In the paper, the large time behavior of solutions of the Cauchy problem for the one dimensional fractal Burgers equation u t + (−∂ 2 x) α/2 u + uu x = 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… (More)
In this paper, we prove the existence and uniqueness of the solutions for the 2D viscous shallow water equations with low regularity assumptions on the initial data as well as the initial height bounded away from zero.
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 Λ α ≡ (−Δ) α… (More)