# American Institute of Mathematical Sciences

March  2019, 18(2): 559-568. doi: 10.3934/cpaa.2019028

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

 1 Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700, Russia 2 RUDN University, 6 Miklukho-Maklay St, Moscow, 117198, Russia 3 Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St, Moscow, 119991

Received  June 2017 Revised  July 2018 Published  October 2018

Fund Project: 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.

Citation: Nikolay A. Gusev. On the one-dimensional continuity equation with a nearly incompressible vector field. Communications on Pure & Applied Analysis, 2019, 18 (2) : 559-568. doi: 10.3934/cpaa.2019028
##### References:
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##### References:
 [1] 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.  Google Scholar [2] 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.  Google Scholar [3] L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math., 158 (2004), 227-260.  doi: 10.1007/s00222-004-0367-2.  Google Scholar [4] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, New York, 2000.  Google Scholar [5] 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.  Google Scholar [6] 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.  Google Scholar [7] Stefano Bianchini and Paolo Bonicatto, A uniqueness result for the decomposition of vector fields in $\mathbb R^d$, SISSA Preprint 15/2017/MATE. Google Scholar [8] 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.  Google Scholar [9] 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.  Google Scholar [10] A. Bressan, An ill posed Cauchy problem for a hyperbolic system in two space dimensions, Rend. Sem. Mat. Univ. Padova, 110 (2003), 103-117.   Google Scholar [11] 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.  Google Scholar [12] G. Crippa, The Flow Associated to Weakly Differentiable Vector Fields, Theses of Scuola Normale Superiore di Pisa (New Series), 12 Edizioni della Normale, 2009.  Google Scholar [13] 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.  Google Scholar [14] 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.  Google Scholar [15] 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.  Google Scholar [16] 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.  Google Scholar [17] Stefano Modena and László Székelyhidi Jr, Non-uniqueness for the transport equation with Sobolev vector fields, preprint, arXiv: 1712.03867. Google Scholar [18] Stefano Modena and László Székelyhidi Jr, Non-renormalized solutions to the continuity equation, preprint, arXiv: 1806.09145. Google Scholar
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