![]() Lions, Ordinary differential equations, transport theory and Sobolev spaces. III (Elsevier/North-Holland, Amsterdam, 2007), pp. De Lellis, Notes on hyperbolic systems of conservation laws and transport equations, Handbook of Differential Equations: Evolutionary Equations, vol. Now we shall define the weak solution to the transport equation. Bressan, An ill posed Cauchy problem for a hyperbolic system in two space dimensions. Gusev, Renormalization for autonomous nearly incompressible BV vector fields in two dimensions. Gusev, Steady nearly incompressible vector fields in two-dimension: chain rule and renormalization. ![]() In particular, u u must be a scalar field for something like comparison with cones or viscosity solutions to make. ![]() They turn out to be equivalent, and both essentially amount to asserting that u u satisfies the correct form of the maximum principle. Pallara, Functions of Bounded Variation and Free Discontinuity Problems (Oxford Science Publications, Clarendon Press, 2000) There are two good notions of weak solution here: viscosity solutions, and comparison with cones. Ambrosio, Transport equation and Cauchy problem for BV vector fields. Crippa, A uniqueness result for the continuity equation in two dimensions. Crippa, Structure of level sets and Sard-type properties of Lipschitz maps. ![]() 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, ) that allows to reduce ( 1) to a family of 1-dimensional equations which can be solved explicitly, thus yielding uniqueness for the original problem. 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 \(\mathrm \) and it is nearly incompressible. Furthermore, from the Lagrangian point of view, this gives insights on the structure of the flow of non-smooth vector fields. Bressan, raised while studying the well-posedness of a class of hyperbolic conservation laws. This problem is related to a conjecture made by A. ![]()
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