This paper establishes the hyper-contractivity in L∞ (ℝd) (it's known as ultra-contractivity) for the multi-dimensional Keller-Segel systems with the dif;fusion exponent m > 1 -2/d. The results show that for the supercritical and critical case 1 - 2/d < m ≤ 2 - 2/d, if || U0||d(2-m)/2) < Cd, m where Cd, m is a universal constant, then for any t > 0, || u (⋅, t)|| ... is bounded and decays as t goes to infinity, For the subcritical case m > 2 - 2/d, the solution u (⋅, t) ∈ L∞ (ℝd) with any initial data U0 ∈ L1+ (ℝd) for any positive time.

We present a unique global solvability and flocking estimate of an entropic weak solution to the one-dimensional pressureless Euler system with a flocking dissipation in all-to-all coupling setting. This model appears naturally as a quasi-equilibrium model for hydrodynamic description of CuckerSmale flocking. For the unique global solvability, we adopt the variation approach from Ding and Huang's work [19] on the inhomogeneous pressureless gas dynamic model. When initial mass and velocity are locally integrable and bounded measurable functions, respectively, we give explicit representations for the global entropic weak solutions. Our results do not require any smallness of initial data except that initial mass density is almost everywhere positive.

Abstract We prove the convergence of the vortex blob method for a family of nonlinear evolutionary partial differential equations (PDEs), the so-called b-equation. This kind of PDEs, including the Camassa-Holm equation and the Degasperis-Procesi equation, has many applications in diverse scientific fields. Our convergence analysis also provides a proof for the existence of the global weak solution to the b-equation when the initial data is a nonnegative Radon measure with compact support.

In this paper, we consider the Cauchy problem to the planar non-resistive magnetohydrodynamic equations without heat conductivity, and establish the global well-posedness of strong solutions with large initial data. The key ingredient of the proof is to establish the a priori estimates on the effective viscous flux and a newly introduced ``transverse effective viscous flux" vector field inducted by the transverse magnetic field. The initial density is assumed only to be uniformly bounded and of finite mass and, in particular, the vacuum and discontinuities of the density are allowed.

It is well-known in the combustion community that curvature effect in general slows down ame propagation speeds because it smooths out wrinkled ames. However, such a folklore has never been justied rigorously. In this paper, as the rst theoretical result in this direction, we prove that the turbulent ame speed (an eective burning velocity) is decreasing with respect to the curvature diusivity (Markstein number) for shear flows in the well-known G-equation model. Our proof involves several novel and rather sophisticated inequalities arising from the nonlinear structure of the equation. On a related fundamental issue, we solve the selection problem of weak solutions or nd the "physical fluctuations" when the Markstein number goes to zero and solutions approach those of the inviscid G-equation model. The limiting solution is given by a closed form analytical formula.