Our objective is to establish a generalized Oleinik-type inequality for a convex scalar balance law with nonlinear source term [Formula: see text], in the spirit of the classical results obtained independently by Lax and Oleinik for scalar conservation laws. From this inequality, and under a generic condition linking the flux function and the source term, we deduce a [Formula: see text] regularity result for the unique entropy solution. If this condition is not satisfied, we obtain instead a fractional regularity result, showing that the solution lies in [Formula: see text] for some [Formula: see text], as in the homogeneous case [Formula: see text].
This paper investigates the large time optimal growth estimates of solutions for an elastic plate system under the Green–Naghdi hyperbolic thermal law in the inviscid case. By employing WKB asymptotic analysis and Fourier analysis, we establish sharp large time estimates in the [Formula: see text] norm for both the vertical deflection of plate and the thermal displacement, which grow in lower dimensions. A key technical tool is the Fourier splitting method equipped with suitably chosen time-dependent splitting functions, which effectively handles non-summable singularities for small frequencies as the viscous dissipation vanishes. Our results show the crucial influence of viscous dissipation and the growth mechanism in the viscoelastic plate system studied by Chen et al. [Large time asymptotic behavior for the elastic plate system with type II heat conduction. Part I: The viscous case, Acta Math. Sin. (Chin. Ser.) (2026)].
In this paper, we study a degenerate high-order parabolic equation that describes the evolution of elastic plates and thin films and membranes of viscous liquid, in which the surface tension the liquid/air interface determines the dynamics. We prove the existence of the solutions of the Cauchy problem associated to this equation.
We consider a class of multi-population pedestrian models consisting in a system of nonlocal conservation laws coupled in the nonlocal components and describing several groups of pedestrians moving toward their respective targets while trying to avoid each other and the obstacles limiting the walking domain. Specifically, the nonlocal operators account for interactions occurring at the microscopic level as a reaction to the presence of other individuals or obstacles along the preferred path. In particular, the presence of obstacles is implemented in the nonlocal terms of the equations and not as classical boundary conditions. This allows to rewrite domain shape optimization problems as PDE-constrained problems. In this paper, we investigate the well-posedness of such optimization problems by proving the stability of solutions with respect to the positions and shapes of the obstacles. A differentiability result in the linear case is also provided. These properties are illustrated with a numerical example.
In this paper, we consider the Cauchy problem for a generic hyperbolic system of conservation laws and assume it is provided by a standard Riemann semigroup of solutions, for summable initial data with small total variation. Then, we introduce an integral functional involving the solutions of the Cauchy problem and investigate when such a functional has a (nontrivial) minimum in case the initial data vary in suitable admissible classes of functions.
Modeling heterogeneous and multi-lane traffic flow is essential for understanding and controlling complex transportation systems. In this work, we consider three vehicle populations: two classes of human-driven vehicles (cars and trucks) and autonomous vehicles, the latter characterized by controlled acceleration. Compared to single-population models, multi-population modeling poses greater challenges, primarily due to the increased number of parameters required to describe lane-changing behavior and the added complexity in passing to the mean-field limit. We model multi-lane traffic as a hybrid dynamical system, combining continuous dynamics within each lane and discrete events corresponding to lane-changing maneuvers. We then formulate and analyze the optimal control problem associated with such hybrid systems from both microscopic and mesoscopic perspectives. Using techniques from Gamma-convergence, we prove the existence of solutions to the optimal control problem in the mean-field limit of a finite-dimensional hybrid system. Finally, we present numerical simulations illustrating the impact of trucks on overall traffic efficiency.
In this paper, we study the quantitative small noise limit in the [Formula: see text] norm of certain time-dependent Hamilton–Jacobi equations equipped with Neumann boundary conditions, depending on the regularity of the data and the geometric properties of the domain. We first provide a [Formula: see text] rate of convergence for Hamilton–Jacobi equations with locally Lipschitz Hamiltonians posed on convex domains of the Euclidean space. We then enhance this speed of convergence in the case of quadratic Hamiltonians proving one-side rates of order [Formula: see text] and [Formula: see text], [Formula: see text]. The results exploit recent [Formula: see text] contraction estimates for Fokker–Planck equations with bounded velocity fields on unbounded domains used to derive differential Harnack estimates for the corresponding Neumann heat flow.
In this paper we investigate the theory of a-contraction with shifts with the intention of extending it to intermediate families. The theory of a-contraction with shifts is used to prove orbital L^2 stability to shock solutions of conservation laws. In this setting there are strong results for scalar laws and the extremal families of n× n systems of conservation laws. The only known results showing contraction of interior families are for the contact family the full Euler system and the case of rich systems due to Serre and Vasseur '16. This investigation culminates in finding necessary and sufficient conditions for which small shocks of general systems are local attractors with respect to the a-contraction theory.
This work studies the dissipative structure of regularizations of any order of hyperbolic systems of conservation laws in several space dimensions. It is proved that the seminal equivalence theorem by Kawashima and Shizuta (Hokkaido Math. J. 14, 1985, no. 2, 249-275), which relates the strictly dissipative structure of second-order (viscous) systems to a genuine coupling condition of algebraic type, can be extended to higher-order multidimensional systems. For that purpose, the symbolic formulation of the genuine coupling condition by Humpherys (J. Hyperbolic Differ. Equ. 2, 2005, no. 4, 963-974) for linear operators of any order in one dimension, is adopted and extrapolated. Therefore, the concepts of symbol symmetrizability and genuine coupling are extended to the most general setting of differential operators of any order in several space dimensions. Applications to many viscous-dispersive systems of physical origin, such as compressible viscous-capillar fluids of Korteweg type, the dispersive Navier-Stokes-Fourier system and the equations of quantum hydrodynamics, illustrate the relevance of this extension.
The paper is concerned with a scalar balance law, where the source term depends on a control function alpha(t). Given a control alpha is an element of L infinity([0,T]), it is proved that, for generic initial data & umacr; is an element of & Cscr;3(& Ropf;), the solution has finitely many shocks, interacting at most two at a time. Moreover, at the terminal time T no shock interaction occurs and no new shock is formed. In addition, a family of optimal control problems is considered, including a running cost and a terminal cost. An example is constructed where the optimal solution contains two shocks merging exactly at the terminal time T. Such behavior persists under any suitably small perturbation of the flux, source and cost functions, and of the initial data. This shows that generic solutions of optimization problems have different qualitative properties, compared with generic solutions to Cauchy problems.
In this paper, we introduce model-based transition rates for controlled compartmental models in mathematical epidemiology, with a focus on the effects of control strategies applied to interacting multi-agent systems describing contact formation dynamics. In the framework of kinetic control problems, we compare two prototypical control protocols: one additive control directly influencing the dynamics and another targeting the interaction strength between agents. The emerging controlled macroscopic models are derived for an SIR compartmentalization to illustrate their impact on epidemic progression and contact interaction dynamics. Numerical results show the effectiveness of this approach in steering the dynamics and controlling epidemic trends, even in scenarios, where contact distributions exhibit an overpopulated tail.
We study the anisotropic, incompressible Cahn-Hilliard-Navier-Stokes system with variable density in a bounded smooth domain Ω⊂ℝ^d. This work extends previous results on the isotropic case by incorporating anisotropic surface energy, represented by 𝔉= ∫_Ω 2 Γ^2(∇ ϕ). The thermodynamic consistency of this system, as well as its modeling background and physical motivation, has been established in . Using a Galerkin approximation scheme, we prove the existence of global weak solutions in both two- and three-dimensions (d=2,3). A key ingredient in extending the local existence of approximate solutions to a global one is the application of Bihari's inequality combined with a fixed-point argument.
We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in a periodic domain in one-space dimension with linear pressure term. The main result is the global existence of periodic entropy weak solutions, for periodic initial data having finite total variation and initial density bounded away from zero.
Stability is a key property of both forward models and inverse problems, and depends on the norms considered in the relevant function spaces. For instance, stability estimates for hyperbolic partial differential equations are often based on energy conservation principles, and are therefore expressed in terms of L2 norms. The focus of this paper is on stability with respect to the L infinity norm, which is more relevant to detect localized phenomena. The linear wave equation is not stable in L infinity, and we design an alternative solution method based on the regularization of Fourier multipliers, which is stable in L infinity. Furthermore, we show how these ideas can be extended to inverse problems, and design a regularization method for the inversion of compact operators that is stable in L infinity. We also discuss the connection with the stability of deep neural networks modeled by hyperbolic PDEs.
Peter D. Lax has passed away recently. His important contributions to partial differential equations, computational mathematics, and connection to science have benefitted us over the past decades. Several well-known mathematical terms are named after him, including the Lax Admissibility Condition for hyperbolic conservation laws, Lax Pairs for completely integrable systems, Lax Equivalence Theorem for numerical schemes, and Lax–Milgram Theorem in functional analysis. Peter’s unique sense of beauty and his philosophy of the universality of mathematics will continue to fascinate us. He is a member of the National Academy of Sciences, was awarded the National Medal of Science in 1986, the Wolf Prize in 1987, and the Abel Prize in 2005.
In this paper, we investigate an abstract model associated with the Timoshenko system, incorporating fractional dissipative effects. The fractional dissipative effect is characterized by the powers of an arbitrary, strictly positive self-adjoint operator, with domain densely embedded in a Hilbert space. Our first main result demonstrates that the operator, derived by reformulating the abstract system as a first-order system, is the generator of a strongly continuous semigroup. The second main result is that the same semigroup can exhibit properties such as exponential stability, analyticity, or belong to a certain Gevrey class, with its precise behavior dependent on the assigned values of these powers. Furthermore, we establish that the Gevrey class obtained is sharp.
This paper investigates the well-posedness of local smooth solutions for the one-dimensional free boundary problem of the Patlak-Keller-Segel (PKS) model. By introducing a mixed velocity, the PKS model can be written into a form of the hyperbolic-Poisson coupled system, which becomes a degenerate system on the free boundary. We introduce the Lagrange transformation and convert the free boundary problem into an initial boundary value problem. Then, the degeneracy of this system can be described by the function rho(gamma-1)(0) of the initial density rho(0) under Lagrangian coordinates, which is equivalent to the distance function near the boundary points. By utilizing the Hardy inequality and weighted Sobolev spaces, we establish high-order a priori estimates of local smooth solutions for any slow diffusion component gamma > 1, overcoming the degeneracy issues at the free boundary. This confirms the existence and uniqueness of the local smooth solutions.
The inverse scattering problem for the two-dimensional nonlinear Klein-Gordon equation utt - Delta u + u = & Nscr;(u) is studied. We assume that the unknown nonlinearity & Nscr; of the equation satisfies & Nscr; is an element of C infinity(& Ropf;; & Ropf;), & Nscr;(k)(y) = O(|y|max{3-k,0}) (y -> 0) and & Nscr;(k)(y) = O(ecy2) (|y|->infinity) for any k = 0, 1, 2,& mldr;. Here, c is a positive constant. We establish a reconstruction formula of & Nscr;(k)(0) (k = 3, 4, 5,& mldr;) by the knowledge of the scattering operator for the equation. As an application, we also give an expression for higher-order G & acirc;teaux differentials of the scattering operator at 0.
