Consider a non-local (i.e., involving a convolution term) conservation law: when the convolution ... more Consider a non-local (i.e., involving a convolution term) conservation law: when the convolution term converges to a Dirac delta, in the limit we formally recover a classical (or "local") conservation law. In this note we overview recent progress on this so-called non-local to local limit and in particular we discuss the case of anistropic kernels, which is extremely relevant in view of applications to traffic models. We also provide a new proof of a related compactness estimate.
For a scalar conservation law with strictly convex flux, by Oleinik's estimates the total variati... more For a scalar conservation law with strictly convex flux, by Oleinik's estimates the total variation of a solution with initial data u ∈ L ∞ (R) decays like t −1. This paper introduces a class of intermediate domains P α , 0 < α < 1, such that for u ∈ P α a faster decay rate is achieved: Tot.Var. u(t, •) ∼ t α−1. A key ingredient of the analysis is a "Fourier-type" decomposition of u into components which oscillate more and more rapidly. The results aim at extending the theory of fractional domains for analytic semigroups to an entirely nonlinear setting.
Springer proceedings in mathematics & statistics, 2018
We provide an informal presentation of the work mainly contained in [3]. We consider the entropy ... more We provide an informal presentation of the work mainly contained in [3]. We consider the entropy solution u of a scalar conservation law in one-space dimension. In particular we prove that the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This follows from a detailed analysis of the structure of the characteristics. We will introduce a few notions of Lagrangian representations and we prove that characteristics are segments outside a countably 1-rectifiable set. MSC: 35L65.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
We consider entropy solutions to the eikonal equation $|\nabla u|=1$ in two-space dimensions. The... more We consider entropy solutions to the eikonal equation $|\nabla u|=1$ in two-space dimensions. These solutions are motivated by a class of variational problems and fail in general to have bounded variation. Nevertheless, they share several of their fine properties with BV functions: we show in particular that the set of non-Lebesgue points has at least one co-dimension.
Calculus of Variations and Partial Differential Equations
We consider weak solutions with finite entropy production to the scalar conservation law ∂tu + di... more We consider weak solutions with finite entropy production to the scalar conservation law ∂tu + divxF (u) = 0 in (0, T) × R d. Building on the kinetic formulation we prove under suitable nonlinearity assumption on f that the set of non Lebesgue points of u has Hausdorff dimension at most d. A notion of Lagrangian representation for this class of solutions is introduced and this allows for a new interpretation of the entropy dissipation measure.
We consider a class of nonlocal conservation laws with exponential kernel and prove that quantiti... more We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term W := 1 (−∞,0] (•) exp(•) * ρ satisfy an Oleȋnik-type entropy condition. More precisely, under different sets of assumptions on the velocity function V , we prove that W satisfies a one-sided Lipschitz condition and that V (W)W ∂xW satisfies a one-sided bound, respectively. As a byproduct, we deduce that, as the exponential kernel is rescaled to converge to a Dirac delta distribution, the weak solution of the nonlocal problem converges to the unique entropy-admissible solution of the corresponding local conservation law, under the only assumption that the initial datum is essentially bounded and not necessarily of bounded variation. 2020 Mathematics Subject Classification. 35L65.
ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
We consider line-energy models of Ginzburg-Landau type in a two-dimensional simplyconnected bound... more We consider line-energy models of Ginzburg-Landau type in a two-dimensional simplyconnected bounded domain. Configurations of vanishing energy have been characterized by Jabin, Otto and Perthame: the domain must be a disk, and the configuration a vortex. We prove a quantitative version of this statement in the class of C 1,1 domains, improving on previous results by Lorent. In particular, the deviation of the domain from a disk is controlled by a power of the energy, and that power is optimal. The main tool is a Lagrangian representation introduced by the second author, which allows to decompose the energy along characteristic curves.
Nonlocal conservation laws (the signature feature being that the flux function depends on the sol... more Nonlocal conservation laws (the signature feature being that the flux function depends on the solution through the convolution with a given kernel) are extensively used in the modeling of vehicular traffic. In this work we discuss the singular local limit, namely the convergence of the nonlocal solutions to the entropy admissible solution of the conservation law obtained by replacing the convolution kernel with a Dirac delta. Albeit recent counterexamples rule out convergence in the general case, in the specific framework of traffic models (with anisotropic convolution kernels) the singular limit has been established under rigid assumptions, i.e. in the case of the exponential kernel (which entails algebraic identities between the kernel and its derivatives) or under fairly restrictive requirements on the initial datum. In this work we obtain general convergence results under assumptions that are entirely natural in view of applications to traffic models, plus a convexity requirement on the convolution kernels. We then provide a general criterion for entropy admissibility of the limit and a convergence rate. We also exhibit a counterexample showing that the convexity assumption is necessary for our main compactness estimate.
Let H ∈ C 1 ∩ W 2,p be an autonomous, non-constant Hamiltonian on a compact 2-dimensional manifol... more Let H ∈ C 1 ∩ W 2,p be an autonomous, non-constant Hamiltonian on a compact 2-dimensional manifold, generating an incompressible velocity field b = ∇ ⊥ H. We give sharp upper bounds on the enhanced dissipation rate of b in terms of the properties of the period T (h) of the close orbits {H = h}. Specifically, if 0 < ν 1 is the diffusion coefficient, the enhanced dissipation rate can be at most O(ν 1/3) in general, the bound improves when H has isolated, non-degenerate elliptic point. Our result provides the better bound O(ν 1/2) for the standard cellular flow given by Hc(x) = sin x1 sin x2, for which we can also prove a new upper bound on its mixing mixing rate and a lower bound on its enhanced dissipation rate. The proofs are based on the use of action-angle coordinates and on the existence of a good invariant domain for the regular Lagrangian flow generated by b.
We prove the stability of entropy solutions of nonlinear conservation laws with respect to pertur... more We prove the stability of entropy solutions of nonlinear conservation laws with respect to perturbations of the initial datum, the space-time dependent flux and the entropy inequalities. Such a general stability theorem is motivated by the study of problems in which the flux P [u](t, x, u) depends possibly non-locally on the solution itself. For these problems we show the conditional existence and uniqueness of entropy solutions. Moreover, the relaxation of the entropy inequality allows to treat approximate solutions arising from various numerical schemes. This can be used to derive the rate of convergence of the recent particle method introduced in [RS21] to solve a one-dimensional model of traffic with congestion, as well as recover already known rates for some other approximation methods.
We consider line-energy models of Ginzburg-Landau type in a two-dimensional simplyconnected bound... more We consider line-energy models of Ginzburg-Landau type in a two-dimensional simplyconnected bounded domain. Configurations of vanishing energy have been characterized by Jabin, Otto and Perthame: the domain must be a disk, and the configuration a vortex. We prove a quantitative version of this statement in the class of C 1,1 domains, improving on previous results by Lorent. In particular, the deviation of the domain from a disk is controlled by a power of the energy, and that power is optimal. The main tool is a Lagrangian representation introduced by the second author, which allows to decompose the energy along characteristic curves.
Nonlocal conservation laws (the signature feature being that the flux function depends on the sol... more Nonlocal conservation laws (the signature feature being that the flux function depends on the solution through the convolution with a given kernel) are extensively used in the modeling of vehicular traffic. In this work we discuss the singular local limit, namely the convergence of the nonlocal solutions to the entropy admissible solution of the conservation law obtained by replacing the convolution kernel with a Dirac delta. Albeit recent counterexamples rule out convergence in the general case, in the specific framework of traffic models (with anisotropic convolution kernels) the singular limit has been established under rigid assumptions, i.e. in the case of the exponential kernel (which entails algebraic identities between the kernel and its derivatives) or under fairly restrictive requirements on the initial datum. In this work we obtain general convergence results under assumptions that are entirely natural in view of applications to traffic models, plus a convexity requirement on the convolution kernels. We then provide a general criterion for entropy admissibility of the limit and a convergence rate. We also exhibit a counterexample showing that the convexity assumption is necessary for our main compactness estimate.
We focus on entropy admissible solutions of scalar conservation laws in one space dimension and e... more We focus on entropy admissible solutions of scalar conservation laws in one space dimension and establish new regularity results with respect to time. First, we assume that the flux function f is strictly convex and show that, for every x ∈ R, the total variation of the composite function f • u(•, x) is controlled by the total variation of the initial datum. Next, we assume that f is monotone and, under no convexity assumption, we show that, for every x, the total variation of the left and right trace u(•, x ±) is controlled by the total variation of the initial datum. We also exhibit a counterexample showing that in the first result the total variation bound does not extend to the function u, or equivalently that in the second result we cannot drop the monotonicity assumption. We then discuss applications to a source-destination model for traffic flows on road networks. We introduce a new approach, based on the analysis of transport equations with irregular coefficients, and, under the assumption that the network only contains so-called T-junctions, we establish existence and uniqueness results for merely bounded data in the class of solutions where the traffic is not congested. Our assumptions on the network and the traffic congestion are basically necessary to obtain wellposedness in view of a counterexample due to Bressan and Yu. We also establish stability and propagation of BV regularity, and this is again interesting in view of recent counterexamples .
We consider bounded weak solutions to the Burgers equation for which every entropy dissipation is... more We consider bounded weak solutions to the Burgers equation for which every entropy dissipation is representable by a measure and we prove that all these measures are concentrated on the graphs of countably many Lipschitz curves. The main tool is the Lagrangian representation, which is an extension of the method of characteristics to the non-smooth setting.
Springer Proceedings in Mathematics & Statistics, 2018
We provide an informal presentation of the work mainly contained in [3]. We consider the entropy ... more We provide an informal presentation of the work mainly contained in [3]. We consider the entropy solution u of a scalar conservation law in one-space dimension. In particular we prove that the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This follows from a detailed analysis of the structure of the characteristics. We will introduce a few notions of Lagrangian representations and we prove that characteristics are segments outside a countably 1-rectifiable set. MSC: 35L65.
We prove the existence of generalized characteristics for weak, not necessarily entropic, solutio... more We prove the existence of generalized characteristics for weak, not necessarily entropic, solutions of Burgers' equation ∂tu + ∂x u 2 2 = 0, whose entropy productions are signed measures. Such solutions arise in connection with large deviation principles for the hydrodynamic limit of interacting particle systems. The present work allows to remove a technical trace assumption in a recent result by the two first authors about the L 2 stability of entropic shocks among such non-entropic solutions. The proof relies on the Lagrangian representation of a solution's hypograph, recently constructed by the third author. In particular, we prove a decomposition formula for the entropy flux across a given hypersurface, which is valid for general multidimensional scalar conservation laws.
We prove that if u is the entropy solution to a scalar conservation law in one space dimension, t... more We prove that if u is the entropy solution to a scalar conservation law in one space dimension, then the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This result is a consequence of a detailed analysis of the structure of the characteristics. In particular the characteristic curves are segments outside a countably 1-rectifiable set and the left and right traces of the solution exist in a C0-sense up to the degeneracy due to the segments where f ′′ = 0. We prove also that the initial data is taken in a suitably strong sense and we give some counterexamples which show that these results are sharp. Preprint SISSA 43/2016/MATE
In this chapter we collect some preliminary and technical results that will be used in the main b... more In this chapter we collect some preliminary and technical results that will be used in the main body of this thesis. More in details, Section 1.1 deals with several independent topics: first we recall the Kuratowski convergence of sets in a metric space, then we introduce a decomposition of piecewise monotone functions in “undulations”. Next we recall the notion of BV(R) space and we give estimates of the generalized variation of piecewise monotone functions in terms of their undulations. Finally we mention some elementary properties on smooth functions for future references. In Section 1.2 we review some result about scalar conservation laws: the general theory is only mentioned, with some emphasis on the more relevant point for the following chapters. After recalling the fundamental theorem by Kruzkov, we introduce the wave-front tracking algorithm and the notion of measure valued entropy solution. Then we consider the problem in bounded domains: the related notion of admissible b...
We consider bounded entropy solutions to the scalar conservation law in one space dimension: \beg... more We consider bounded entropy solutions to the scalar conservation law in one space dimension: \begin{equation*} u_t+f(u)_x=0. \end{equation*} We quantify the regularizing effect of the non linearity of the flux $f$ on the solution $u$ in terms of spaces of functions with bounded generalized variation.
We introduce a notion of Lagrangian representation for entropy solutions to scalar conservation l... more We introduce a notion of Lagrangian representation for entropy solutions to scalar conservation laws in several space dimension { ∂tu + divx(f(u)) = 0 (t, x) ∈ (0,+∞)× Rd, u(0, x) = u0 t = 0. The construction is based on the transport collapse method introduced by Brenier. As a first application we show that if the solution u is continuous, then it is hypograph is given by the set { (t, x, h) : h ≤ u0(x− f(h)t) } , i.e. it is the translation of each level set of u0 by its characteristic speed. Preprint SISSA 36/2017/MATE
Consider a non-local (i.e., involving a convolution term) conservation law: when the convolution ... more Consider a non-local (i.e., involving a convolution term) conservation law: when the convolution term converges to a Dirac delta, in the limit we formally recover a classical (or "local") conservation law. In this note we overview recent progress on this so-called non-local to local limit and in particular we discuss the case of anistropic kernels, which is extremely relevant in view of applications to traffic models. We also provide a new proof of a related compactness estimate.
For a scalar conservation law with strictly convex flux, by Oleinik's estimates the total variati... more For a scalar conservation law with strictly convex flux, by Oleinik's estimates the total variation of a solution with initial data u ∈ L ∞ (R) decays like t −1. This paper introduces a class of intermediate domains P α , 0 < α < 1, such that for u ∈ P α a faster decay rate is achieved: Tot.Var. u(t, •) ∼ t α−1. A key ingredient of the analysis is a "Fourier-type" decomposition of u into components which oscillate more and more rapidly. The results aim at extending the theory of fractional domains for analytic semigroups to an entirely nonlinear setting.
Springer proceedings in mathematics & statistics, 2018
We provide an informal presentation of the work mainly contained in [3]. We consider the entropy ... more We provide an informal presentation of the work mainly contained in [3]. We consider the entropy solution u of a scalar conservation law in one-space dimension. In particular we prove that the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This follows from a detailed analysis of the structure of the characteristics. We will introduce a few notions of Lagrangian representations and we prove that characteristics are segments outside a countably 1-rectifiable set. MSC: 35L65.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
We consider entropy solutions to the eikonal equation $|\nabla u|=1$ in two-space dimensions. The... more We consider entropy solutions to the eikonal equation $|\nabla u|=1$ in two-space dimensions. These solutions are motivated by a class of variational problems and fail in general to have bounded variation. Nevertheless, they share several of their fine properties with BV functions: we show in particular that the set of non-Lebesgue points has at least one co-dimension.
Calculus of Variations and Partial Differential Equations
We consider weak solutions with finite entropy production to the scalar conservation law ∂tu + di... more We consider weak solutions with finite entropy production to the scalar conservation law ∂tu + divxF (u) = 0 in (0, T) × R d. Building on the kinetic formulation we prove under suitable nonlinearity assumption on f that the set of non Lebesgue points of u has Hausdorff dimension at most d. A notion of Lagrangian representation for this class of solutions is introduced and this allows for a new interpretation of the entropy dissipation measure.
We consider a class of nonlocal conservation laws with exponential kernel and prove that quantiti... more We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term W := 1 (−∞,0] (•) exp(•) * ρ satisfy an Oleȋnik-type entropy condition. More precisely, under different sets of assumptions on the velocity function V , we prove that W satisfies a one-sided Lipschitz condition and that V (W)W ∂xW satisfies a one-sided bound, respectively. As a byproduct, we deduce that, as the exponential kernel is rescaled to converge to a Dirac delta distribution, the weak solution of the nonlocal problem converges to the unique entropy-admissible solution of the corresponding local conservation law, under the only assumption that the initial datum is essentially bounded and not necessarily of bounded variation. 2020 Mathematics Subject Classification. 35L65.
ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
We consider line-energy models of Ginzburg-Landau type in a two-dimensional simplyconnected bound... more We consider line-energy models of Ginzburg-Landau type in a two-dimensional simplyconnected bounded domain. Configurations of vanishing energy have been characterized by Jabin, Otto and Perthame: the domain must be a disk, and the configuration a vortex. We prove a quantitative version of this statement in the class of C 1,1 domains, improving on previous results by Lorent. In particular, the deviation of the domain from a disk is controlled by a power of the energy, and that power is optimal. The main tool is a Lagrangian representation introduced by the second author, which allows to decompose the energy along characteristic curves.
Nonlocal conservation laws (the signature feature being that the flux function depends on the sol... more Nonlocal conservation laws (the signature feature being that the flux function depends on the solution through the convolution with a given kernel) are extensively used in the modeling of vehicular traffic. In this work we discuss the singular local limit, namely the convergence of the nonlocal solutions to the entropy admissible solution of the conservation law obtained by replacing the convolution kernel with a Dirac delta. Albeit recent counterexamples rule out convergence in the general case, in the specific framework of traffic models (with anisotropic convolution kernels) the singular limit has been established under rigid assumptions, i.e. in the case of the exponential kernel (which entails algebraic identities between the kernel and its derivatives) or under fairly restrictive requirements on the initial datum. In this work we obtain general convergence results under assumptions that are entirely natural in view of applications to traffic models, plus a convexity requirement on the convolution kernels. We then provide a general criterion for entropy admissibility of the limit and a convergence rate. We also exhibit a counterexample showing that the convexity assumption is necessary for our main compactness estimate.
Let H ∈ C 1 ∩ W 2,p be an autonomous, non-constant Hamiltonian on a compact 2-dimensional manifol... more Let H ∈ C 1 ∩ W 2,p be an autonomous, non-constant Hamiltonian on a compact 2-dimensional manifold, generating an incompressible velocity field b = ∇ ⊥ H. We give sharp upper bounds on the enhanced dissipation rate of b in terms of the properties of the period T (h) of the close orbits {H = h}. Specifically, if 0 < ν 1 is the diffusion coefficient, the enhanced dissipation rate can be at most O(ν 1/3) in general, the bound improves when H has isolated, non-degenerate elliptic point. Our result provides the better bound O(ν 1/2) for the standard cellular flow given by Hc(x) = sin x1 sin x2, for which we can also prove a new upper bound on its mixing mixing rate and a lower bound on its enhanced dissipation rate. The proofs are based on the use of action-angle coordinates and on the existence of a good invariant domain for the regular Lagrangian flow generated by b.
We prove the stability of entropy solutions of nonlinear conservation laws with respect to pertur... more We prove the stability of entropy solutions of nonlinear conservation laws with respect to perturbations of the initial datum, the space-time dependent flux and the entropy inequalities. Such a general stability theorem is motivated by the study of problems in which the flux P [u](t, x, u) depends possibly non-locally on the solution itself. For these problems we show the conditional existence and uniqueness of entropy solutions. Moreover, the relaxation of the entropy inequality allows to treat approximate solutions arising from various numerical schemes. This can be used to derive the rate of convergence of the recent particle method introduced in [RS21] to solve a one-dimensional model of traffic with congestion, as well as recover already known rates for some other approximation methods.
We consider line-energy models of Ginzburg-Landau type in a two-dimensional simplyconnected bound... more We consider line-energy models of Ginzburg-Landau type in a two-dimensional simplyconnected bounded domain. Configurations of vanishing energy have been characterized by Jabin, Otto and Perthame: the domain must be a disk, and the configuration a vortex. We prove a quantitative version of this statement in the class of C 1,1 domains, improving on previous results by Lorent. In particular, the deviation of the domain from a disk is controlled by a power of the energy, and that power is optimal. The main tool is a Lagrangian representation introduced by the second author, which allows to decompose the energy along characteristic curves.
Nonlocal conservation laws (the signature feature being that the flux function depends on the sol... more Nonlocal conservation laws (the signature feature being that the flux function depends on the solution through the convolution with a given kernel) are extensively used in the modeling of vehicular traffic. In this work we discuss the singular local limit, namely the convergence of the nonlocal solutions to the entropy admissible solution of the conservation law obtained by replacing the convolution kernel with a Dirac delta. Albeit recent counterexamples rule out convergence in the general case, in the specific framework of traffic models (with anisotropic convolution kernels) the singular limit has been established under rigid assumptions, i.e. in the case of the exponential kernel (which entails algebraic identities between the kernel and its derivatives) or under fairly restrictive requirements on the initial datum. In this work we obtain general convergence results under assumptions that are entirely natural in view of applications to traffic models, plus a convexity requirement on the convolution kernels. We then provide a general criterion for entropy admissibility of the limit and a convergence rate. We also exhibit a counterexample showing that the convexity assumption is necessary for our main compactness estimate.
We focus on entropy admissible solutions of scalar conservation laws in one space dimension and e... more We focus on entropy admissible solutions of scalar conservation laws in one space dimension and establish new regularity results with respect to time. First, we assume that the flux function f is strictly convex and show that, for every x ∈ R, the total variation of the composite function f • u(•, x) is controlled by the total variation of the initial datum. Next, we assume that f is monotone and, under no convexity assumption, we show that, for every x, the total variation of the left and right trace u(•, x ±) is controlled by the total variation of the initial datum. We also exhibit a counterexample showing that in the first result the total variation bound does not extend to the function u, or equivalently that in the second result we cannot drop the monotonicity assumption. We then discuss applications to a source-destination model for traffic flows on road networks. We introduce a new approach, based on the analysis of transport equations with irregular coefficients, and, under the assumption that the network only contains so-called T-junctions, we establish existence and uniqueness results for merely bounded data in the class of solutions where the traffic is not congested. Our assumptions on the network and the traffic congestion are basically necessary to obtain wellposedness in view of a counterexample due to Bressan and Yu. We also establish stability and propagation of BV regularity, and this is again interesting in view of recent counterexamples .
We consider bounded weak solutions to the Burgers equation for which every entropy dissipation is... more We consider bounded weak solutions to the Burgers equation for which every entropy dissipation is representable by a measure and we prove that all these measures are concentrated on the graphs of countably many Lipschitz curves. The main tool is the Lagrangian representation, which is an extension of the method of characteristics to the non-smooth setting.
Springer Proceedings in Mathematics & Statistics, 2018
We provide an informal presentation of the work mainly contained in [3]. We consider the entropy ... more We provide an informal presentation of the work mainly contained in [3]. We consider the entropy solution u of a scalar conservation law in one-space dimension. In particular we prove that the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This follows from a detailed analysis of the structure of the characteristics. We will introduce a few notions of Lagrangian representations and we prove that characteristics are segments outside a countably 1-rectifiable set. MSC: 35L65.
We prove the existence of generalized characteristics for weak, not necessarily entropic, solutio... more We prove the existence of generalized characteristics for weak, not necessarily entropic, solutions of Burgers' equation ∂tu + ∂x u 2 2 = 0, whose entropy productions are signed measures. Such solutions arise in connection with large deviation principles for the hydrodynamic limit of interacting particle systems. The present work allows to remove a technical trace assumption in a recent result by the two first authors about the L 2 stability of entropic shocks among such non-entropic solutions. The proof relies on the Lagrangian representation of a solution's hypograph, recently constructed by the third author. In particular, we prove a decomposition formula for the entropy flux across a given hypersurface, which is valid for general multidimensional scalar conservation laws.
We prove that if u is the entropy solution to a scalar conservation law in one space dimension, t... more We prove that if u is the entropy solution to a scalar conservation law in one space dimension, then the entropy dissipation is a measure concentrated on countably many Lipschitz curves. This result is a consequence of a detailed analysis of the structure of the characteristics. In particular the characteristic curves are segments outside a countably 1-rectifiable set and the left and right traces of the solution exist in a C0-sense up to the degeneracy due to the segments where f ′′ = 0. We prove also that the initial data is taken in a suitably strong sense and we give some counterexamples which show that these results are sharp. Preprint SISSA 43/2016/MATE
In this chapter we collect some preliminary and technical results that will be used in the main b... more In this chapter we collect some preliminary and technical results that will be used in the main body of this thesis. More in details, Section 1.1 deals with several independent topics: first we recall the Kuratowski convergence of sets in a metric space, then we introduce a decomposition of piecewise monotone functions in “undulations”. Next we recall the notion of BV(R) space and we give estimates of the generalized variation of piecewise monotone functions in terms of their undulations. Finally we mention some elementary properties on smooth functions for future references. In Section 1.2 we review some result about scalar conservation laws: the general theory is only mentioned, with some emphasis on the more relevant point for the following chapters. After recalling the fundamental theorem by Kruzkov, we introduce the wave-front tracking algorithm and the notion of measure valued entropy solution. Then we consider the problem in bounded domains: the related notion of admissible b...
We consider bounded entropy solutions to the scalar conservation law in one space dimension: \beg... more We consider bounded entropy solutions to the scalar conservation law in one space dimension: \begin{equation*} u_t+f(u)_x=0. \end{equation*} We quantify the regularizing effect of the non linearity of the flux $f$ on the solution $u$ in terms of spaces of functions with bounded generalized variation.
We introduce a notion of Lagrangian representation for entropy solutions to scalar conservation l... more We introduce a notion of Lagrangian representation for entropy solutions to scalar conservation laws in several space dimension { ∂tu + divx(f(u)) = 0 (t, x) ∈ (0,+∞)× Rd, u(0, x) = u0 t = 0. The construction is based on the transport collapse method introduced by Brenier. As a first application we show that if the solution u is continuous, then it is hypograph is given by the set { (t, x, h) : h ≤ u0(x− f(h)t) } , i.e. it is the translation of each level set of u0 by its characteristic speed. Preprint SISSA 36/2017/MATE
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