NDLI: Linear transport equations for vector fields with subexponentially integrable divergence

Content Provider	SpringerLink
Author	Jiang, Renjin Orobitg, Joan Clop, Albert Mateu, Joan
Copyright Year	2016
Abstract	We face the well-posedness of linear transport Cauchy problems $$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\partial u}{\partial t} + b\cdot \nabla u + c\,u = f&{}(0,T)\times \mathbb {R}^n\\ u(0,\cdot )=u_0\in L^\infty &{}\mathbb {R}^n \end{array}\right. } \end{aligned}$$ under borderline integrability assumptions on the divergence of the velocity field b. For $$W^{1,1}_{loc}$$ vector fields b satisfying $$\frac{\|b(x,t)\|}{1+\|x\|}\in L^1(0,T; L^1)+L^1(0,T; L^\infty )$$ and $$\begin{aligned} {\text {div }}b\in L^1(0,T; L^\infty ) + L^1\left( 0,T; {{\text {Exp}}\;}\left( \frac{L}{\log L}\right) \right) , \end{aligned}$$ we prove existence and uniqueness of weak solutions. Moreover, optimality is shown in the following way: for every $$\gamma >1$$ , we construct an example of a bounded autonomous velocity field b with $$\begin{aligned} {\text {div }}(b)\in {{\text {Exp}}\;}\left( \frac{L}{\log ^\gamma L}\right) \end{aligned}$$ for which the associate Cauchy problem for the transport equation admits infinitely many solutions. Stability questions and further extensions to the BV setting are also addressed.
Ending Page	30
Page Count	30
Starting Page	1
File Format	PDF
ISSN	09442669
e-ISSN	14320835
Journal	Calculus of Variations and Partial Differential Equations
Issue Number	1
Volume Number	55
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2016-02-09
Publisher Place	Berlin, Heidelberg
Access Restriction	Subscribed
Subject Keyword	Systems Theory, Control Calculus of Variations and Optimal Control; Optimization Initial value problems for linear first-order equations Analysis Theoretical, Mathematical and Computational Physics Linear first-order equations
Content Type	Text
Resource Type	Article
Subject	Analysis Applied Mathematics

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