Weakly singular initial values for the Stokes equation on a half space

We study the extension of Ukai's formula to the case of singular initial values for the Stokes problem on the half space.     

A REMARK ON THE BLOW-UP CRITERION OF STRONG SOLUTIONS AND REGULARITY FOR WEAK SOLUTIONS OF NAVIER-STOKES EQUATIONS

We give blow-up criteria of strong solutions of Navier-Stokes equations with its initial data in Besov spaces and consider the regularity of Leray-Hopf solutions of the equation.     

Weighted inequalities for multilinear commutators of some sublinear operators

We establish sharp estimates for some multilinear commutators related to the Littlewood-Paley and Marcinkiewicz operators. As an application, we obtain the weighted norm inequalities and L log L type estimate for the multilinear commutators.      

Calderón-Zygmund operators on product Hardy spaces

Let T be a product Calderón-Zygmund singular integral introduced by Journé. Using an elegant rectangle atomic decomposition of Hp(Rn×Rm) and Journé's geometric covering lemma, R. Fefferman proved the remarkable Hp(Rn×Rm)−Lp(Rn×Rm) boundedness of T. In this paper we apply vector-valued singular integral, Calderón's identity, Littlewood-Paley theory and the almost orthogonality together with Fefferman's rectangle atomic decomposition and Journé's covering lemma to show that T is bounded on product Hp(Rn×Rm) for if and only if , where ε is the regularity exponent of the kernel of T.     

Well-posedness of thermal boundary layer equation in two-dimensional incompressible heat conducting flow with analytic datum

In this paper, we study the well-posedness of the thermal boundary layer equation in two-dimensional incompressible heat conducting flow. The thermal boundary layer equation describes the behavior of thermal layer and viscous layer for the two-dimensional incompressible viscous flow with heat conduction in the small viscosity and heat conductivity limit. When the initial datum are analytic, with respect to the tangential variable of the boundary, and without the monotonicity condition of the tangential velocity, by using the Littlewood-Paley theory, we obtain the local-in-time existence and uniqueness of solution to this thermal boundary layer problem.     

On the Radius of Analyticity of Solutions to 3D Navier-Stokes System with Initial Data in Lp

Given initial data u₀ ∈ L^p(R³) for some p in[3, 18/5[, the auhtors ?rst prove that 3D incompressible Navier-Stokes system has a unique solution u = uL+v with uL def = et^?u0 and v ∈ e L^∞([0, T]; ˙ H^{5/2 ? 6/p} ) ∩ L¹(]0, T[; ˙H^{9/2 ? 6/p} ) for some positive time T. Then they derive an explicit lower bound for the radius of space analyticity of v, which in particular extends the corresponding results in [Chemin, J.-Y., Gallagher, I. and Zhang, P., On the radius of analyticity of solutions to semi-linear parabolic system, Math. Res. Lett., 27, 2020, 1631– 1643, Herbst, I. and Skibsted, E., Analyticity estimates for the Navier-Stokes equations, Adv. in Math., 228, 2011, 1990–2033] with initial data in ˙H^s(R³) for s∈[1/2,3/2[.     

Endpoint estimates of generalized homogeneous Littlewood–Paley <Emphasis Type="Italic">g</Emphasis>-functions over non-homogeneous metric measure spaces

Let (χ, d, μ) be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions. Under the weak reverse doubling condition, the authors prove that the generalized homogeneous Littlewood-Paley g-function_r (r∈[2,∞)) is bounded from Hardy space H¹(μ) into L¹(μ). Moreover, the authors show that, if f ∈ RBMO(μ), then[_r(f)]^r is either infinite everywhere or finite almost everywhere, and in the latter case,[_r(f)]^r belongs to RBLO(μ) with the norm no more than ‖f‖_RBMO(μ) multiplied by a positive constant which is independent of f. As a corollary, the authors obtain the boundedness of_r from RBMO(μ) into RBLO(μ). The vector valued Calderón-Zygmund theory over (χ, d, μ) is also established with details in this paper.     

On the Small Ball Inequality in all dimensions

Let hR denote an L^∞ normalized Haar function adapted to a dyadic rectangle R⊂d[0,1]. We show that for choices of coefficients α(R), we have the following lower bound on the L^∞ norms of the sums of such functions, where the sum is over rectangles of a fixed volume:

     

Local existence and blow-up criterion for the Euler equations in Besov spaces of weak type

We introduce Besov type function spaces, based on the weak L ^p -spaces instead of the standard L ^p -spaces, and prove a local-in-time unique existence and a blow-up criterion of solutions in those spaces for the Euler equations of perfect incompressible fluid in . For the proof, we establish the Beale-Kato-Majda type logarithmic inequality and commutator type estimates in our weak spaces. Abbreviate title: Euler equations in Besov spaces of weak type     

On a Littlewood-Paley type inequality

The following is proved: If is a function harmonic in the unit ball and if then the inequality

holds, where is the nontangential maximal function of This improves a recent result of Stoll. This inequality holds for polyharmonic and hyperbolically harmonic functions as well.