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.  Google Scholar

[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.  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. 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.  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. 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.  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

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.  Google Scholar

[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.  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. 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.  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. 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.  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

[1]

Mourad Bellassoued, Ibtissem Ben Aïcha, Zouhour Rezig. Stable determination of a vector field in a non-Self-Adjoint dynamical Schrödinger equation on Riemannian manifolds. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020042

[2]

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

[3]

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

[4]

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

[5]

Robert Roussarie. A topological study of planar vector field singularities. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5217-5245. doi: 10.3934/dcds.2020226

[6]

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

[7]

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

[8]

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

[9]

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

[10]

Angela Aguglia, Antonio Cossidente, Giuseppe Marino, Francesco Pavese, Alessandro Siciliano. Orbit codes from forms on vector spaces over a finite field. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020105

[11]

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

[12]

Yanfei Lu, Qingfei Yin, Hongyi Li, Hongli Sun, Yunlei Yang, Muzhou Hou. Solving higher order nonlinear ordinary differential equations with least squares support vector machines. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1481-1502. doi: 10.3934/jimo.2019012

[13]

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

[14]

Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108

[15]

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

[16]

Yin Yang, Sujuan Kang, Vasiliy I. Vasil'ev. The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutions. Electronic Research Archive, 2020, 28 (3) : 1161-1189. doi: 10.3934/era.2020064

[17]

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

[18]

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

[19]

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

[20]

Daniel G. Alfaro Vigo, Amaury C. Álvarez, Grigori Chapiro, Galina C. García, Carlos G. Moreira. Solving the inverse problem for an ordinary differential equation using conjugation. Journal of Computational Dynamics, 2020, 7 (2) : 183-208. doi: 10.3934/jcd.2020008

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (108)
  • HTML views (214)
  • Cited by (0)

Other articles
by authors

[Back to Top]