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In this paper we address the regularity issues of drift-diffusion equation with nonlocal diffusion, where the diffusion operator is in the realm of stable-type Lévy operator and the velocity field is defined from the considered quantity by a zero-order pseudo-differential operator. Through using the method of nonlocal maximum principle in a unified way, we prove the eventual regularity result in the supercritical type cases and the global regularity at some logarithmically supercritical cases. The feature of these results is that the time after which the solution is smoothly regular in the supercritical type cases can be evaluated explicitly. [ABSTRACT FROM AUTHOR]

We construct global-in-time, unique solutions of the twodimensional Euler equations in a Yudovich type space and a bmo-type space. First, we show the regularity of solutions for the two-dimensional Euler equations in the Spanne space involving an unbounded and nondecaying vorticity. Next, we establish an estimate with a logarithmic loss of regularity for the transport equation in a bmo-type space by developing classical analysis tool such as the John-Nirenberg inequality. We also optimize estimates of solutions to the vorticity-stream formulation of the two-dimensional Euler equations with a bi-Lipschitz vector field in bmotype space by combining an observation introduced by Yodovich with the so-called "quasi-conformal property" of the incompressible flow. [ABSTRACT FROM AUTHOR]

It is proved that the local smoothing conjecture for wave equations implies certain improvements on Stein's analytic family of maximal spherical means. Some related problems are also discussed. [ABSTRACT FROM AUTHOR]

We reprove Wolff's L5/2-bound for the R³-Kakeya maximal function without appealing to the argument of induction on scales. The main ingredient in our proof is an adaptation of Sogge's strategy used in the work on Nikodym-type sets in curved spaces. Although the equivalence between these two type maximal functions is well known, our proof may shed light on some new geometric observations which is interesting in its own right. [ABSTRACT FROM AUTHOR]

The two-dimensional incompressible Boussinesq equations with partial or fractional dissipation have recently attracted considerable attention and the global regularity issue has been extensively investigated. This paper aims at the global regularity in the case when the dissipation is critical. The critical dissipation refers to α + β = 1 when Λα ≡ (-Δ) α/2 and Λ β represent the fractional Laplacian dissipation in the velocity and the temperature equations, respectively. When α = 1 and β = 0 or when α = 0 and β = 1, the global regularity was obtained in [T. Hmidi, S. Keraani, and F. Rousset, J. Differential Equations, 249 (2010), pp. 2147-2174; T. Hmidi, S. Keraani, and F. Rousset, Comm. Partial Differential Equations, 36 (2011), pp. 420-445]. However, the approaches there do not apply to the situation when α + β = 1 with both α > 0 and β > 0. The novelty here is to reduce the critical Boussinesq system to a critical active scalar equation or, more precisely, the generalized critical surface quasi-geostrophic equation. When α is restricted to a suitable range, the global regularity of the critical Boussinesq system can be obtained by exploiting the global regularity of this scalar equation and the global bound for a combined quantity of the vorticity and the temperature. [ABSTRACT FROM AUTHOR]

We consider a class of defocusing energy-supercritical nonlinear Schrödinger equations in four space dimensions. Following a concentration-compactness approach, we show that for 1

In this paper, we study the Strichartz-type estimates of the solution for the linear wave equation with inverse square potential. Assuming the initial data possesses additional angular regularity, especially the radial initial data, the range of admissible pairs is improved. As an application, we show the global well-posedness of the semi-linear wave equation with inverse-square potential ... for power p being in some regime when the initial data are radial. This result extends the well-posedness result in Planchón, Stalker, and Tahvildar-Zadeh. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the restriction problem in harmonic analysis. Based on the spherical harmonics expansion and analyzing the asymptotic behavior of the Bessel function, we show that a modified linear adjoint restriction estimate holds for all Schwartz functions compactly supported on the cone, which generalizes Shao's result. [ABSTRACT FROM AUTHOR]

In this paper, we prove the local well-posedness for the ideal MHD equations in the Triebel–Lizorkin spaces and obtain a blow-up criterion of smooth solutions. Specifically, we fill a gap in a step of the proof of the local well-posedness part for the incompressible Euler equation in Chae (Comm Pure Appl Math 55:654–678 2002). [ABSTRACT FROM AUTHOR]

We improve and extend some known regularity criterion of the weak solution for the 3D viscous Magneto-hydrodynamics equations by means of the Fourier localization technique and Bony’s para-product decomposition. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and uniqueness of the solutions for the two-dimensional viscous shallow water equations with low regularity assumptions on the initial data as well as the initial height bounded away from zero. [ABSTRACT FROM AUTHOR]

In this paper, the large time behavior of solutions of the Cauchy problem for the one-dimensional fractal Burgers equation ut + (-∂x2)α/2u+ uux = 0 with α ∈ (1, 2) is studied. It is shown that if the nondecreasing initial datum approaches the constant states u± (u_ < u+) as x →±∞, respectively, then the corresponding solution converges toward the rarefaction wave, i.e., the unique entropy solution of the Riemann problem for the nonviscous Burgers equation. [ABSTRACT FROM AUTHOR]

We study the blow-up criterion of smooth solutions to the 3D MHD equations. By means of the Littlewood-Paley decomposition, we prove a Beale-Kato-Majda type blow-up criterion of smooth solutions via the vorticity of velocity only, namely where Δ j is the frequency localization operator in the Littlewood-Paley decomposition. [ABSTRACT FROM AUTHOR]

We show a new Bernstein’s inequality which generalizes the results of Cannone-Planchon, Danchin and Lemarié-Rieusset. As an application of this inequality, we prove the global well-posedness of the 2D quasi-geostrophic equation with the critical and super-critical dissipation for the small initial data in the critical Besov space, and local well-posedness for the large initial data. [ABSTRACT FROM AUTHOR]

In this paper, we study the global well-posedness and scattering theory for the defocusing fourth-order nonlinear Schrödinger equation (FNLS) in dimensions . We prove that if the solution u is apriorily bounded in the critical Sobolev space, that is, with all if p is an even integer or otherwise, then u is global and scatters. We will give a uniform way to treat the energy-subcritical, energy-critical and energy-supercritical FNLS by making use of the strategy derived from concentration compactness ideas, and we are able to overcome the logarithmic blowup in the double Duhamel trick in dimension eight by exploiting the refined dispersive estimate which is in sharp contrast to the Schrödinger equation. [ABSTRACT FROM AUTHOR]

In this paper we study the blow-up criterion of smooth solution to the ideal MHD equations in Rn. By means of the Fourier frequency localization and Bony paraproduct decomposition, we show a losing estimate for the ideal MHD equations and apply it to establish an improved blow-up criterion of smooth solutions. As a special case, we recover a previous result of Planchon for the incompressible Euler equations. [ABSTRACT FROM AUTHOR]