Abstract
We make use of the method of modulus of continuity [A. Kiselev, F. Nazarov, R. Shterenberg, Blow up and regularity for fractal Burgers equation, Dyn. Partial Differ. Equ. 5 (2008) 211–240] and Fourier localization technique [H. Abidi, T. Hmidi, On the global well-posedness of the critical quasi-geostrophic equation, SIAM J. Math. Anal. 40 (1) (2008) 167–185] [H. Abidi, T. Hmidi, On the global well-posedness of the critical quasi-geostrophic equation, SIAM J. Math. Anal. 40 (1) (2008) 167–185] to prove the global well-posedness of the critical Burgers equation ∂ t u + u ∂ x u + Λ u = 0 in critical Besov spaces B ˙ p , 1 1 p ( R ) with p ∈ [ 1 , ∞ ) , where Λ = − Δ .
Highlights
We consider the Burgers equation with fractional dissipation in R,∂tu + u∂xu + Λαu = 0 u(x, 0) = u0(x), where 0 ≤ α ≤ 2 and the operator Λα is defined by Fourier transform F(Λαu)(ξ) = |ξ|αFu(ξ). (1.1)The Burgers equation (1.1) with α = 0 and α = 2 has received an extensive amount of attention since the studies by Burgers in the 1940s
If α = 0, the equation is perhaps the most basic example of a PDE evolution leading to shocks; if α = 2, it provides an accessible model for studying the interaction between nonlinear and dissipative phenomena
For α = 1, with help of the method of modulus of continuity they proved the global well-posedness of the equation in the critical
Summary
For α = 1, with help of the method of modulus of continuity they proved the global well-posedness of the equation in the critical. Making use of Fourier localization technique and the method of modulus of continuity [12], we prove the global well-posedness of the critical Burgers equation (1.2) in critical Besov spaces Bpp,1(R) with p ∈ [1, ∞). Because of the restriction of the smooth index s stemming from the a priori estimate for the transport-diffusion equation (see Theorem 1.2), we can not get the result for the limit case p = ∞. The key of proving the local well-posedness is an optimal a priori estimate for the following transport-diffusion equation in RN :. We shall sometimes use the notation A B instead of A ≤ CB and A ≈ B means that A B and B A
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