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: |
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.
![]() |