Linear transport equations for vector fields with subexponentially integrable divergence

  • Albert Clop
  • Renjin Jiang
  • Joan Mateu
  • Joan Orobitg
Article

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.

Mathematics Subject Classification

Primary 35F05 Secondary 35F10 

Notes

Acknowledgments

The authors are grateful to Gianluca Crippa for interesting remarks which improved the paper. Albert Clop, Joan Mateu and Joan Orobitg were partially supported by Generalitat de Catalunya (2014SGR75) and Ministerio de Economía y Competitividad (MTM2013-44699). Albert Clop was partially supported by the Programa Ramón y Cajal. Renjin Jiang was partially supported by National Natural Science Foundation of China (NSFC 11301029). All authors were partially supported by Marie Curie Initial Training Network MAnET (FP7-607647).

References

  1. 1.
    Ambrosio, L.: Transport equation and Cauchy problem for \(BV\) vector fields. Invent. Math. 158, 227–260 (2004)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Ambrosio, L., Colombo, M., Figalli, A.: Existence and uniqueness of maximal regular flows for non-smooth vector fields. Arch. Ration. Mech. Anal. 218, 1043–1081 (2015)Google Scholar
  3. 3.
    Ambrosio, L., Crippa, G., Figalli, A., Spinolo, L.V.: Some new well-posedness results for continuity and transport equations, and applications to the chromatography system. SIAM J. Math. Anal. 41, 1890–1920 (2009)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Ambrosio, L., Figalli, A.: On flows associated to Sobolev vector fields in Wiener spaces: an approach à la DiPerna–Lions. J. Funct. Anal. 256, 179–214 (2009)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)Google Scholar
  6. 6.
    Cipriano, F., Cruzeiro, A.B.: Flows associated with irregular \({\mathbb{R}}^d\)-vector fields. J. Differ. Equ. 219, 183–201 (2005)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Colombini, F., Crippa, G., Rauch, J.: A note on two-dimensional transport with bounded divergence. Commun. Partial Differ. Equ. 31, 1109–1115 (2006)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Colombini, F., Lerner, N.: Uniqueness of continuous solutions for \(BV\) vector fields. Duke Math. J. 111, 357–384 (2002)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Colombo, M., Crippa, G., Spirito, S.: Logarithmic estimates for continuity equations Institute of Mathematics, University of Basel. Preprint No. 2015–08Google Scholar
  10. 10.
    Colombo M., Crippa G., Spirito S., Renormalized solutions to the continuity equation with an integrable damping term. Calc. Var. Partial Differ. Equ. 54(2), 1831–1845 (2015)Google Scholar
  11. 11.
    Crippa, G.: The Flow Associated to Weakly Differentiable Vector Fields. Tesi. Scuola Normale Superiore di Pisa (Nuova Series) [Theses of Scuola Normale Superiore di Pisa (New Series)], vol. 12, pp. xvi+167. Edizioni della Normale, Pisa, (2009)Google Scholar
  12. 12.
    Crippa, G., De Lellis, C.: Estimates and regularity results for the DiPerna–Lions flow. J. Reine Angew. Math. 616, 15–46 (2008)MathSciNetMATHGoogle Scholar
  13. 13.
    Desjardins, B.: A few remarks on ordinary differential equations. Commun. Partial Differ. Equ. 21, 1667–1703 (1996)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    DiPerna, R.J., Lions, P.L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton (1992)Google Scholar
  16. 16.
    Fang, S.Z., Luo, D.J.: Transport equations and quasi-invariant flows on the Wiener space. Bull. Sci. Math. 134, 295–328 (2010)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Mucha, P.B.: Transport equation: extension of classical results for \({\rm div} \, b\in BMO\). J. Differ. Equ. 249, 1871–1883 (2010)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Rao, M., Ren, Z.: Theory of Orlicz Spaces. Dekker, New York (1991)MATHGoogle Scholar
  20. 20.
    Subko, P.: A remark on the transport equation with \(b\in BV\) and \({\rm div} _xb\in BMO\). Colloq. Math. 135, 113–125 (2014)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Albert Clop
    • 1
  • Renjin Jiang
    • 1
    • 2
  • Joan Mateu
    • 1
  • Joan Orobitg
    • 1
  1. 1.Departament de Matemàtiques, Facultat de CiènciesUniversitat Autònoma de BarcelonaBellaterra (Barcelona)Spain
  2. 2.Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Ministry of EducationBeijing Normal UniversityBeijingChina

Personalised recommendations