Differentiability in Measure of the Flow Associated with a Nearly Incompressible BV Vector Field

Abstract

We study the regularity of the flow {varvec{X}}(t,y), which represents (in the sense of Smirnov or as regular Lagrangian flow of Ambrosio) a solution rho in L^infty ({mathbb {R}}^{d+1}) of the continuity equation ∂tρ+div(ρb)=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\partial _t \\rho + {{\\,\ extrm{div}\\,}}(\\rho {\\varvec{b}}) = 0, \\end{aligned}$$\\end{document}with {varvec{b}}in L^1_t {{,textrm{BV},}}_x. We prove that {varvec{X}} is differentiable in measure in the sense of Ambrosio–Malý, that is X(t,y+rz)-X(t,y)r→r→0W(t,y)zin measure,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{{\\varvec{X}}(t,y+rz) - {\\varvec{X}}(t,y)}{r} \\underset{r \\rightarrow 0}{\\rightarrow }\\ W(t,y) z \\quad \ ext {in measure}, \\end{aligned}$$\\end{document}where the derivative W(t, y) is a BV function satisfying the ODE ddtW(t,y)=(Db)y(dt)J(t-,y)W(t-,y),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{\ extrm{d}}{\ extrm{d}t} W(t, y) = \\frac{(D {\\varvec{b}})_y(\ extrm{d}t)}{J(t-,y)} W(t-, y), \\end{aligned}$$\\end{document}where (D{varvec{b}})_y(textrm{d}t) is the disintegration of the measure int D {varvec{b}}(t,cdot ) , textrm{d}t with respect to the partition given by the trajectories {varvec{X}}(t, y) and the Jacobian J(t, y) solves ddtJ(t,y)=(divb)y(dt)=Tr(Db)y(dt).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{\ extrm{d}}{\ extrm{d}t} J(t,y) = ({{\\,\ extrm{div}\\,}}{\\varvec{b}})_y(\ extrm{d}t) = \ extrm{Tr}(D{\\varvec{b}})_y(\ extrm{d}t). \\end{aligned}$$\\end{document}The proof of this regularity result is based on the theory of Lagrangian representations and proper sets introduced by Bianchini and Bonicatto in [16], on the construction of explicit approximate tubular neighborhoods of trajectories, and on estimates that take into account the local structure of the derivative of a {{,textrm{BV},}} vector field.