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Differential Equations with singular fields
"... Abstract. This paper investigates the well posedness of ordinary differential equations and more precisely the existence (or uniqueness) of a flow through explicit compactness estimates. Instead of assuming a bounded divergence condition on the vector field, a compressibility condition on the flow ( ..."
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Cited by 12 (4 self)
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Abstract. This paper investigates the well posedness of ordinary differential equations and more precisely the existence (or uniqueness) of a flow through explicit compactness estimates. Instead of assuming a bounded divergence condition on the vector field, a compressibility condition on the flow (bounded jacobian) is considered. The main result provides existence under the condition that the vector field belongs to BV in dimension 2 and SBV in higher dimensions. 1
Wellposedness in any dimension for Hamiltonian flows with non BV force terms
, 2009
"... We study existence and uniqueness for the classical dynamics of a particle in a force field in the phase space. Through an explicit control on the regularity of the trajectories, we show that this is well posed if the force belongs to the Sobolev space H 3/4. ..."
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Cited by 10 (4 self)
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We study existence and uniqueness for the classical dynamics of a particle in a force field in the phase space. Through an explicit control on the regularity of the trajectories, we show that this is well posed if the force belongs to the Sobolev space H 3/4.
Uniqueness and nonuniqueness for nonsmooth divergence free transport
 Séminaire XEDP, Ecole Polytechnique
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Transport equations with partially BV velocities
 ANNALI SCUOLA NORMALE SUPERIORE
, 2004
"... We prove the uniqueness of weak solutions for the Cauchy problem for a class of transport equations whose velocities are partially with bounded variation. Our result deals with the initial value problem ∂tu + Xu = f, ut=0 = g, where X is the vector field a1(x1) · ∂x1 + a2(x1, x2) · ∂x2, a1 ∈ BV ..."
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Cited by 8 (0 self)
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We prove the uniqueness of weak solutions for the Cauchy problem for a class of transport equations whose velocities are partially with bounded variation. Our result deals with the initial value problem ∂tu + Xu = f, ut=0 = g, where X is the vector field a1(x1) · ∂x1 + a2(x1, x2) · ∂x2, a1 ∈ BV (RN1x1), a2 ∈ L1x1
A note on twodimensional transport with bounded divergence
 Comm. PDE
"... Abstract. We prove uniqueness for two dimensional transport across a noncharacteristic curve, under the hypothesis that the vector field is autonomous, bounded and with bounded divergence. We also obtain uniqueness for the Cauchy problem in Rt×R 2 x under an additional condition on the local directi ..."
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Cited by 8 (1 self)
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Abstract. We prove uniqueness for two dimensional transport across a noncharacteristic curve, under the hypothesis that the vector field is autonomous, bounded and with bounded divergence. We also obtain uniqueness for the Cauchy problem in Rt×R 2 x under an additional condition on the local direction of the vector field. 1.
The ordinary differential equation with nonLipschitz vector fields
, 2008
"... In this note we survey some recent results on the wellposedness of the ordinary differential equation with nonLipschitz vector fields. We introduce the notion of regular Lagrangian flow, which is the right concept of solution in this framework. We present two different approaches to the theory of ..."
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Cited by 4 (0 self)
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In this note we survey some recent results on the wellposedness of the ordinary differential equation with nonLipschitz vector fields. We introduce the notion of regular Lagrangian flow, which is the right concept of solution in this framework. We present two different approaches to the theory of regular Lagrangian flows. The first one is quite general and is based on the connection with the continuity equation, via the superposition principle. The second one exploits some quantitative apriori estimates and provides stronger results in the case of Sobolev regularity of the vector field.
C 1 Measure Respecting Maps Preserve BV Iff they have Bounded Derivative
"... If Ωj ∈ R d are bounded open subsets and Φ ∈ C 1 (Ω1; Ω2) respects Lebesgue measure and satisfies F ◦ Φ ∈ BV (Ω1) for all F ∈ BV (Ω2) then Φ is uniformly Lipshitzean. The problem addressed in this note was motivated by the study of the propagation of regularity in the transport by vector fields with ..."
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If Ωj ∈ R d are bounded open subsets and Φ ∈ C 1 (Ω1; Ω2) respects Lebesgue measure and satisfies F ◦ Φ ∈ BV (Ω1) for all F ∈ BV (Ω2) then Φ is uniformly Lipshitzean. The problem addressed in this note was motivated by the study of the propagation of regularity in the transport by vector fields with bounded divergence, ∂u ∂t + d∑ aj(x, t) ∂u j=1 where x = (x1, x2, · · · , xd) and, ∂xj = 0, x ∈ R d, d ≥ 2, t> 0, (1) divxa = d∑ j=1 ∂xj aj(x, t) ∈ L ∞ ( [0, T] × R d) in the sense of distribution. To guarantee the uniqueness of L ∞ solutions of Cauchy problem it suffices to assume that (cf. [Am]) a = (a1, a2, · · · , ad) ∈ L 1 ([0, T], BVloc(R d)) ∩ L 1 ([0, T], L ∞ (R d)). Then for arbitrary initial data u0(x) ∈ L ∞ (R d) there is a unique solution u(x, t) ∈ L ∞ ( [0, T] × R d) with ut=0 = u0. With the same hypotheses, there is a well defined flow Φt and the solution is given by u(t) = u0 ◦ Φ−t. The flow respects Lebesgue measure in the sense of (3) below.