Abstract

Given a bounded autonomous vector field $b \colon \mathbb{R}^2 \to \mathbb{R}^2$, we study the uniqueness of bounded solutions to the initial value problem for the related transport equation $\partial_t u + b \cdot \nabla u= 0.$ Assuming that $b$ is of class BV and it is nearly incompressible, we prove uniqueness of weak solutions to the transport equation. The present work extends the result obtained in [S. Bianchini and N. A. Gusev, Steady nearly incompressible vector fields in $2$D: Chain rule and Renormalization, preprint, 2014], where the steady nearly incompressible case is treated. Our proof is based on splitting the equation onto a suitable partition of the plane: this technique was introduced in [G. Alberti, S. Bianchini, and G. Crippa, J. Eur. Math. Soc. (JEMS), 16 (2014), pp. 201--234], using the results on the structure of level sets of Lipschitz maps obtained in [G. Alberti, S. Bianchini, and G. Crippa, Ann. Sc. Norm. Super. Pisa Cl. Sci.(s), 12 (2013), pp. 863--902]. Furthermore, in order to construct the partition, we use Ambrosio's superposition principle [L. Ambrosio, Invent. Math., 158 (2004), pp. 227--260].

Keywords

  1. transport equation
  2. continuity equation
  3. renormalization
  4. disintegration of measures
  5. Lipschitz functions
  6. superposition principle

MSC codes

  1. 35F10
  2. 35L03
  3. 28A50
  4. 35D30

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References

1.
G. Alberti, S. Bianchini, and G. Crippa, Structure of level sets and Sard-type properties of Lipschitz maps, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 12 (2013), pp. 863--902.
2.
G. Alberti, S. Bianchini, and G. Crippa, A uniqueness result for the continuity equation in two dimensions, J. Eur. Math. Soc. (JEMS), 16 (2014), pp. 201--234.
3.
L. Ambrosio, Transport equation and cauchy problem for $bv$ vector fields, Invent. Math., 158 (2004), pp. 227--260.
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L. Ambrosio, F. Bouchut, and C. De Lellis, Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions, Comm. Partial Differential Equations, 29 (2004), pp. 1635--1651.
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L. Ambrosio, G. Crippa, and S. Maniglia, Traces and fine properties of a BD class of vector fields and applications, Ann. Fac. Sci. Toulouse Math. (6), 14 (2005), pp. 527--561.
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L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Science Publications, Clarendon Press, Oxford, UK, 2000.
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S. Bianchini and N. A. Gusev, Steady Nearly Incompressible Vector Fields in 2D: Chain Rule and Renormalization, preprint, 2014.
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C. De Lellis, Notes on Hyperbolic Systems of Conservation Laws and Transport Equations, in Handbook of Differential Equations: Evolutionary Equations., Vol. 3, Elsevier/North-Holland, Amsterdam, 2007, pp. 277--382.
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B. L. Keyfitz and H. C. Kranzer, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Ration. Mech. Anal., 72 (1979/80), pp. 219--241.

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Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 1 - 33
ISSN (online): 1095-7154

History

Submitted: 6 February 2015
Accepted: 5 October 2015
Published online: 6 January 2016

Keywords

  1. transport equation
  2. continuity equation
  3. renormalization
  4. disintegration of measures
  5. Lipschitz functions
  6. superposition principle

MSC codes

  1. 35F10
  2. 35L03
  3. 28A50
  4. 35D30

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