Well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum under the non-collinearity condition
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March 2019, 18(2): 559-568. doi: 10.3934/cpaa.2019028
On the one-dimensional continuity equation with a nearly incompressible vector field
Nikolay A. Gusev 1,2,3,
Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700, Russia
RUDN University, 6 Miklukho-Maklay St, Moscow, 117198, Russia
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.
Keywords: Continuity equation, nearly incompressible vector field, non-smooth vector field, ordinary differential equation.
Mathematics Subject Classification: Primary: 35D30, 34A12; Secondary: 34A36.
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
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
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
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
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford, New York, 2000. Google Scholar
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
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
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
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
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
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
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
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
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
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
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
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
Stefano Modena and László Székelyhidi Jr, Non-uniqueness for the transport equation with Sobolev vector fields, preprint, arXiv: 1712.03867. Google Scholar
Stefano Modena and László Székelyhidi Jr, Non-renormalized solutions to the continuity equation, preprint, arXiv: 1806.09145. Google Scholar
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
Franz W. Kamber and Peter W. Michor. The flow completion of a manifold with vector field. Electronic Research Announcements, 2000, 6: 95-97.
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
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
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
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
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Nikolay A. Gusev