American Institute of Mathematical Sciences

Originally introduced in the fluid mechanics community, dynamic mode decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of nonlinear systems. However, existing DMD theory deals primarily with sequential time series for which the measurement dimension is much larger than the number of measurements taken. We present a theoretical framework in which we define DMD as the eigendecomposition of an approximating linear operator. This generalizes DMD to a larger class of datasets, including nonsequential time series. We demonstrate the utility of this approach by presenting novel sampling strategies that increase computational efficiency and mitigate the effects of noise, respectively. We also introduce the concept of linear consistency, which helps explain the potential pitfalls of applying DMD to rank-deficient datasets, illustrating with examples. Such computations are not considered in the existing literature but can be understood using our more general framework. In addition, we show that our theory strengthens the connections between DMD and Koopman operator theory. It also establishes connections between DMD and other techniques, including the eigensystem realization algorithm (ERA), a system identification method, and linear inverse modeling (LIM), a method from climate science. We show that under certain conditions, DMD is equivalent to LIM.

Stability analyses and error estimates are carried out for a number of commonly usednumerical schemes for the Allen-Cahn and Cahn-Hilliard equations. It is shown thatall the schemes we considered are either unconditionally energy stable, orconditionally energy stable with reasonable stability conditions in thesemi-discretized versions. Error estimates for selected schemes with aspectral-Galerkin approximation are also derived. The stability analyses and errorestimates are based on a weak formulation thus the results can be easily extended toother spatial discretizations, such as Galerkin finite element methods, which arebased on a weak formulation.

In this paper we study the existence of non-trivial solutions forequations driven by a non-local integrodifferentialoperator $\mathcal L_K$ with homogeneous Dirichlet boundaryconditions. More precisely, we consider the problem$$ \left\{\begin{array}{ll}\mathcal L_K u+\lambda u+f(x,u)=0 in Ω \\u=0 in \mathbb{R}^n \backslash Ω ,\end{array} \right.$$where $\lambda$ is a real parameter and the nonlinear term $f$satisfies superlinear and subcritical growth conditions at zero andat infinity. This equation has a variational nature, and so itssolutions can be found as critical points of the energy functional$\mathcal J_\lambda$ associated to the problem. Here we get suchcritical points using both the Mountain Pass Theorem and the LinkingTheorem, respectively when $\lambda<\lambda_1$ and $\lambda\geq\lambda_1$\,, where $\lambda_1$ denotes the first eigenvalue of theoperator $-\mathcal L_K$.As a particular case, we derive an existence theorem for thefollowing equation driven by the fractional Laplacian$$ \left\{\begin{array}{ll}(-\Delta)^s u-\lambda u=f(x,u) in Ω \\u=0 in \mathbb{R}^n \backslash Ω.\end{array} \right.$$Thus, the results presented here may be seen as the extensionof some classical nonlinear analysis theorems to the case of fractionaloperators.

In this paper, we mainly review recent results on mathematical theory andnumerical methods for Bose-Einstein condensation (BEC),based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are thencarried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models forspin-1 BEC, Bogoliubov excitation and BEC at finite temperature.

We discuss the Cucker-Smale's (C-S) particle model for flocking, deriving precise conditions for flocking to occur when pairwise interactions are sufficiently strong long range. We then derive a Vlasov-type kinetic model for the C-S particle model and prove it exhibits time-asymptotic flocking behavior for arbitrary compactly supported initial data. Finally, we introduce a hydrodynamic description of flocking based on the C-S Vlasov-type kinetic model and prove flocking behavior without closure of higher moments.

To understand the impact of spatial heterogeneity of environment and movement of individuals on the persistence and extinction of a disease, a spatial SIS reaction-diffusion model is studied, with the focus on the existence, uniqueness and particularly the asymptotic profile of the steady-states. First, the basic reproduction number $\R_{0}$ is defined for this SIS PDE model. It is shown that if $\R_{0} 1$, the disease-free equilibrium is unstable and there is a unique endemic equilibrium. A domain is called high (low) risk if the average of the transmission rates is greater (less) than the average of the recovery rates. It is shown that the disease-free equilibrium is always unstable $(\R_{0} > 1)$ for high-risk domains. For low-risk domains, the disease-free equilibrium is stable $(\R_{0} < 1)$ if and only if infected individuals have mobility above a threshold value. The endemic equilibrium tends to a spatially inhomogeneous disease-free equilibrium as the mobility of susceptible individuals tends to zero. Surprisingly, the density of susceptibles for this limiting disease-free equilibrium, which is always positive on the subdomain where the transmission rate is less than the recovery rate, must also be positive at some (but not all) places where the transmission rates exceed the recovery rates.

The existence of a global attractor in the natural energy space is proved for the semilinear wave equation $u_{t t}+\beta u_t -\Delta u + f(u)=0$ on a bounded domain $\Omega\subset\mathbf R^n$ with Dirichlet boundary conditions. The nonlinear term $f$ is supposed to satisfy an exponential growth condition for $n=2$, and for $n\geq 3$ the growth condition $|f(u)|\leq c_0(|u|^{\gamma}+1)$, where $1\leq\gamma\leq\frac{n}{n-2}$. No Lipschitz condition on $f$ is assumed, leading to presumed nonuniqueness of solutions with given initial data. The asymptotic compactness of the corresponding generalized semiflow is proved using an auxiliary functional. The system is shown to possess Kneser's property, which implies the connectedness of the attractor. In the case $n\geq 3$ and $\gamma>\frac{n}{n-2}$ the existence of a global attractor is proved under the (unproved) assumption that every weak solution satisfies the energy equation.

We prove the global well-posedness of the viscousincompressible Boussinesq equations in two spatial dimensions for generalinitial data in $H^m$ with $m\ge 3$.It is known that when both the velocity and thedensity equations have finite positive viscosity,the Boussinesq system does not develop finite time singularities.We consider here the challenging case when viscosity enters only in the velocityequation, but there is no viscosity in the density equation.Using sharp and delicate energy estimates, we prove global existenceand strong regularity of this viscous Boussinesq system for generalinitial data in $H^m$ with $m \ge 3$.

We propose a regularization algorithm for color/vectorial images which is fast, easy to code and mathematically well-posed. More precisely, the regularization model is based on the dual formulation of the vectorial Total Variation (VTV) norm and it may be regarded as the vectorial extension of the dual approach defined by Chambolle in [13] for gray-scale/scalar images. The proposed model offers several advantages. First, it minimizes the exact VTV norm whereas standard approaches use a regularized norm. Then, the numerical scheme of minimization is straightforward to implement and finally, the number of iterations to reach the solution is low, which gives a fast regularization algorithm. Finally, and maybe more importantly, the proposed VTV minimization scheme can be easily extended to many standard applications. We apply this $L^1$ vectorial regularization algorithm to the following problems: color inverse scale space, color denoising with the chromaticity-brightness color representation, color image inpainting, color wavelet shrinkage, color image decomposition, color image deblurring, and color denoising on manifolds. Generally speaking, this VTV minimization scheme can be used in problems that required vector field (color, other feature vector) regularization while preserving discontinuities.

In this article, we study generalized fractional derivatives that contain kernels depending on a function on the space of absolute continuous functions. We generalize the Laplace transform in order to be applicable for the generalized fractional integrals and derivatives and apply this transform to solve some ordinary differential equations in the frame of the fractional derivatives under discussion.

We prove the persistence of boundary smoothness of vortex patches for the quasi-geostrophic shallow-water (QGSW) equations. The QGSW equations generalize the Euler equations by including an additional parameter, the Rossby radius $ \varepsilon^{-1} $, which modifies the relationship between the streamfunction and the (potential) vorticity. Furthermore, for compactly supported initial data in the little Hölder space $ {c^{\gamma}}_c $, $ 0<\gamma<1 $, we prove that solutions of the QGSW equations converge in the $ {\mathcal{C}^\gamma} $ norm to the corresponding two-dimensional Euler solution as $ \varepsilon\to0 $, uniformly on a time interval independent of $ \varepsilon $.

This paper studies a nonlinear clamped plate equation with frictional damping, a logarithmic source and an internal tempered Caputo fractional operator acting on a time-varying delayed state. By introducing a normalized delay variable and a diffusive memory variable, the problem is reformulated as a non-autonomous first-order evolution system. Local well-posedness is proved in the natural plate energy space by Kato's non-autonomous evolution theory and a locally Lipschitz treatment of the logarithmic source. Under suitable structural conditions on the damping, delay and memory parameters, global existence is obtained through energy estimates and a potential-well argument. Exponential stability is derived by a Lyapunov functional, while finite-time blow-up is proved for negative initial energy by the concavity method. Numerical simulations based on finite differences, upwind delay discretization and diffusive-memory approximation illustrate the stable and blow-up regimes, limiting cases and parameter sensitivity.

This paper investigates a delayed reaction-diffusion-advection mussel-algae model defined on a rectangular domain, subject to mixed boundary conditions which are Dirichlet on one side and Neumann on the other three. The main novelty of this work lies in the adoption of a two-dimensional rectangular domain that better aligns with practical application scenarios, and the introduction of advection direction as a key parameter, thus better reflecting realistic ecological and aquaculture scenarios. Starting with a systematic analysis of the associated eigenvalue problem, this work elucidates the regulatory effects of diffusion, advection velocity, and advection direction on the principal eigenvalue. Building upon these results, we investigate delay-induced Hopf bifurcations and spatial Turing instabilities, highlighting the pivotal roles of the advection direction and the domain's aspect ratio in stability transitions and pattern selection. Furthermore, by comparing distinct boundary configurations, the study clarifies the coupling mechanisms between boundary constraints and directional flow in driving periodic oscillations and spatiotemporal pattern dynamics. From the perspective of bifurcation theory in dynamical systems, this study provides theoretical foundations and decision support for practical issues such as flow direction regulation, aquaculture pond design, and controlled boundary design and stocking management in mussel-algae farming systems.

In this paper, we investigate the initial condition control problem for backward-in-time semilinear parabolic equations. A variational approach with Tikhonov regularization is proposed to recover the unknown initial state from noisy final-time observations. Furthermore, employing the adjoint method, we explicitly derive the first-order necessary optimality criterion and the second-order optimality condition. An illustrative numerical example is presented to demonstrate the effectiveness, stability, and convergence of the approach.

We are interested in the dynamics of a population structured by a phenotypic trait. Individuals reproduce sexually, which is represented by a non-linear integral operator. This operator is combined to a multiplicative operator representing selection. When the strength of selection is small, we show that the dynamics of the population is governed by a simple macroscopic differential equation, and that solutions converge exponentially to steady states that are locally unique. The analysis is based on Wasserstein distance inequalities using a uniform lower bound on distributions. These inequalities are coupled to tail estimates to show the stability of the steady states.

We study the optimal regularizing effect for scalar balance laws with strictly convex fluxes and time-dependent nonlinear source terms. In a recent work, we proved that a suitable transversality condition between the source term and the degeneracy set of the flux restores full $BV_{\mathrm{loc}}$ regularity. In this note, we show that this condition is optimal. If the flux has polynomial degeneracy and the source vanishes along a degenerate state on a nontrivial time interval, then the full $BV_{\mathrm{loc}}$ regularizing effect cannot be expected in general. We construct an entropy solution showing that the fractional threshold $BV_{\mathrm{loc}}^{1/p}$, already known from the regularizing theory, cannot be improved. The same construction also yields the corresponding sharp obstruction in the Sobolev scale.

At ASIACRYPT 2022, Coutinho et al. proposed the stream cipher Forró as a successor to ChaCha. ChaCha is a well-known ARX-based cipher widely deployed in many practical applications, including TLS 1.3, SSH, Noise, S/MIME 4.0, and Google Chrome. The goal of the Forró design is to achieve a security level comparable to ChaCha while requiring fewer operations. Although Forró follows a structure similar to ChaCha, its modified round function provides stronger diffusion. The designers claim that 14 rounds of Forró offer a security level comparable to ChaCha20.In this work, we study the security of Forró using MILP-based differential-linear cryptanalysis. We model the ARX structure of the cipher using Mixed Integer Linear Programming (MILP) and automatically search for optimal differential trails. Using the best one-round trail, we construct a 3-round differential-linear distinguisher. Furthermore, we introduce the concept of probabilistic neutral vectors (PNVs), which improves the selection of effective neutral key-bit positions compared to traditional probabilistic neutral bit (PNB) methods. By combining our MILP-derived trails, the newly found differential-linear distinguisher, and the proposed PNV technique, we obtain improved key-recovery attacks on 5 and 5.5 rounds of Forró, improving the attack time by approximately $ 2^5 $ and $ 2^{30} $, respectively. In addition, to the best of our knowledge, we present the first key-recovery attacks on 5.75, $ 6^{*} $, and 6 rounds.

NASA's SCOLE (Spacecraft Control Laboratory Experiment) model is studied in the fully inertial regime, where an Euler-Bernoulli beam is coupled with both a translational tip mass and a rotational tip inertia. A mixed lower-order/higher-order boundary feedback law is imposed at the free end. For the continuous model, exponential decay of the total energy is established by a Lyapunov functional under explicit gain conditions. For the approximation, the beam equation is first reduced to a first-order system with respect to the spatial variable, and a staggered finite-difference scheme is then designed for the reduced model. The resulting discrete energy parallels its continuous counterpart and permits the construction of a discrete Lyapunov functional. The resulting semi-discrete family satisfies an exponential energy estimate with constants independent of the spatial mesh parameter. In this way, a rigorous first-order staggered finite-difference framework is obtained for the fully inertial SCOLE system.

In this work we study homoclinic classes for some classes of flows, extending to flows certain results previously obtained, by other authors, for diffeomorphisms. More precisely, we prove that given a conservative $ C^2 $ flow $ \varphi_t:M\to M $ over a smooth compact Riemannian manifold $ M $, if both the stable and unstable homoclinic classes of a certain periodic hyperbolic orbit $ \gamma $ have positive Lebesgue measure, then their intersection is an ergodic component of the flow. We also obtain analogous results for invariant SRB measures.

This paper studies a class of mixed mean-field jump processes on an abstract state space $ \Pi $, together with their associated $ N $-particle systems. The dynamics consist of the superposition of an independent Markovian component and a bounded mean-field jump interaction; in particular, piecewise deterministic Markov processes (PDMPs) with mean-field interactions are covered by this framework. A key feature of our setting is that the jump kernel may depend nonlinearly on the mean-field law, and correspondingly on the empirical measure at the particle level. Under a second-order bounded difference condition on the mean-field jump kernel, we establish entropic propagation of chaos as $ N \to \infty $. In particular, we obtain an explicit quantitative bound on the relative entropy between the law of the $ N $-particle system and the product measure induced by the mean-field limit. The proof relies on the second-order concentration inequality introduced in [19].