coupling terms of the system in the continuous case. Concerning global results, in the continuous case, we establish the global well-posedness in Hs(R)×Hs(R), for some negatives indexes s. The proof of our global result uses the I-method introduced by Colliander et al.

Sharp global well‐posedness for the fractional BBM equation

The Cauchy problem of the fractional Benjamin‐Bona‐Mahony equation with exponent α ∈ (1,2] on the real line is studied. First, some bilinear estimates for the fractional Benjamin‐Bona‐Mahony are proved, and the estimates are shown to be sharp by providing counter‐examples. Second, the local well‐posedness in Hs with is proved by the contraction principle. Finally, the local solution is extended to a global one by the I‐method. The well‐posedness result turns out to be sharp and fills the gap between the results in Discrete Contin. Dyn. Syst, 23 (2009) 1253‐1275 and that in Discrete Contin. Dyn. Syst., 30 (2011) 253‐259.

The Cauchy problem for the Ostrovsky equation with positive dispersion

This paper is devoted to studying the Cauchy problem for the Ostrovsky equation $$\begin{aligned} \partial _{x}\left( u_{t}-\beta \partial _{x}^{3}u +\frac{1}{2}\partial _{x}(u^{2})\right) -\gamma u=0, \end{aligned}$$with positive \(\beta \) and \(\gamma \). This equation describes the propagation of surface waves in a rotating oceanic flow. We first prove that the problem is locally well-posed in \(H^{-\frac{3}{4}}(\text{ R })\). Then we reestablish the bilinear estimate, by means of the Strichartz estimates instead of calculus inequalities and Cauchy–Schwartz inequalities. As a byproduct, this bilinear estimate leads to the proof of the local well-posedness of the problem in \(H^{s}(\text{ R })\) for \( s>-\frac{3}{4}\), with help of a fixed point argument.

Transference of bilinear restriction estimates to quadratic variation norms and the Dirac–Klein–Gordon system

Firstly, bilinear Fourier Restriction estimates --which are well-known for free waves-- are extended to adapted spaces of functions of bounded quadratic variation, under quantitative assumptions on the phase functions. This has applications to nonlinear dispersive equations, in particular in the presence of resonances. Secondly, critical global well-posedness and scattering results for massive Dirac-Klein-Gordon systems in dimension three are obtained, in resonant as well as in non-resonant regimes. The results apply to small initial data in scale-invariant Sobolev spaces exhibiting a small amount of angular regularity.

On a bilinear Strichartz estimate on irrational tori

We prove a bilinear Strichartz-type estimate for irrational tori via a decoupling-type argument, as used by Bourgain and Demeter (2015), recovering and generalizing a result of De Silva, Pavlovic, Staffilani and Tzirakis (2007). As a corollary, we derive a global well-posedness result for the cubic defocusing NLS on two-dimensional irrational tori with data of infinite energy.

A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates

A new decomposition for frequency-localized solutions to the Schrodinger equation is given which describes the evolution of the wavefunction using a weighted sum of Lipschitz tubes. As an application of this decomposition, we provide a new proof of the bilinear Strichartz estimate as well as the multilinear restriction theorem for the paraboloid.

QUARKS: Identification of Large-Scale Kronecker Vector-AutoRegressive Models

In this paper, we address the identification of two-dimensional (2-D) spatial-temporal dynamical systems described by the Vector-AutoRegressive (VAR) form. The coefficient-matrices of the VAR model are parametrized as sums of Kronecker products. When the number of terms in the sum is small compared to the size of the matrices, such a Kronecker representation efficiently models large-scale VAR models. Estimating the coefficient matrices in least-squares sense gives rise to a bilinear estimation problem that is tackled using an Alternating Least Squares (ALS) algorithm. Regularization or parameter constraints on the coefficient-matrices allows to induce temporal network properties, such as stability, as well as spatial properties, such as sparsity or Toeplitz structure. Convergence of the regularized ALS is proved using fixed-point theory. A numerical example demonstrates the advantages of the new modeling paradigm. It leads to comparable variance of the prediction error with the unstructured least-squares estimation of VAR models. However, the number of parameters grows only linearly with respect to the number of nodes in the 2-D sensor network instead of quadratically in the case of fully unstructured coefficient-matrices.