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Hybrid inverse problems and internal functionals
 of Math. Sci. Res. Inst. Publ
, 2013
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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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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.
G.Crippa: Equations de transport à coefficient dont le gradient est donné par une intégrale singuliére. Séminaire Équations aux dérivées partielles
 Exp
, 2009
"... Abstract. Nous rappelons tout d’abord l’approche maintenant classique de renormalisation pour établir l’unicité des solutions faibles des équations de transport linéaires, en mentionnant les résultats récents qui s’y rattachent. Ensuite, nous montrons comment l’approche alternative introduite par Cr ..."
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Cited by 6 (1 self)
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Abstract. Nous rappelons tout d’abord l’approche maintenant classique de renormalisation pour établir l’unicité des solutions faibles des équations de transport linéaires, en mentionnant les résultats récents qui s’y rattachent. Ensuite, nous montrons comment l’approche alternative introduite par Crippa et DeLellis estimant directement le flot lagrangien permet d’obtenir des résultats nouveaux. Nous établissons l’existence et l’unicité du flot associé à une équation de transport dont le coefficient a un gradient donné par l’intégrale singulière d’une fonction intégrable. L’application au système d’Euler bidimensionnel des fluides incompressibles et au système de VlasovPoisson permet d’obtenir des résultats nouveaux de convergence forte pour des suites de solutions. 1. Equations de transport linéaires scalaires ⊲ Nous considérons des équations de transport multidimensionnelles ∂tu + divx(bu) + lu = f, (1)
LAGRANGIAN FLOWS FOR VECTOR FIELDS WITH GRADIENT GIVEN BY A SINGULAR INTEGRAL
"... Abstract. We prove quantitative estimates on flows of ordinary differential equations with vector field with gradient given by a singular integral of an L 1 function. Such estimates allow to prove existence, uniqueness, quantitative stability and compactness for the flow, going beyond the BV theory. ..."
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Abstract. We prove quantitative estimates on flows of ordinary differential equations with vector field with gradient given by a singular integral of an L 1 function. Such estimates allow to prove existence, uniqueness, quantitative stability and compactness for the flow, going beyond the BV theory. We illustrate the related wellposedness theory of Lagrangian solutions to the continuity and transport equations. 1.
Some new wellposedness results for continuity and transport equations, and applications to the chromatography system
 SIAM J. MATH. ANAL
"... We obtain various new wellposedness results for continuity and transport equations, among them an existence and uniqueness theorem (in the class of strongly continuous solutions) in the case of nearly incompressible vector fields, possibly having a blowup of the BV norm at the initial time. We a ..."
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Cited by 3 (1 self)
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We obtain various new wellposedness results for continuity and transport equations, among them an existence and uniqueness theorem (in the class of strongly continuous solutions) in the case of nearly incompressible vector fields, possibly having a blowup of the BV norm at the initial time. We apply these results (valid in any space dimension) to the k × k chromatography system of conservation laws and to the k × k Keyfitz and Kranzer system, both in one space dimension.
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 2 (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.
Compactness for nonlinear continuity equations
, 2012
"... Abstract. We prove compactness and hence existence for solutions to a class of non linear transport equations. The corresponding models combine the features of linear transport equations and scalar conservation laws. We introduce a new method which gives quantitative compactness estimates compatible ..."
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Abstract. We prove compactness and hence existence for solutions to a class of non linear transport equations. The corresponding models combine the features of linear transport equations and scalar conservation laws. We introduce a new method which gives quantitative compactness estimates compatible with both frameworks.
Inside Out II MSRI Publications
"... Hybrid inverse problems and internal functionals GUILLAUME BAL This paper reviews recent results on hybrid inverse problems, which are also called coupledphysics inverse problems of multiwave inverse problems. Inverse problems tend to be most useful in, e.g., medical and geophysical imaging, when t ..."
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Hybrid inverse problems and internal functionals GUILLAUME BAL This paper reviews recent results on hybrid inverse problems, which are also called coupledphysics inverse problems of multiwave inverse problems. Inverse problems tend to be most useful in, e.g., medical and geophysical imaging, when they combine high contrast with high resolution. In some settings, a single modality displays either high contrast or high resolution but not both. In favorable situations, physical effects couple one modality with high contrast with another modality with high resolution. The mathematical analysis of such couplings forms the class of hybrid inverse problems. Hybrid inverse problems typically involve two steps. In a first step, a wellposed problem involving the highresolution lowcontrast modality is solved from knowledge of boundary measurements. In a second step, a quantitative reconstruction of the parameters of interest is performed from knowledge of the pointwise, internal, functionals of the parameters reconstructed during the first step. This paper reviews mathematical techniques that have been developed in recent years to address the second step. Mathematically, many hybrid inverse problems find interpretations in terms of linear and nonlinear (systems of) equations. In the analysis of such equations, one often needs to verify that qualitative properties of solutions to elliptic linear equations are satisfied, for instance the absence of any critical points. This paper reviews several methods to prove that such qualitative properties hold, including the method based on the construction of complex geometric optics solutions.