A general result on the approximation of local conservation laws by nonlocal conservation laws: The singular limit problem for exponential kernels - Coclite, Giuseppe Maria; Coron, Jean Michel; De Nitti, Nicola; Keimer, Alexander; Pflug, Lukas (Annales de l'Institut Henri Poincaré - Analyse Non Linéaire, 2023)
We deal with the problem of approximating a scalar conservation law by a conservation law with nonlocal flux. As convolution kernel in the nonlocal flux, we consider an exponential-type approximation of the Dirac distribution. We then obtain a total variation bound on the nonlocal term and can prove that the (unique) weak solution of the nonlocal problem converges strongly in C.L1loc/ to the entropy solution of the local conservation law. We conclude with several numerical illustrations which underline the main results and, in particular, the difference between the solution and the nonlocal term.
CRIS-Team Scopus Importer:2023-11-17
Keywords: approximation of local conservation laws; balance laws; entropy solution; Nonlocal conservation laws; nonlocal flux; singular limits

Boundary Controllability and Asymptotic Stabilization of a Nonlocal Traffic Flow Model - Bayen, Alexandre; Coron, Jean-Michel; De Nitti, Nicola; Keimer, Alexander; Pflug, Lukas (Vietnam Journal of Mathematics, 2021)
We study the exact boundary controllability of a class of nonlocal conservation laws modeling traffic flow. The velocity of the macroscopic dynamics depends on a weighted average of the traffic density ahead and the averaging kernel is of exponential type. Under specific assumptions, we show that the boundary controls can be used to steer the system towards a target final state or out-flux. The regularizing effect of the nonlocal term, which leads to the uniqueness of weak solutions, enables us to prove that the exact controllability is equivalent to the existence of weak solutions to the backwards-in-time problem. We also study steady states and the long-time behavior of the solution under specific boundary conditions.
CRIS-Team Scopus Importer:2021-06-25
Keywords: Boundary controllability; Characteristics; Conservation laws; Exact controllability; Nonlocal flux; Stabilization; Traffic flow

Control of Hyperbolic and Parabolic Equations on Networks and Singular limits - Bárcena-Petisco, Jon Asier; Cavalcante, M; Coclite, Giuseppe Maria; De Nitti, Nicola; Zuazua Iriondo, Enrique (Mathematical Control and Related Fields, 2025)
We study the controllability properties of the transport equation and of parabolic equations posed on a tree. Using a control localized on the exterior nodes, we prove that the hyperbolic and the parabolic systems are null-controllable. The hyperbolic proof relies on the method of characteristics, the parabolic one on duality arguments and Carleman inequalities. We also show that the parabolic system may not be controllable if we do not act on all exterior vertices because of symmetries. Moreover, we estimate the cost of the null-controllability of transport-diffusion equations with diffusivity ε > 0 and study its asymptotic behavior when ε → 0 +. We prove that the cost of the controllability decays for a time sufficiently large and explodes for short times. This is done by duality arguments allowing to reduce the problem to obtain observability estimates which depend on the viscosity parameter. These are derived by using Agmon and Carleman inequalities.<br
CRIS-Team Scopus Importer:2025-01-31
Keywords: Controllability; cost of controllability; advection-diffusion equations; singular limits; vanishing viscosity; networks

Critical functions and blow-up asymptotics for the fractional Brezis-Nirenberg problem in low dimension - De Nitti, Nicola; Konig, Tobias (Calculus of Variations and Partial Differential Equations, 2023)
For s?(0,1),N>2s, and a bounded open set (O)?R(N)with C(2)boundary, we study the fractional Brezis-Nirenberg type minimization problem of finding
CRIS-Team WoS Importer:2023-05-05

Differentiability in Measure of the Flow Associated with a Nearly Incompressible BV Vector Field - Bianchini, Stefano; De Nitti, Nicola (Archive for Rational Mechanics and Analysis, 2022)
We study the regularity of the flow X(t, y) , which represents (in the sense of Smirnov or as regular Lagrangian flow of Ambrosio) a solution ρ∈ L∞(Rd+1) of the continuity equation ∂tρ+div(ρb)=0,with b∈Lt1BVx. We prove that X is differentiable in measure in the sense of Ambrosio–Malý, that is X(t,y+rz)-X(t,y)r→r→0W(t,y)zin measure,where the derivative W(t, y) is a BV function satisfying the ODE ddtW(t,y)=(Db)y(dt)J(t-,y)W(t-,y),where (Db) y(d t) is the disintegration of the measure ∫Db(t,·)dt with respect to the partition given by the trajectories X(t, y) and the Jacobian J(t, y) solves ddtJ(t,y)=(divb)y(dt)=Tr(Db)y(dt).The proof of this regularity result is based on the theory of Lagrangian representations and proper sets introduced by Bianchini and Bonicatto in [16], on the construction of explicit approximate tubular neighborhoods of trajectories, and on estimates that take into account the local structure of the derivative of a BV vector field.
CRIS-Team Scopus Importer:2022-10-28

Exponential convergence to steady-states for trajectories of a damped dynamical system modeling adhesive strings - Coclite, Giuseppe Maria; De Nitti, Nicola; Maddalena, Francesco; Orlando, Gianluca; Zuazua Iriondo, Enrique (Mathematical Models & Methods in Applied Sciences, 2024)
We study the global well-posedness and asymptotic behavior for a semilinear damped wave equation with Neumann boundary conditions, modeling a one-dimensional linearly elastic body interacting with a rigid substrate through an adhesive material. The key feature of of the problem is that the interplay between the nonlinear force and the boundary conditions allows for a continuous set of equilibrium points. We prove an exponential rate of convergence for the solution towards a (uniquely determined) equilibrium point.
CRIS-Team Scopus Importer:2024-05-03
Keywords: adhesion phenomena; Damped wave equation; decay; long-time asymptotics

Exponential convergence to steady-states for trajectories of a damped dynamical system modelling adhesive strings - Coclite, Giuseppe Maria; De Nitti, Nicola; Maddalena, Francesco; Orlando, Gianluca; Zuazua Iriondo, Enrique (2024)
We study the global well-posedness and asymptotic behavior for a semilinear damped wave equation with Neumann boundary conditions, modelling a one-dimensional linearly elastic body interacting with a rigid substrate through an adhesive material. The key feature of of the problem is that the interplay between the nonlinear force and the boundary conditions allows for a continuous set of equilibrium points. We prove an exponential rate of convergence for the solution towards a (uniquely determined) equilibrium point.
https://cris.fau.de/converis/publicweb/Publication/313437958
Keywords: Damped wave equation; adhesion phenomena; long-time asymptotics; decay

Fractional Hardy–Rellich inequalities via integration by parts - De Nitti, Nicola; Djitté, Sidy Moctar (Nonlinear Analysis - Theory Methods & Applications, 2024)
We prove a fractional Hardy–Rellich inequality with an explicit constant in bounded domains of class C1,1. The strategy of the proof generalizes an approach pioneered by E. Mitidieri (Mat. Zametki, 2000) by relying on a Pohozaev-type identity.
CRIS-Team Scopus Importer:2024-03-15
Keywords: Fractional Lapalcian; Fractional Sobolev spaces; Hardy inequality; Pohozaev identity; Rellich inequality

INTERFACE PROPAGATION PROPERTIES FOR A NONLOCAL THIN-FILM EQUATION - De Nitti, Nicola; Taranets, Roman M. (SIAM Journal on Mathematical Analysis, 2024)
We consider a degenerate nonlocal parabolic equation in a one-dimensional domain introduced to model hydraulic fractures. The nonlocal operator is given by a fractional power of the Laplacian and the degenerate mobility exponent corresponds to a ``strong slippage" regime with ``complete wetting" interfacial conditions for local thin-film equations. Using a localized entropy estimate and a Stampacchia-type lemma, we establish a finite speed of propagation result and sufficient conditions (and lower bounds) for the waiting-time phenomenon.
CRIS-Team Scopus Importer:2024-02-02
Keywords: higher order degenerate parabolic equations; hydraulic fractures; nonlocal thin-film equation; spectral fractional Laplacian

Long-time convergence of a nonlocal Burgers' equation towards the local N-wave - Coclite, Giuseppe Maria; De Nitti, Nicola; Keimer, Alexander; Pflug, Lukas; Zuazua Iriondo, Enrique (Nonlinearity, 2023)
We study the long-time behavior of the unique weak solution of a nonlocal regularization of the (inviscid) Burgers' equation where the velocity is approximated by a one-sided convolution with an exponential kernel. The initial datum is assumed to be positive, bounded, and integrable. The asymptotic profile is given by the "N -wave'' entropy solution of the Burgers' equation. The key ingredients of the proof are a suitable scaling argument and a nonlocal Oleinik-type estimate.
Keywords: Nonlocal conservation laws; nonlocal flux; Burgers’ equation; approximation of local conservation laws; N-waves; source-type solutions; entropy solutions

On existence and uniqueness of weak solutions to nonlocal conservation laws with BV kernels - Coclite, Giuseppe Maria; De Nitti, Nicola; Keimer, Alexander; Pflug, Lukas (Zeitschrift für Angewandte Mathematik und Physik, 2022)
In this note, we extend the known results on the existence and uniqueness of weak solutions to conservation laws with nonlocal flux. In case the nonlocal term is given by a convolution γ∗ q, we weaken the standard assumption on the kernel γ∈ L∞((0 , T) ; W1,∞(R)) to the substantially more general condition γ∈ L∞((0 , T) ; BV(R)) , which allows for discontinuities in the kernel.
CRIS-Team Scopus Importer:2022-11-04
Keywords: BV kernels; Nonlocal conservation laws; Nonlocal flux; Well-posedness of weak solutions

On Liouville-type theorems for the 2D stationary MHD equations - De Nitti, Nicola; Hounkpe, Francis; Schulz, Simon (Nonlinearity, 2022)
We establish new Liouville-type theorems for the two-dimensional stationary magneto-hydrodynamic incompressible system assuming that the velocity and magnetic field have bounded Dirichlet integral. The key tool in our proof is observing that the stream function associated to the magnetic field satisfies a simple drift-diffusion equation for which a maximum principle is available.
CRIS-Team WoS Importer:2022-01-14

On the Controllability of Entropy Solutions of Scalar Conservation Laws at a Junction via Lyapunov Methods - De Nitti, Nicola; Zuazua Iriondo, Enrique (Vietnam Journal of Mathematics, 2023)
In this note, we prove a controllability result for entropy solutions of scalar conservation laws on a star-shaped graph. Using a Lyapunov-type approach, we show that, under a monotonicity assumption on the flux, if u and v are two entropy solutions corresponding to different initial data and same in-flux boundary data (at the exterior nodes of the star-shaped graph), then u ≡ v for a sufficiently large time. In order words, we can drive u to the target profile v in a sufficiently large control time by inputting the trace of v at the exterior nodes as in-flux boundary data for u. This result can also be shown to hold on tree-shaped networks by an inductive argument. We illustrate the result with some numerical simulations.<br
Keywords: Scalar conservation laws; Entropy solutions; Controllability; Lyapunov; Networks; Star-shaped graphs; Tree-shaped graphs

On the decay of one-dimensional locally and partially dissipated and hyperbolic systems - Crin-Barat, Timothee; De Nitti, Nicola; Zuazua Iriondo, Enrique (2025)
We study the time-asymptotic behavior of linear hyperbolic systems under partial dissipation which is localized in suitable subsets of the domain. More precisely, we recover the classical decay rates of partially dissipative systems satisfying the stability condition (SK) with a time-delay depending only on the velocity of each component and the size of the undamped region. To quantify this delay, we assume that the undamped region is a bounded space-interval and that the system without spacerestriction on the dissipation satisfies the stability condition (SK). The former assumption ensures that the time spent by the characteristics of the system in the undamped region is finite and the latter that whenever the damping is active the solutions decay. Our approach consists in reformulating the system into n coupled transport equations and showing that the time-decay estimates are delayed by the sum of the times each characteristics spend in the undamped region.<br
https://cris.fau.de/converis/publicweb/Publication/276246693
Keywords: Hyperbolic systems; partially dissipative hyperbolic systems; localized damping; time-decay; Shizuta-Kawashima condition

Sharp criteria for the waiting time phenomenon in solutions to the thin-film equation - De Nitti, Nicola; Fischer, Julian (Communications in Partial Differential Equations, 2022)
We establish sharp criteria for the instantaneous propagation of free boundaries in solutions to the thin-film equation. The criteria are formulated in terms of the initial distribution of mass (as opposed to previous almost-optimal results), reflecting the fact that mass is a locally conserved quantity for the thin-film equation. In the regime of weak slippage, our criteria are at the same time necessary and sufficient. The proof of our upper bounds on free boundary propagation is based on a strategy of "propagation of degeneracy" down to arbitrarily small spatial scales: We combine estimates on the local mass and estimates on energies to show that "degeneracy" on a certain space-time cylinder entails "degeneracy" on a spatially smaller space-time cylinder with the same time horizon. The derivation of our lower bounds on free boundary propagation is based on a combination of a monotone quantity and almost optimal estimates established previously by the second author with a new estimate connecting motion of mass to entropy production.
CRIS-Team WoS Importer:2022-06-17

Singular limits with vanishing viscosity for nonlocal conservation laws - Coclite, Giuseppe Maria; De Nitti, Nicola; Keimer, Alexander; Pflug, Lukas (Nonlinear Analysis - Theory Methods & Applications, 2021)
We consider a class of nonlocal conservation laws with a second-order viscous regularization term which finds an application in modelling macroscopic traffic flow. The velocity function depends on a weighted average of the density ahead, where the averaging kernel is of exponential type. We show that, as the nonlocal impact and the viscosity parameter simultaneously tend to zero (under a suitable balance condition), the solution of the nonlocal problem converges to the entropy solution of the corresponding local conservation law. The key ideas of our proof are to observe that the nonlocal term satisfies a third-order equation with diffusive and dispersive effects and to deduce a suitable energy estimate on the nonlocal term. The convergence result is then based on the compensated compactness theory.
CRIS-Team Scopus Importer:2021-06-11
Keywords: Compensated compactness; Conservation laws; Nonlocal flux; Nonlocal-to-local convergence; Singular limits; Traffic flow; Vanishing viscosity

Singular limits with vanishing viscosity for nonlocal conservation laws - Coclite, Giuseppe Maria; De Nitti, Nicola; Keimer, Alexander; Pflug, Lukas (Nonlinear Analysis - Theory Methods & Applications, 2021)

Stability with explicit constants of the critical points of the fractional Sobolev inequality and applications to fast diffusion - De Nitti, Nicola; König, Tobias (Journal of Functional Analysis, 2023)
We study the quantitative stability of critical points of the fractional Sobolev inequality. We show that, for a non-negative function u∈H˙s(RN) whose energy satisfies [Formula presented]SN,s[Formula presented]≤‖u‖H˙s(RN)≤[Formula presented]SN,s[Formula presented], where SN,s is the optimal Sobolev constant, the bound ‖u−U[z,λ]‖H˙s(RN)≲‖(−Δ)su−u2s⁎−1‖H˙−s(RN), holds for a suitable fractional Talenti bubble U[z,λ]. For functions u which are close to Talenti bubbles, we give the sharp asymptotic value of the implied constant in this inequality. As an application of this, we derive an explicit polynomial extinction rate for positive solutions to a fractional fast diffusion equation.
CRIS-Team Scopus Importer:2023-08-04
Keywords: Extinction profiles; Fractional fast diffusion equation; Fractional Sobolev inequality; Stability of critical points

Vanishing viscosity for a 2 x 2 system modeling congested vehicular traffic - Coclite, Giuseppe Maria; De Nitti, Nicola; Garavello, Mauro; Marcellini, Francesca (Networks and Heterogeneous Media, 2021)
We prove the convergence of the vanishing viscosity approximation for a class of 2x2 systems of conservation laws, which includes a model of traffic flow in congested regimes. The structure of the system allows us to avoid the typical constraints on the total variation and the L-1 norm of the initial data. The key tool is the compensated compactness technique, introduced by Murat and Tartar, used here in the framework developed by Panov. The structure of the Riemann invariants is used to obtain the compactness estimates.
CRIS-Team WoS Importer:2021-08-20