Commutator Estimates Research Articles

Compressible Euler equation with damping on Torus in arbitrary dimensions

We study the exponential stability of constant steady state of isentropic compressible Euler equation with damping on $\mathbb T^n$. The local existence of solutions is based on semigroup theory and some commutator estimates. We propose a new method instead of energy estimates to study the stability, which works equally well for any spatial dimensions.

A Young-like inequality with applications to the commutator estimates

A Young-like inequality with applications to the commutator estimates

On the Existence and Stability of Standing Waves for 2-Coupled Nonlinear Fractional Schrödinger System

We study a system of 2-coupled nonlinear fractional Schrödinger equations. Firstly, we construct constrained minimization problem to the system. Next, we prove the existence of standing waves for the system by using the concentration-compactness and commutator estimates method. Lastly, we also consider the set of minimizers of the constrained minimization problem. We prove that it is a stable set for initial value of the problem; that is, a solution to the system with initial value which is near the set will remain near it for all time.

Existence of pulses in excitable media with nonlocal coupling

We prove the existence of fast traveling pulse solutions in excitable media with non-local coupling. Existence results had been known, until now, in the case of local, diffusive coupling and in the case of a discrete medium, with finite-range, non-local coupling. Our approach replaces methods from geometric singular perturbation theory, that had been crucial in previous existence proofs, by a PDE oriented approach, relying on exponential weights, Fredholm theory, and commutator estimates.

On an endpoint Kato-Ponce inequality

We prove that the $L^{\infty}$ end-point Kato-Ponce inequality (Leibniz rule) holds for the fractional Laplacian operators $D^s=(-\Delta)^{s/2}$, $J^s=(1-\Delta)^{s/2}$, $s>0$. This settles a conjecture by Grafakos, Maldonado and Naibo [7]. We also establish a family of new refined Kato-Ponce commutator estimates. Some of these inequalities are in borderline spaces.

Well-posedness of Lagrangian flows and continuity equations in metric measure spaces

We establish, in a rather general setting, an analogue of DiPerna-Lions theory on well-posedness of flows of ODE's associated to Sobolev vector fields. Key results are a well-posedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into $\mathbb{R}^\infty$. When specialized to the setting of Euclidean or infinite dimensional (e.g. Gaussian) spaces, large parts of previously known results are recovered at once. Moreover, the class of ${\sf RCD}(K,\infty)$ metric measure spaces object of extensive recent research fits into our framework. Therefore we provide, for the first time, well-posedness results for ODE's under low regularity assumptions on the velocity and in a nonsmooth context.

Weighted endpoint estimates for multilinear commutator of singular integral operators with non-smooth kernels

In this paper, we prove the weighted endpoint estimates for multilinear commutator of singular integral operators with non-smooth kernels.

The Weyl Symbol of Schrödinger Semigroups

In this paper, we study the Weyl symbol of the Schrodinger semigroup e−tH, H = −Δ + V, t > 0, on \({L^2(\mathbb{R}^n)}\) , with nonnegative potentials V in \({L^1_{\rm loc}}\) . Some general estimates like the L∞ norm concerning the symbol u are derived. In the case of large dimension, typically for nearest neighbor or mean field interaction potentials, we prove estimates with parameters independent of the dimension for the derivatives \({\partial_x^\alpha\partial_\xi^\beta u}\) . In particular, this implies that the symbol of the Schrodinger semigroups belongs to the class of symbols introduced in Amour et al. (To appear in Proceedings of the AMS) in a high-dimensional setting. In addition, a commutator estimate concerning the semigroup is proved.

Gevrey regularity for the supercritical quasi-geostrophic equation

In this paper, following the techniques of Foias and Temam, we establish suitable Gevrey class regularity of solutions to the supercritical quasi-geostrophic equations in the whole space, with initial data in “critical” Sobolev spaces. Moreover, the Gevrey class that we obtain is “near optimal” and as a corollary, we obtain temporal decay rates of higher order Sobolev norms of the solutions. Unlike the Navier–Stokes or the subcritical quasi-geostrophic equations, the low dissipation poses a difficulty in establishing Gevrey regularity. A new commutator estimate in Gevrey classes, involving the dyadic Littlewood–Paley operators, is established that allow us to exploit the cancellation properties of the equation and circumvent this difficulty.

Some new estimates for the commutators of n-dimensional Hausdorff operator

In this paper, we discuss the boundedness of commutators of high dimensional Hausdorff operator H Φ on Herz type spaces. In addition, central BMO estimates for such commutators are also presented.

Uniqueness for continuity equations in Hilbert spaces with weakly differentiable drift

We prove uniqueness for continuity equations in Hilbert spaces $$H$$ . The corresponding drift $$F$$ is assumed to be in a first order Sobolev space with respect to some Gaussian measure. As in previous work on the subject, the proof is based on commutator estimates which are infinite dimensional analogues to the classical ones due to DiPerna–Lions. Our general approach is, however, quite different since, instead of considering renormalized solutions, we prove a dense range condition implying uniqueness. In addition, compared to known results by Ambrosio–Figalli and Fang–Luo, we use a different approximation procedure, based on a more regularizing Ornstein–Uhlenbeck semigroup and consider Sobolev spaces of vector fields taking values in $$H$$ rather than the Cameron–Martin space of the Gaussian measure. This leads to different conditions on the derivative of $$F$$ , which are incompatible with previous work on the subject. Furthermore, we can drop the usual exponential integrability conditions on the Gaussian divergence of $$F$$ , thus improving known uniqueness results in this respect.

Higher order commutator estimates and local existence for the non-resistive MHD equations and related models

This paper establishes the local-in-time existence and uniqueness of strong solutions in Hs for s>n/2 to the viscous, non-resistive magnetohydrodynamics (MHD) equations in Rn, n=2,3, as well as for a related model where the advection terms are removed from the velocity equation. The uniform bounds required for proving existence are established by means of a new estimate, which is a partial generalisation of the commutator estimate of Kato and Ponce (1988) [13].

Some estimates for commutators of Riesz transforms associated with Schrödinger operators

We consider the Schrödinger operator L=−Δ+V on Rn, where the nonnegative potential V belongs to the reverse Hölder class Bq1 for some q1≥n2. Let q2=1 when q1≥n and 1q2=1−1q1+1n when n2<q1<n. Set δ′=min{1,2−nq1}. Let HLp(Rn) be the Hardy space related to the Schrödinger operator L for nn+δ′<p≤1. The commutator [b,R] is generated by a function b∈Λνθ, where Λνθ is a function space which is larger than the classical Companato space, and the Riesz transform R≐∇(−Δ+V)−12. We show that the commutator [b,R] is bounded from Lp(Rn) into Lq(Rn) for 1<p<q2′, where 1q=1p−νn and bounded from HLp(Rn) into Lq(Rn) for nn+ν<p≤1, where 1q=1p−νn. Moreover, we prove that the commutator [b,R] maps HLnn+ν(Rn) continuously into weak L1(Rn). At last, we give a characterization for the boundedness of the commutator [b,R] in an extreme case.

Well-posedness and time-decay for compressible viscoelastic fluids in critical Besov space

In this paper, we construct local solution with highly oscillating initial velocity and then get the global strong solution in the Lp based Besov space which improves the result of J. Qian, Z. Zhang (2010) [25] and X. Hu, D. Wang (2011) [14]. The local existence and uniqueness lies on the Lagrange coordinate transform and the contraction mapping theorem. The global result lies on a decomposition of the system and some commutator estimates. In the last part, we prove a time-decay in the critical Besov space framework which seems to have little investigation. The proof is based on the properties of the Green's matrix and various interpolations between Besov type spaces.

GENERALIZED FRACTIONAL INTEGRALS AND THEIR COMMUTATORS OVER NON-HOMOGENEOUS METRIC MEASURE SPACES

Let $({\mathcal X},d,\mu)$ be a metric measure space satisfying both the upper doubling and the geometrically doubling conditions. In this paper, the authors establish some equivalent characterizations for the boundedness of fractional integrals over $({\mathcal X},d,\mu)$. The authors also prove that multilinear commutators of fractional integrals with RBMO(μ) functions are bounded on Orlicz spaces over $({\mathcal X},d,\mu)$, which include Lebesgue spaces as special cases. The weak type endpoint estimates for multilinear commutators of fractional integrals with functions in the Orlicz-type space ${\mathrm{Osc}_{\exp L^r}(\mu)}$, where $r\in [1,\infty)$, are also presented. Finally, all these results are applied to a specific example of fractional integrals over non-homogeneous metric measure spaces.

Incompressible limit of Oldroyd-B fluids in the whole space

This paper is dedicated to the study of compressible Oldroyd-B model in the framework of critical spaces. We first prove the local well-posedness for the compressible Oldroyd-B model with initial data (ρ0−1,u0,τ0)∈B˙2,1d2×(B˙2,1d2−1)d×(B˙2,1d2)d×d. The global well-posedness is also established provided the initial data and coupling constant are sufficiently small. On the basis of this, we also consider the incompressible limit problem for ill prepared initial data. We prove that as the Mach number tends to zero, the global solution to the compressible Oldroyd-B fluids converges to the solution to the corresponding incompressible model in some function spaces. In the meanwhile, the converge rates are obtained in some sense. To fulfill our proof, we give a new commutator estimate.

Endpoint Estimates for Fractional Hardy Operators and Their Commutators on Hardy Spaces

(Hpℝn,Lqℝn)bounds of fractional Hardy operators are obtained. Moreover, the estimates for commutators of fractional Hardy operators on Hardy spaces are worked out. It is also proved that the commutators of fractional Hardy operators are mapped from the Herz-type Hardy spaces into the Herz spaces. The estimates for multilinear commutators of fractional Hardy operators are also discussed.

Dependence of High-Frequency Waves with Respect to Potentials

In this article, we consider the wave equation in a bounded domain $\Omega$ of $\mathbb{R}^d$ with a potential $q$. Our goal then is to show that the high-frequency part of the corresponding solutions weakly depends on the potential. We will in particular focus on two instances of interest arising in data assimilation and control theory, respectively corresponding to the problem of recovering an initial data from a measurement and to the problem of computing a control. In these two cases, we derive an explicit bound on the error of the high-frequency part of the solution induced by a $W^{s, p} (\Omega)$-error on the potential for $s \in (0,1]$ and $p\in (\max\{d,2\},\infty]$. In order to do that and to express it in a quantified form, we introduce spectral truncations. Our main tool is a commutator estimate.

Weighted estimates for vector-valued commutators of generalized fractional integrals

In this paper, the authors study the vector-valued commutator of generalized fractional integral operator Ik α,b,q where the kernel Kα satisfies some conditions associated with the Young functions. The authors prove the two-weight norm inequalities for Ik α,b,q where the weight ω is only a local integrable function. As an application of the main theorems in this paper, the weighted boundedness for vector-valued commutators of fractional integral with a rough kernel is also given. Mathematics subject classification (2010): 42B20, 42B25.

Weighted Lipschitz estimates for commutators of one-sided operators on one-sided Triebel-Lizorkin spaces

Weighted Lipschitz estimates for commutators of one-sided operators on one-sided Triebel-Lizorkin spaces