Flow With Density and Transport Equation In

Abstract

We show that, if b ∈ L 1 (0, T ; L 1 loc (R)) has spatial derivative in the John-Nirenberg space BMO(R), then it generalizes a unique flow φ(t, ·) which has an A ∞ (R) density for each time t ∈ [0, T ]. Our condition on the map b is optimal and we also get a sharp quantitative estimate for the density. As a natural application we establish a well-posedness for the Cauchy problem of the transport equation in BMO(R). Statement of main resultsGiven an integer n ≥ 1, a real T ≥ t > 0 and an evolutionary self-map b(t, ·) of R n with b ∈ L 1 (0, T ; L 1 loc (R n )), consider the flow φ(t, x) = x + t 0

Sharp Reverse Hölder Inequality for $$C_p$$ Weights and Applications

“…This quantification has two main properties: the constant on the right hand side is uniform for all weights and the dependence on the A ∞ constant of the exponent is sharp. These sharp RHI inequalities have been used in different applications: for obtaining quantitative estimates for norms of some singular integral operators [9,15,19], or for obtaining sharp estimates for solutions of certain PDE [16], among many others.…”

Classical flows of vector fields with exponential or sub-exponential summability

“…Sobolev regularity stated in Theorem C is sharp, as we show in Example 8.2 below. Moreover it can be compared with the result in [27], which proves existence and uniqueness of a unique flow X, with D [22,Chap. 7] for the definition of the class of functions BM O(R n )).…”

“…A fairly complete account of the development in this topic can be found in [5] and references therein. A sample of the literature on this subject is [4,8,9,10,13,11,12,14,15,16,27,29,28].…”

New John--Nirenberg--Campanato-Type Spaces Related to Both Maximal Functions and Their Commutators

“…It is a byproduct appearing in the study of John and Nirenberg [19] on functions with bounded mean oscillation (the celebrated space BMO), and was further used in the interpolation theory by Stampacchia [24]. Both JN p and BMO are function spaces based on mean oscillations of functions, and we refer the reader to [3,6,7,8,12,13,14,18,21,22] for more related researches. Obviously, we have JN 1 (Q 0 ) = L 1 (Q 0 ) and JN ∞ (Q 0 ) = BMO (Q 0 ); see, for instance, [27].…”