For a semiconcave function with linear modulus, we introduce the cut locus through a touching approach and prove that it coincides with the variational cut locus defined via the Lax–Oleinik semigroup, independently of the Hamiltonian. Cut points are characterized by the emptiness of the proximal subdifferential of ϕ, which we use as the analytic criterion behind the touching definition. We introduce the degree of regularity R_ϕ, which measures the C^1,1 deviation of ϕ, and prove the quantitative relation R_ϕ(x)∼ 1/τ_ϕ,H(x) with the cut time function, valid pointwise up to explicit truncation constants. From this relation we derive a separation estimate for calibrated curves, showing that no conjugate points occur before the cut locus. This estimate supplies the tools for the propagation results. Cut points propagate globally along generalized characteristics for the evolutionary Hamilton–Jacobi equation, and Alexandrov points propagate forward along calibrated curves, with the second derivative satisfying a matrix Riccati equation. We also show that the cut locus is a Lebesgue null set and give a streamlined proof of Alexandrov's theorem. Finally, we give necessary and sufficient conditions for the cut locus of a weak KAM solution to be closed, in terms of the cut time function, the C^1,1 support, and the degree of regularity.

We study a two-layer one-dimensional energy balance model, which allows for vertical energy exchanges between a surface layer and the atmosphere, as well as meridional energy transport across latitudes via a diffusion law. The evolution equations of the surface temperature and the atmospheric temperature are coupled by exchange of infrared radiation as well as other non-radiative energy exchanges. The energy enters the system as solar radiation, which is partially absorbed and partially reflected by the two layers. The system is then composed of two degenerate parabolic equations coupled by nonlinear terms, the growth of these terms being crucial for the choice of the functional setting. An essential parameter is the absorptivity of the atmosphere, denoted ε _a, whose value depends critically on greenhouse gases. We prove that blow up in finite time occurs if ε _a >2, while global existence of solutions and the existence of a global attractor hold when ε _a ∈ (0,2). Proofs are based on comparison principles that derive from the cooperative structure of the problem, and that provide invariant rectangles for smooth initial conditions, and on regularity properties.

In this paper, we study the robustness of exponential stability for semigroups generated by linear operators under perturbations. Extending a classical result of Gibson's Stability Theorem, we show that if the generator of an analytic exponentially stable semigroup is perturbed by a class of relatively bounded operators satisfying certain assumptions, then exponential stability is preserved, provided the perturbed semigroup is strongly stable. We also show that, for a restricted class of perturbations, the analyticity requirement can be relaxed to Gevrey regularity. Moreover, we present applications to uniformly parabolic equations, degenerate/singular parabolic equations, coupled hyperbolic plate systems, and generalized coupled systems of Kirchhoff-Love plates and a membrane-like electric network.

We study the long-time behavior of the generalized gradient flow associated with solutions of the critical Hamilton–Jacobi equation for mechanical Hamiltonians on the flat torus. For any semiconcave function, we show that its critical set — points whose superdifferential contains the zero vector — acts as an approximate attractor for the flow. When the function is a solution of the critical equation, the critical set decomposes into regular and singular parts, and we establish a dichotomy describing which part trajectories approach as [Formula: see text]. Our analysis uses limiting occupational measures, a class of invariant measures capturing the asymptotic distribution of the flow. An essential ingredient is a complete proof of the global invariance of the singular set, a result previously announced by Albano [Global propagation of singularities for solutions of Hamilton–Jacobi equations, J. Math. Anal. Appl. 444(2) (2016) 1462–1478] but not fully established.

We consider a first-order transport equation partial derivative(t)u(x,t)+(H(x)& sdot;del u(x,t))+p(x)u(x,t)=F(x,t) for x is an element of Omega subset of R-d, where d >= 2, Omega is a bounded domain and 0 < t < T. We prove a Carleman estimate for more general condition on the principal coefficients H(x) than in the existing works. The key is the construction of a piecewise smooth weight function in x according to a suitable decomposition of Omega. Our assumptions on H generalize the conditions in the existing articles, and require that a directed graph created by the corresponding stream field has no closed loops. Then, we apply our Carleman estimate to two inverse problems of determination of an initial value and one of a spatial factor of a source term, so that we establish Lipschitz stability estimates for the inverse problems.

We address the inverse problem of recovering a degeneracy point within the diffusion coefficient of a one-dimensional complex parabolic equation by observing the normal derivative at one point of the boundary. The strongly degenerate case is analyzed. In particular, we derive sufficient conditions on the initial data that guarantee the stability and uniqueness of the solution obtained from a one-point measurement. Moreover, we present more general uniqueness theorems, which also cover the identification of the initial data, the coefficient of the zero order term and the degeneracy power, using measurements taken over time. Our method is based on a careful analysis of the spectral problem and relies on an explicit form of the solution in terms of Bessel functions. Our investigation also covers the case of real 1-D degenerate parabolic systems of equations coupled with a specific structure. Theoretical results are also supported by numerical simulations.

We study the Navier-Stokes equations on the two-dimensional torus 𝕋^2 with a single input bilinear control term which forces the evolution to preserve hyperplanes. We reduce the problem to the solution of the Navier-Stokes system perturbed by a certain nonlinear nonlocal term and show the well-posedness of the initial value problem for such a system.

We prove that singularities propagate globally for viscosity solutions of Hamilton-Jacobi equations related to magnetic mechanical systems on closed Riemannian manifolds. Our main result shows that for any weak KAM solution u, the singular set Sing (u) remains invariant under the generalized gradient flow dynamics. The proof combines three key elements: (1) reduction from magnetic to Riemannian systems, (2) analysis of reparameterized flows, and (3) regularization techniques. Compared to previous analytic approaches, our geometric method provides clearer insights into the underlying Riemannian structure. We also establish necessary conditions for singularity existence, particularly when the Euler characteristic is nonzero and the magnetic form is non-exact. This approach does not extend directly to Finsler metrics due to structural differences.

In this paper, we study an optimal exit time problem with general running and terminal costs and a target 𝒮⊂ℝ^d having an inner ball property for a nonlinear control system that satisfies mild controllability assumptions. In particular, Petrov’s condition at the boundary of 𝒮 is not required and the value function V may fail to be locally Lipschitz. In such a weakened set-up, we first establish a representation formula for proximal (horizontal) supergradients of V by using transported proximal normal vectors. This allows us to obtain an external sphere condition for the hypograph of V which yields several regularity properties. In particular, V is almost everywhere twice differentiable and the Hausdorff dimension of its singularities is not greater than d-1/2 . Furthermore, besides optimality conditions for trajectories of the optimal control problem, we extend the analysis to propagation of singularities and differentiability properties of the value function. An upper bound for the Hausdorff measure of the singular set is also studied, which implies that V belongs to W^1,1_loc .

In the context of weak KAM theory, we discuss the commutators $\{T^-_t\circ T^+_t\}_{t\geqslant0}$ and $\{T^+_t\circ T^-_t\}_{t\geqslant0}$ of Lax-Oleinik operators. We characterize the relation $T^-_t\circ T^+_t=Id$ for both small time and arbitrary time $t$. We show this relation characterizes controllability for evolutionary Hamilton-Jacobi equation. Based on our previous work on the cut locus of viscosity solution, we refine our analysis of the cut time function $\tau$ in terms of commutators $T^+_t\circ T^-_t-T^+_t\circ T^-_t$ and clarify the structure of the super/sub-level set of the cut time function $\tau$.

In this work, we extend Aubry-Mather theory to the case of control systems with nonholonomic constraints. In this framework, we consider an optimal control problem where admissible trajectories are solutions of a control-affine equation. Such an equation is associated with a family of smooth vector fields that satisfy the Hormander condition, which implies the controllability of the system. In this case, the Hamiltonian fails to be coercive, so results for Tonelli Hamiltonians cannot be applied. To overcome these obstacles, we develop an intrinsic approach based on the metric properties of the geometry induced on the state space by the sub-Riemannian structure.

In this paper, we investigate the singularities and their propagation properties of potential energy functionals, in the space of all Borel probability measures with compact support 𝒫_c(ℝ^m) , Φ (μ ):=∫ _ℝ^mϕ dμ , ∀μ∈𝒫_c(ℝ^m), (we denote Φ (· ) by ϕ (· ) for brevity hereafter) associated with semiconcave functions ϕ defined on ℝ^m . Our study covers two cases: when ϕ is a semiconcave function and when u is a weak KAM solution of the Hamilton–Jacobi equation H(x, Du(x)) = c[0] on a smooth closed manifold. By applying previous work on Hamilton–Jacobi equations in the Wasserstein space, we prove that the singularities of u(· ) (the potential energy associated with u ) will propagate globally when u is a weak KAM solution, and the dynamical cost function C^t is the associated fundamental solution. We also demonstrate the existence of solutions evolving along the cut locus, governed by an irregular Lagrangian semiflow on the cut locus of u .

Granular materials are everywhere, in the environment but also in our pantry. Their properties are different from those of any solid material, due to the possibility of sudden phenomena such as avalanches or landslides. Here we present a brief survey on their characteristics and on what can be found (from the past thirty years) in the recent mathematics literature in order to reproduce their behavior. We discuss, in particular, differential models proposed for the growth of a sandpile on a table and, when wind comes into play, for the formation and dynamics of sand dunes. This field of research is still of great interest since there is no consolidated general model for the dynamics of granular matter, but rather only standalone models adapted to specific situations.

We study the exact controllability of the evolution equation \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0 \end{equation*} where $A$ is a nonnegative self-adjoint operator on a Hilbert space $X$ and $B$ is an unbounded linear operator on $X$, which is dominated by the square root of $A$. The control action is bilinear and only of scalar-input form, meaning that the control is the scalar function $p$, which is assumed to depend only on time. Furthermore, we only consider square-integrable controls. Our main result is the local exact controllability of the above equation to the ground state solution, that is, the evolution through time, of the first eigenfunction of $A$, as initial data. The analogous problem (in a more general form) was addressed in our previous paper [Exact controllablity to eigensolutions for evolution equations of parabolic type via bilinear control, Alabau-Boussouira F., Cannarsa P. and Urbani C., Nonlinear Diff. Eq. Appl. (2022)] for a bounded operator $B$. The current extension to unbounded operators allows for many more applications, including the Fokker-Planck equation in one space dimension, and a larger class of control actions.

We consider an inverse problem of reconstructing a degeneracy point in the diffusion coefficient in a one-dimensional parabolic equation by measuring the normal derivative on one side of the domain boundary. We analyze the sensitivity of the inverse problem to the initial data. We give sufficient conditions on the initial data for uniqueness and stability for the one-point measurement and show some examples of positive and negative results. On the other hand, we present more general uniqueness results, also for the identification of an initial data by measurements distributed over time. The proofs are based on an explicit form of the solution by means of Bessel functions of the first type. Finally, the theoretical results are supported by numerical experiments.

A classical problem in ergodic continuous time control consists of studying the limit behavior of the optimal value of a discounted cost functional with infinite horizon as the discount factor $\lambda$ tends to zero. In the literature, this problem has been addressed under various controllability or ergodicity conditions ensuring that the rescaled value function converges uniformly to a constant limit. In this case the limit can be characterized as the unique constant such that a suitable Hamilton-Jacobi equation has at least one continuous viscosity solution. In this paper, we study this problem without such conditions, so that the aforementioned limit needs not be constant. Our main result characterizes the uniform limit (when it exists) as the maximal subsolution of a system of Hamilton-Jacobi equations. Moreover, when such a subsolution is a viscosity solution, we obtain the convergence of optimal values as well as a rate of convergence. This mirrors the analysis of the discrete time case, where we characterize the uniform limit as the supremum over a set of sub-invariant half-lines of the dynamic programming operator. The emerging structure in both discrete and continuous time models shows that the supremum over sub-invariato half-lines with respect to the Lax-Oleinik semigroup/dynamic programming operator, captures the behavior of the limit cost as discount vanishes.

On a smooth closed manifold M, we introduce a novel theory of maximal slope curves for any pair (ϕ,H) with ϕ a semiconcave function and H a Hamiltonian. By using the notion of maximal slope curve from gradient flow theory, the intrinsic singular characteristics constructed in [Cannarsa, P.; Cheng, W., Generalized characteristics and Lax-Oleinik operators: global theory. Calc. Var. Partial Differential Equations 56 (2017), no. 5, 56:12], the smooth approximation method developed in [Cannarsa, P.; Yu, Y. Singular dynamics for semiconcave functions. J. Eur. Math. Soc. 11 (2009), no. 5, 999–1024], and the broken characteristics studied in [Khanin, K.; Sobolevski, A., On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations. Arch. Ration. Mech. Anal. 219 (2016), no. 2, 861–885], we prove the existence and stability of such maximal slope curves and discuss certain new weak KAM features. We also prove that maximal slope curves for any pair (ϕ,H) are exactly broken characteristics which have right derivatives everywhere. Applying this theory, we establish a global variational construction of strict singular characteristics and broken characteristics. Moreover, we prove a result on the global propagation of cut points along generalized characteristics, as well as a result on the propagation of singular points along strict singular characteristics, for weak KAM solutions. We also obtain the continuity equation along strict singular characteristics which clarifies the mass transport nature in the problem of propagation of singularities.

This work focuses on the rate of convergence for singular perturbation problems for first-order Hamilton–Jacobi equations. We use the nonlinear adjoint method to analyze how the Hamiltonian’s regularizing effect on the initial data influences the convergence rate. As an application we derive the rate of convergence for singularly perturbed two-players zero-sum deterministic differential games (i.e., leading to Hamilton–Jacobi–Isaacs equations) and, subsequently, in case of singularly perturbed mean field games of acceleration. Namely, we show that in both the models the rate of convergence is ε .

We consider a nonlinear parabolic equation with a nonlocal term which preserves the L^2 -norm of the solution. We study the local and global well-posedness on a bounded domain, as well as the whole Euclidean space, in H^1 . Then we study the asymptotic behavior of solutions. In general, we obtain weak convergence in H^1 to a stationary state. For a ball, we prove strong convergence to the ground state when the initial condition is positive.

We consider the linear degenerate wave equation, on the interval (0,1) w(tt)-(x(alpha)wx)(x)=p(t)mu(x)w, with bilinear control p and Neumann boundary conditions. We study the controllability of this nonlinear control system, locally around a constant reference trajectory, the "ground state." Under some classical and generic assumptions on mu, we prove that there exists a threshold value for time, T-alpha=4 / 2-alpha, such that the reachable set is a neighborhood of the ground state. If T<T-alpha it is contained in a C-1-submanifold of infinite codimension. Finally, it T = T-alpha and alpha is an element of [0,1) it is a C-1-submanifold of codimension 1 and if alpha is an element of(1,2) the reachable set is a neighborhood of the ground state. The case alpha=1 remains open. This extends to the degenerate case the work [K. Beauchard, J. Differential Equations, 250 (2011), pp. 2064-2098] and adapts to bilinear controls the work [F. Alabau-Boussouira, P. Cannarsa, and G. Leugering, SIAM J. Control Optim., 55 (2017), pp. 2052-2087]. Our proofs are based on a careful analysis of the spectral problem and on Ingham type results, which are extensions of Kadec's 1/4 theorem.