Article Contents
Article Contents

# On the one-dimensional continuity equation with a nearly incompressible vector field

The publication was prepared with the support of the "RUDN University Program 5-100"

• We consider the Cauchy problem for the continuity equation with a bounded nearly incompressible vector field $b\colon (0,T) × \mathbb{R}^d \to \mathbb{R}^d$, $T>0$. This class of vector fields arises in the context of hyperbolic conservation laws (in particular, the Keyfitz-Kranzer system, which has applications in nonlinear elasticity theory).

It is well known that in the generic multi-dimensional case ($d≥ 1$) near incompressibility is sufficient for existence of bounded weak solutions, but uniqueness may fail (even when the vector field is divergence-free), and hence further assumptions on the regularity of $b$ (e.g. Sobolev regularity) are needed in order to obtain uniqueness.

We prove that in the one-dimensional case ($d = 1$) near incompressibility is sufficient for existence and uniqueness of locally integrable weak solutions. We also study compactness properties of the associated Lagrangian flows.

Mathematics Subject Classification: Primary: 35D30, 34A12; Secondary: 34A36.

 Citation:

•  G. Alberti , S. Bianchini  and  G. Crippa , A uniqueness result for the continuity equation in two dimensions, J. Eur. Math. Soc. (JEMS), 16 (2014) , 201-234.  doi: 10.4171/JEMS/431. Debora Amadori , Seung-Yeal Ha  and  Jinyeong Park , On the global well-posedness of BV weak solutions to the Kuramoto–Sakaguchi equation, Journal of Differential Equations, 262 (2017) , 978-1022.  doi: 10.1016/j.jde.2016.10.004. L. Ambrosio , Transport equation and Cauchy problem for BV vector fields, Invent. Math., 158 (2004) , 227-260.  doi: 10.1007/s00222-004-0367-2. L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, New York, 2000. S. Bianchini , On Bressan's conjecture on mixing properties of vector fields, Banach Center Publications, 74 (2006) , 13-31.  doi: 10.4064/bc74-0-1. S. Bianchini , P. Bonicatto  and  N. A. Gusev , Renormalization for autonomous nearly incompressible bv vector fields in two dimensions, SIAM Journal on Mathematical Analysis, 48 (2016) , 1-33.  doi: 10.1137/15M1007380. Stefano Bianchini and Paolo Bonicatto, A uniqueness result for the decomposition of vector fields in $\mathbb R^d$, SISSA Preprint 15/2017/MATE. V. I. Bogachev , G. Da Prato , M. Röckner  and  S. V. Shaposhnikov , On the uniqueness of solutions to continuity equations, J. Differential Equations, 259 (2015) , 3854-3873.  doi: 10.1016/j.jde.2015.05.003. F. Bouchut  and  F. James , One-dimensional transport equations with discontinuous coefficients, Nonlinear Analysis: Theory, Methods and Applications, 32 (1998) , 891-933.  doi: 10.1016/S0362-546X(97)00536-1. A. Bressan , An ill posed Cauchy problem for a hyperbolic system in two space dimensions, Rend. Sem. Mat. Univ. Padova, 110 (2003) , 103-117. Laura Caravenna  and  Gianluca Crippa , Uniqueness and Lagrangianity for solutions with lack of integrability of the continuity equation, Comptes Rendus Mathematique, 354 (2016) , 1168-1173.  doi: 10.1016/j.crma.2016.10.009. G. Crippa, The Flow Associated to Weakly Differentiable Vector Fields, Theses of Scuola Normale Superiore di Pisa (New Series), 12 Edizioni della Normale, 2009. G. Crippa , Lagrangian flows and the one-dimensional Peano phenomenon for ODEs, J. Differential Equations, 250 (2011) , 3135-3149.  doi: 10.1016/j.jde.2010.12.007. C. De Lellis , Notes on hyperbolic systems of conservation laws and transport equations, Journal: Handbook of Differential Equations: Evolutionary Differential Equations, 3 (2006) , 277-383.  doi: 10.1016/S1874-5717(07)80007-7. R. J. DiPerna  and  P.-L. Lions , Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989) , 511-547.  doi: 10.1007/BF01393835. B. L. Keyfitz  and  H. C. Kranzer , A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Ration. Mech. Anal., 72 (1979) , 219-241.  doi: 10.1007/BF00281590. Stefano Modena and László Székelyhidi Jr, Non-uniqueness for the transport equation with Sobolev vector fields, preprint, arXiv: 1712.03867. Stefano Modena and László Székelyhidi Jr, Non-renormalized solutions to the continuity equation, preprint, arXiv: 1806.09145.