# On Uniqueness of Weak Solutions to Transport Equation with Non-smooth Velocity Field

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## Abstract

Given a bounded, autonomous vector field \(\mathbf b :\mathbb {R}^d \rightarrow \mathbb {R}^d\), we study the uniqueness of bounded solutions to the initial value problem for the associated transport equation This problem is related to a conjecture made by A. Bressan, raised while studying the well-posedness of a class of hyperbolic conservation laws. Furthermore, from the Lagrangian point of view, this gives insights on the structure of the flow of non-smooth vector fields. In this work, we will discuss the two-dimensional case and we prove that, if \(d=2\), uniqueness of weak solutions for (1) holds under the assumptions that \(\mathbf b\) is of class \(\mathrm {BV}\) and it is

$$\begin{aligned} \partial _t u + \mathbf b \cdot \nabla u = 0. \end{aligned}$$

(1)

*nearly incompressible*. Our proof is based on a splitting technique (introduced previously by Alberti, Bianchini and Crippa in J Eur Math Soc (JEMS) 16(2):201–234, 2014, [2]) that allows to reduce (1) to a family of 1-dimensional equations which can be solved explicitly, thus yielding uniqueness for the original problem. This is joint work with S. Bianchini and N.A. Gusev (SIAM J Math Anal 48(1):1–33, 2016), [6].## Keywords

Transport equation Continuity equation Lipschitz functions Superposition Principle## MSC (2010)

35F10 35L03 28A50 35D30## References

- 1.G. Alberti, S. Bianchini, G. Crippa, Structure of level sets and Sard-type properties of Lipschitz maps. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)
**12**(4), 863–902 (2013)MathSciNetzbMATHGoogle Scholar - 2.G. Alberti, S. Bianchini, G. Crippa, A uniqueness result for the continuity equation in two dimensions. J. Eur. Math. Soc. (JEMS)
**16**(2), 201–234 (2014)MathSciNetCrossRefGoogle Scholar - 3.L. Ambrosio, Transport equation and Cauchy problem for BV vector fields. Invent. Math.
**158**(2), 227–260 (2004)MathSciNetCrossRefGoogle Scholar - 4.L. Ambrosio, N. Fusco, D. Pallara,
*Functions of Bounded Variation and Free Discontinuity Problems*(Oxford Science Publications, Clarendon Press, 2000)zbMATHGoogle Scholar - 5.S. Bianchini, N.A. Gusev, Steady nearly incompressible vector fields in two-dimension: chain rule and renormalization. Arch. Ration. Mech. Anal.
**222**(2), 451–505 (2016)MathSciNetCrossRefGoogle Scholar - 6.S. Bianchini, P. Bonicatto, N.A. Gusev, Renormalization for autonomous nearly incompressible BV vector fields in two dimensions. SIAM J. Math. Anal.
**48**(1), 1–33 (2016)MathSciNetCrossRefGoogle Scholar - 7.A. Bressan, An ill posed Cauchy problem for a hyperbolic system in two space dimensions. Rend. Semin. Mat. Univ. Padova
**110**, 103–117 (2003)MathSciNetzbMATHGoogle Scholar - 8.C. De Lellis, Notes on hyperbolic systems of conservation laws and transport equations,
*Handbook of Differential Equations: Evolutionary Equations*, vol. III (Elsevier/North-Holland, Amsterdam, 2007), pp. 277–382CrossRefGoogle Scholar - 9.R.J. DiPerna, P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math.
**98**(3), 511–547 (1989)MathSciNetCrossRefGoogle Scholar

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