arXiv subjects

Luca Talamini

Publications and source records attributed to Luca Talamini.

8 recordsLinked to original sources

arXiv2026

Strong traces for solutions of $2\times 2$ nonlinear hyperbolic systems

In this paper we show that bounded entropy solutions of genuinely nonlinear $2\times 2$ systems of hyperbolic conservation laws admit a strong trace on Lipschitz curves. The main argument relies on a new half-space Liouville-type theorem for isentropic solutions with constant normal traces, providing the first extension to general genuinely nonlinear systems of the strong trace properties available for scalar conservation laws.

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arXiv2026

Measure preserving maps with bounded total variation

Consider a piecewise affine Lipschitz map $\phi : \Omega \to \mathbb R$, where $\Omega \subset \mathbb R^d$ is an open set, and assume that $x \mapsto x + t \nabla \phi(x)$ is injective for almost every $t > 0$. In (J.-G. Liu, R.~L. Pego, \emph{Rigidly breaking potential flows and a countable Alexandrov theorem for polytopes}, Pure Appl. Anal., \textbf{7}(4), 2025) the authors conjecture that every such $\phi$ must be locally convex. We prove the result assuming additionally $\nabla \phi \in BV_{loc}(\Omega)$, for a more general class of measure preserving maps.

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arXiv2026

On the structure of entropy dissipation and regularity for quasi-entropy solutions to 1d scalar conservation laws and to isentropic Euler system with $\gamma=3$

In this paper, we first investigate quasi-entropy solutions to scalar conservation laws in several space dimensions. In this setting, we introduce a suitable Lagrangian representation for such solutions. Next, we prove that, in one space dimension and for fluxes $f$ satisfying a general non-degeneracy condition, the entropy dissipation measures of quasi-entropy solutions are concentrated on a 1-rectifiable set. The same result is obtained for the isentropic Euler system with $\gamma = 3$, for which we also slightly improve the available fractional regularity by exploiting the sign of the kinetic measures.

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arXiv2025

Strong time regularity and decay of $L^\infty$ solutions to $2\times 2$ systems of conservation laws

We consider $\mathbf L^\infty$ solutions to $2\times 2$ systems of conservation laws. For genuinely nonlinear systems we prove that finite entropy solutions (in particular entropy solutions, if a uniformly convex entropy exists) belong to $C^0(\mathbb R^+; \mathbf L^1_{loc}(\mathbb R))$. Our second result establishes a dispersive-type decay estimate for vanishing viscosity solutions. Both results are unified by the use of a kinetic formulation.

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arXiv2025

Liouville type theorem and kinetic formulation for 2x2 systems of conservation laws

We study $\mathbf L^\infty$ entropy solutions to $2\times 2$ systems of conservation laws. We show that, if a uniformly convex entropy exists, these solutions satisfy a pair of kinetic equations (nonlocal in velocity), which are then shown to characterize all solutions with finite entropy production. Next, we prove a Liouville-type theorem for genuinely nonlinear systems, which is the main result of the paper. This implies in particular that for every finite entropy solution, every point $(t,x) \in \mathbb R^+\times \mathbb R\setminus \br J$ is of vanishing mean oscillation, where $\br J \subset \mathbb R^+\times \mathbb R$ is a set of Hausdorff dimension at most 1.

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arXiv2024

Initial Data Identification for Conservation Laws with Spatially Discontinuous Flux

We consider a scalar conservation law with a spatially discontinuous flux at a single point $x=0$, and we study the initial data identification problem for $AB$-entropy solutions associated to an interface connection $(A,B)$. This problem consists in identifying the set of initial data driven by the corresponding $AB$-entropy solution to a given target profile~$\omega^T$, at a time horizon $T>0$. We provide a full characterization of such a set in terms of suitable integral inequalities, and we establish structural and geometrical properties of this set. A distinctive feature of the initial set is that it is in general not convex, differently from the case of conservation laws with convex flux independent on the space variable. The results rely on the properties of the $AB$-backward-forward evolution operator introduced in~\cite{talamini_ancona_attset}, and on a proper concept of $AB$-genuine/interface characteristic for $AB$-entropy solutions provided in this paper.

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arXiv2024

Intermediate Domains for Scalar Conservation Laws

For a scalar conservation law with strictly convex flux, by Oleinik's estimates the total variation of a solution with initial data $\overline{u}\in \bf{L}^\infty(\mathbb R)$ decays like $t^{-1}$. This paper introduces a class of intermediate domains $\mathcal P_\alpha$, $0<\alpha<1$, such that for $\overline u\in \mathcal P_\alpha$ a faster decay rate is achieved: $\mathrm{Tot.Var.}\bigl\{ u(t,\cdot)\bigr\}\sim t^{\alpha-1}$. A key ingredient of the analysis is a ``Fourier-type" decomposition of $\overline u$ 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.

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arXiv2024

Backward-forward characterization of attainable set for conservation laws with spatially discontinuous flux

Consider a scalar conservation law with a spatially discontinuous flux at a single point x=0, and assume that the flux is uniformly convex when x\neq 0. Given an interface connection (A,B), we define a backward solution operator consistent with the concept of AB-entropy solution [4,13,16]. We then analyze the family A^{[AB]}(T) of profiles that can be attained at time T>0 by AB-entropy solutions with L^\infty-initial data. We provide a characterization of A^{[AB]}(T) as fixed points of the backward-forward solution operator. As an intermediate step we establish a full characterization of A^{[AB]}(T) in terms of unilateral constraints and Ole\v{\i}nik-type estimates, valid for all connections. Building on such a characterization we derive uniform BV bounds on the flux of AB-entropy solutions, which in turn yield the L^1_{loc}-Lipschitz continuity in time of these solutions.

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