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:
[1]

G. AlbertiS. 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.

[2]

Debora AmadoriSeung-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.

[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.

[4]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, New York, 2000.

[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.

[6]

S. BianchiniP. 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.

[7]

Stefano Bianchini and Paolo Bonicatto, A uniqueness result for the decomposition of vector fields in $ \mathbb R^d$, SISSA Preprint 15/2017/MATE.

[8]

V. I. BogachevG. Da PratoM. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

Stefano Modena and László Székelyhidi Jr, Non-uniqueness for the transport equation with Sobolev vector fields, preprint, arXiv: 1712.03867.

[18]

Stefano Modena and László Székelyhidi Jr, Non-renormalized solutions to the continuity equation, preprint, arXiv: 1806.09145.

show all references

References:
[1]

G. AlbertiS. 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.

[2]

Debora AmadoriSeung-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.

[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.

[4]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, New York, 2000.

[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.

[6]

S. BianchiniP. 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.

[7]

Stefano Bianchini and Paolo Bonicatto, A uniqueness result for the decomposition of vector fields in $ \mathbb R^d$, SISSA Preprint 15/2017/MATE.

[8]

V. I. BogachevG. Da PratoM. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

Stefano Modena and László Székelyhidi Jr, Non-uniqueness for the transport equation with Sobolev vector fields, preprint, arXiv: 1712.03867.

[18]

Stefano Modena and László Székelyhidi Jr, Non-renormalized solutions to the continuity equation, preprint, arXiv: 1806.09145.

[1]

Giuseppe Tomassetti. Smooth and non-smooth regularizations of the nonlinear diffusion equation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1519-1537. doi: 10.3934/dcdss.2017078

[2]

Franz W. Kamber and Peter W. Michor. The flow completion of a manifold with vector field. Electronic Research Announcements, 2000, 6: 95-97.

[3]

Constantin Christof, Christian Meyer, Stephan Walther, Christian Clason. Optimal control of a non-smooth semilinear elliptic equation. Mathematical Control & Related Fields, 2018, 8 (1) : 247-276. doi: 10.3934/mcrf.2018011

[4]

Chao Zhang, Lihe Wang, Shulin Zhou, Yun-Ho Kim. Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2559-2587. doi: 10.3934/cpaa.2014.13.2559

[5]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155

[6]

Mikhail I. Belishev, Aleksei F. Vakulenko. Non-smooth unobservable states in control problem for the wave equation in $\mathbb{R}^3$. Evolution Equations & Control Theory, 2014, 3 (2) : 247-256. doi: 10.3934/eect.2014.3.247

[7]

Tomasz Kaczynski, Marian Mrozek, Thomas Wanner. Towards a formal tie between combinatorial and classical vector field dynamics. Journal of Computational Dynamics, 2016, 3 (1) : 17-50. doi: 10.3934/jcd.2016002

[8]

Paul Glendinning. Non-smooth pitchfork bifurcations. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 457-464. doi: 10.3934/dcdsb.2004.4.457

[9]

Kazuhisa Ichikawa, Mahemauti Rouzimaimaiti, Takashi Suzuki. Reaction diffusion equation with non-local term arises as a mean field limit of the master equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 115-126. doi: 10.3934/dcdss.2012.5.115

[10]

Yangdong Xu, Shengjie Li. Continuity of the solution mappings to parametric generalized non-weak vector Ky Fan inequalities. Journal of Industrial & Management Optimization, 2017, 13 (2) : 967-975. doi: 10.3934/jimo.2016056

[11]

Lam Quoc Anh, Pham Thanh Duoc, Tran Ngoc Tam. Continuity of approximate solution maps to vector equilibrium problems. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1685-1699. doi: 10.3934/jimo.2017013

[12]

Biao Ou. Examinations on a three-dimensional differentiable vector field that equals its own curl. Communications on Pure & Applied Analysis, 2003, 2 (2) : 251-257. doi: 10.3934/cpaa.2003.2.251

[13]

Luis Bayón, Jose Maria Grau, Maria del Mar Ruiz, Pedro Maria Suárez. A hydrothermal problem with non-smooth Lagrangian. Journal of Industrial & Management Optimization, 2014, 10 (3) : 761-776. doi: 10.3934/jimo.2014.10.761

[14]

Richard A. Norton, G. R. W. Quispel. Discrete gradient methods for preserving a first integral of an ordinary differential equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1147-1170. doi: 10.3934/dcds.2014.34.1147

[15]

Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709

[16]

Tomás Caraballo, Renato Colucci, Luca Guerrini. Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-17. doi: 10.3934/dcdsb.2018268

[17]

Yubo Yuan, Weiguo Fan, Dongmei Pu. Spline function smooth support vector machine for classification. Journal of Industrial & Management Optimization, 2007, 3 (3) : 529-542. doi: 10.3934/jimo.2007.3.529

[18]

Alain Miranville, Costică Moroşanu. Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 537-556. doi: 10.3934/dcdss.2016011

[19]

Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure & Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577

[20]

Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the Kuramoto model on graphs Ⅰ. The mean field equation and transition point formulas. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 131-155. doi: 10.3934/dcds.2019006

2017 Impact Factor: 0.884

Article outline

[Back to Top]