Special Session 134: Recent advances in wavelet analysis, PDEs and dynamical systems - part II

Read-Bajraktarevi\`c equation on Hardy-Orlicz spaces and symmetries

waldo w arriagada Wenzhou-Kean University Peoples Rep of China Co-Author(s):    Emanuel Guariglia

Abstract: In this short note we prove the existence of local fractal functions of the Hardy-Orlicz class. The graph of a local fractal function coincides with the attractor of an appropriate iterated function system, whose construction is fairly standard. Local fractal functions appear naturally as the fixed points of the Read-Bajraktarevi\`c operator when restricted to a suitable Orlicz-Sobolev space. Our results extend some of the outcomes obtained by Massopust on Lebesgue and Sobolev spaces to higher order, dimension and function spaces (where the role of the norm is now played by a Young function).

HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS WITH DISCRETE EFFECT MEMORY AND BOUNDARY VALUE PROBLEMS FOR ITS

Anar Assanova Institute of Mathematics and Mathematical Modeling Kazakhstan Co-Author(s):

Abstract: In the present communication we study a questions for existence and uniqueness of solution to the boundary value problem for the system of hyperbolic partial differential equations with discrete effect memory on the rectangular domain. Considered problem is transferred to family of problems for differential equations with discrete effect memory and integral condition by introducing a new functions. Further, introducing functional parameters as the values of the desired solution along the lines of the domain partition with respect to the time variable, we obtain an equivalent problem for the system of differential equations with initial conditions and functional relations with respect to the introduced parameters. We have developed a two-stage procedure to approximately solve the latter problem. We have obtained some conditions for the convergence of approximate solutions to the exact solution of the problem under study in terms of input data and proved that these conditions guarantee the existence of a unique solution of the equivalent problem. Finally, we have established coefficient conditions for the unique solvability of the problem for the system of hyperbolic partial differential equations with discrete effect memory subject initial and integral conditions. This research is funded by the Science Committee of the Ministry of Science and Higher Education of Republic of Kazakhstan (Grant no. AP19675193).

Perturbations of non-autonomous second-order abstract Cauchy problems

Christian Budde University of the Free State So Africa Co-Author(s):    Christian Seifert

Abstract: In this talk we present time-dependent perturbations of second-order non-autonomous abstract Cauchy problems associated to a family of operators with constant domain. We make use of the equivalence to a first-order non-autonomous abstract Cauchy problem in a product space, which we elaborate in full detail. As an application we provide a perturbed non-autonomous wave equation. Autonomous second-order abstract Cauchy problems which often occur in the context of wave equations, have been studied intensively by several authors in the past. In contrast to the first-order problem, where (classical) solutions are given by C0-semigroups, one needs another solution concept for the second-order case, the so-called cosine and sine families. Similar to the Hille-Yosida generation theorem for strongly continuous semigroups, one can also characterize generators of cosine families. Non-autonomous second-order abstract Cauchy problems have been studied first by Kozak and later on by Bochenek, Winiarska and Lan, just to mention a few. The classical idea helps to reduce the non-autonomous second-order abstract Cauchy problem again to a first-order problem. The goal is to establish a bounded perturbation result for non-autonomous second-order abstract Cauchy problems. As mentioned above, we also discuss the non-autonomous wave equation as an example.

Support Vector Regression Estimator with Kalman Filtering for Testing Chaotic dynamic System via lyapunov Exponents

Slim cho Chokri Mocfine laboratory ISCAE Manouba University Tunisia Co-Author(s):

Abstract: Support vector machines (SVMs) are a recent supervised learning approach towards function estimation. They combine several results from statistical learning theory, optimisation theory, and machine learning, and employ kernels as one of their most important ingredients. In this regard we propose a novel methodologie to derives a formal test from the nonparametric support vector regression estimator of the Lyapunov exponent in a noisy system with Kalman filtering (SVREKF). Amongst others the advantage of SVREKF compared to the widely used estimators (which is implemented using Artifcial Neural Network (ANN)) is, implicit nonlinear mapping and better regularization capability. In this work, we make use of Kalman recursions instead of quadratic programming which is generally used in kernel methods. We introduce a statistical framework for testing the chaotic hypothesis based on the estimated Lyapunov exponents and a consistent variance estimator. We apply our test to some of the standard chaotic systems and the financial time series. The performance of the test is very satisfactory in the presence of noise as well as with limited number of observations.We also discuss some of the limitations of our fndings.

Fractality in prime distribution

Emanuel Guariglia Wenzhou-Kean University Peoples Rep of China Co-Author(s):    Shengyi Qi

Abstract: This work concerns the fractal-like behavior of prime subsets. Numerical simulations indicate that some prime subsets (e.g., Chen primes, Gaussian primes) resemble a fractal-like behavior. Our simulations are based on the construction of binary images based on prime numbers. Indeed, two-integer sequences can easily be converted into a two-color image. In particular, the Cantor set seems to cover a relevant role in our analysis. It seems that the Cantor set has a sort of relevant role in prime number theory. In addition, our results have potential applications in chaotic dynamical systems and cryptography.

Global dynamics of a tumor growth model with three mechanisms

Mahmoud A. Ibrahim Bolyai Institute, university of Szeged Hungary Co-Author(s):    Attila D\`enes and Gergely R\ost

Abstract: Understanding the emergence of chemotherapy resistance in cancer patients, whether driven by Darwinian evolution, gene expression changes, or the transfer of microvesicles from resistant to sensitive cells, is crucial as it significantly impacts treatment outcomes by promoting the survival and spread of resistant cells. We have developed a mathematical model to describe the evolution of tumor cells that are either sensitive or resistant to chemotherapy and to make it more realistic by including a separate equation for the number of microvesicles. This model accounts for three resistance mechanisms: Darwinian selection, Lamarckian induction, and resistance via microvesicle transfer, mimicking infectious spread. Our analysis identifies three key threshold parameters that determine the stability and existence of different equilibria within the system. We provide a comprehensive description of the global dynamics, including the existence of global attractors, depending on these threshold values. Additionally, we explore the effects of varying drug concentrations and characterize potential bifurcation sequences that lead to either successful treatment or therapeutic failure. Lastly, we identify the factor that exerts the most significant influence on cancer cell growth.

Non-Stationary Signal Analysis in Voice Disorders Using Wavelet Transform

Gabriel Jose Pellisser Dalalana University of Sao Paulo Brazil Co-Author(s):    Lianglin Li, Rodrigo Capobianco Guido

Abstract: This talk presents the application of wavelet theory in the analysis of voice pathologies, focusing on its advantages for handling non-stationary biomedical signals. Voice disorders, such as Reinke`s Edema and Dysphonia, exhibit complex patterns that can be effectively characterized using wavelet-based methods. We explore multiresolution wavelet analysis to dissect voice signals into frequency components, enabling a detailed examination of time-frequency characteristics. This approach facilitates the identification of pathological indicators that are otherwise challenging to detect with traditional methods. By employing specific wavelet filters, including Daubechies and Symlets, we demonstrate the capacity to isolate crucial features, such as irregular vibratory patterns and spectral shifts, providing valuable insights into the underlying pathological conditions. The application of wavelet analysis enhances diagnostic accuracy and provides a robust framework for understanding the dynamics of voice disorders. This talk aims to highlight recent advancements, challenges, and opportunities in using wavelets for the non-invasive evaluation of vocal health, promoting interdisciplinary collaboration between mathematics, biomedical engineering, and clinical practice.

Vortex dynamics for the Gross-Pitaevskii equation

Rowan Juneman University of Bath England Co-Author(s):    Manuel del Pino, Monica Musso

Abstract: The Gross-Pitaevskii equation in the plane arises as a physical model for an idealized, two-dimensional superfluid. We construct solutions to this equation with multiple vortices of degree \(\pm1\), corresponding to concentration points of the associated fluid vorticity. The vortex dynamics is described on any finite time interval, and at leading order is governed by the classical Helmholtz-Kirchhoff system. Compared to previous rigorous results of Bethuel-Jerrard-Smets and Jerrard-Spirn, we use a different method based on linearization around an approximate solution. This approach provides a very precise description of the solutions near the vortex set and information on lower order corrections to the vortex dynamics. Moreover, our analysis of the linearized problem is potentially of independent interest in the study of long-time dynamics. This is joint work with Manuel del Pino and Monica Musso.

Geometrical equivalence of global attractors of reaction diffusion equations under Lipschitz perturbations

Jihoon Lee Chonnam National University Korea Co-Author(s):

Abstract: In this talk, we explore the geometrical equivalence among the global attractors of reaction diffusion equations under Lipschitz perturbations of the domain and equation. Using the facts, we obtain the continuity of the global attractors of reaction diffusion equations if every equilibrium point of the system is hyperbolic. These extend the recent results L.Pires. {\it Joint work with N.Nguyen and L.Pires.}

On the generalization of IFSs

Lianglin Li Wenzhou-Kean University Peoples Rep of China Co-Author(s):    Emanuel Guariglia, Jiayi Wei

Abstract: This talk concerns the generalization of iterated function systems. In fractal geometry, iterated function systems have already been generalized for superfractals by the concept of superIFS. Here, we propose another generalization of iterated function systems with an application in signal theory.

Global existence for the 2D Kuramoto-Sivashinsky equation

Anna L Mazzucato Penn State University USA Co-Author(s):    David Ambrose

Abstract: I will present recent results concerning global existence for the Kuramoto-Sivashinsky equation in 2 space dimensions in the presence of growing modes. The KSE is a model of long-wave instability in dissipative systems.

Extension of wavelets/PDEs to topologically complicated domains

Mani Mehra Department of Mathematics, Indian Institute of Technology Delhi, IITD, India India Co-Author(s):

Abstract: Differential equations on topologically complicated domains is a relatively new branch in the theory of differential equations. Some of the examples include differential equations on manifolds or irregularly shaped domains and differential equation on network-like structure. Differential equations on manifolds arises in the areas of mathematical physics, fluid dynamics, image processing, medical imaging etc.. Differential equations on network-like structure also play a fundamental role in many problems in science and engineering. The aim of this talk is to show how wavelets could be extended to network to solve partial differential equations on network like structure using spectral graph wavelet.

Incompressible MHD Without Resistivity: Structure and regularity

Ronghua Pan Georgia Institute of Technology USA Co-Author(s):

Abstract: We study the global existence of classical solutions to the incompressible MHD system without magnetic diffusion in 2D and 3D. The lack of resistivity or magnetic diffusion poses a major challenge to a global regularity theory even for small smooth initial data. However, the interesting nonlinear structure of the system not only leads to some significant challenges, but some interesting stabilization properties. This helps the establishment of the global regularity and the existence of global strong solutions.

Dynamics of the Navier-Stokes equations in critical spaces

Gabriela Planas Universidade Estadual de Campinas Brazil Co-Author(s):    M. Ikeda, L. Kosloff, C. Niche

Abstract: We discuss the large-time dynamics for the Navier-Stokes equations in the critical space $\dot{H}^{1/2}$. Known decay estimates merely provide decay to zero with no explicit rates. We show an algebraic upper bound for the decay rate of solutions.

Regularity procedure for solutions of PDEs having discontinous coefficients

Maria Alessandra Ragusa University of Catania Italy Co-Author(s):

Abstract: Sharp inequalities have a rich tradition in harmonic analysis, going back to the epoch-making works of Hardy--Littlewood--Sobolev inequalities. In this talk, we survey selected highlights from the past decades, describe some of our own contributions, and pose a few open problems which lie at the interface of euclidean harmonic analysis, regularity of solutions of PDEs and systems.

Algorithmic detection of conserved quantities for finite- difference schemes

Ricardo L Ribeiro KAUST Saudi Arabia Co-Author(s):    Diogo A. Gomes, Friedemann Krannich, Bashayer Majrashi

Abstract: Many partial differential equations (PDEs) admit conserved quantities such as mass for the heat equation or energy for the advection equation. These are often essential for establishing well-posedness results. When approximating a PDE with a finite-difference scheme, it is crucial to determine whether related discretized quantities remain conserved by the scheme. Such conservation may ensure the stability of the numerical scheme. We present an algorithm for verifying the preservation of a polynomial quantity under a polynomial finite-difference scheme. Our schemes can be explicit or implicit, have higher-order time and space derivatives, and have an arbitrary number of variables. Additionally, we introduce an algorithm for finding conserved quantities. We illustrate our algorithm in several finite-difference schemes. Our approach incorporates a naive implementation of Comprehensive Grobner Systems to handle parameters, ensuring accurate computation of conserved quantities.

Interior estimates for elliptic equations in Morrey-type spaces

Andrea Scapellato University of Catania Italy Co-Author(s):

Abstract: The talk deals with some interior estimates for the solutions of elliptic equations in Morrey-type spaces. We collect some results related to the boundedness of fractional integral operators in several Morrey-type spaces and we show some applications related to the regularity theory for elliptic equations with discontinuous coefficients both in non-divergence and divergence form.

Homogenization of elliptic operators with coefficients in variable exponent Lebesgue spaces

Adisak Seesanea Sirindhorn International Institute of Technology, Thammasat University Thailand Co-Author(s):

Abstract: We shall discuss homogenization problems involving second-order elliptic operators in the divergence form whose drift and potential terms belong to variable exponent Lebesgue spaces $L^{p(\cdot)}$. Our techniques rely on a study of the periodic unfolding method in the $L^{p(\cdot)}$ setting. This is a joint work with Mya Hnin Lwin.

Axisymmetric flows with swirl for Euler and Navier-Stokes equations

Athanasios Tzavaras King Abdullah University of Science and Technology Saudi Arabia Co-Author(s):    Theodoros Katsaounis and Ioanna Mousikou

Abstract: We consider the incompressible axisymmetric Navier-Stokes equations with swirl as an idealized model for tornado-like flows. Assuming an infinite vortex line which interacts with a boundary surface resembles the tornado core, we look for stationary self-similar solutions of the axisymmetric Euler and axisymmetric Navier-Stokes equations. We are particularly interested in the connection of the two problems in the zero-viscosity limit. First, we construct a class of explicit stationary self-similar solutions for the axisymmetric Euler equations. Second, we consider the possibility of discontinuous solutions and prove that there do not exist self-similar stationary Euler solutions with slip discontinuity. This nonexistence result is extended to a class of flows where there is mass input or mass loss through the vortex core. Third, we consider solutions of the Euler equations as zero-viscosity limits of solutions to Navier-Stokes. Using techniques from the theory of Riemann problems for conservation laws, we prove that, under certain assumptions, stationary self-similar solutions of the axisymmetric Navier-Stokes equations converge to stationary self-similar solutions of the axisymmetric Euler equations as the viscosity tends to zero. This allows to characterize the type of Euler solutions that arise via viscosity limits.

Blow up solutions on critical problems

Giusi Vaira University of Bari Aldo Moro Italy Co-Author(s):

Abstract: In this talk I will consider some critical problems on domans and on manifold with boundary discussing the existence and qualitative properties of solutions.

Generalized eigenvalue problem of quantum walks in 1-dimension

Kazuyuki Wada Hokkaido University of Education Japan Co-Author(s):    Masaya Maeda, Akito Suzuki

Abstract: This talk is concerned with the discrete-time quantum walks in 1-dimension. This model is corresponding to the space-time discretized Dirac equation. We consider the generalized eigenvalue problem under long-range perturbations and construct Jost solutions. After that, we apply it to spectral theory.

Exponential spectral process (ESP): High order temporal discretization for semilinear PDEs

Xiang Wang Jilin University Peoples Rep of China Co-Author(s):

Abstract: We propose an exponential spectral process (ESP) method for time discretization of spatial-temporal equations. The proposed ESP method uses explicit iterations at each time step, which allows us to use simple initializations at each iteration. This method has the capacity to obtain high accuracy (up to machine precision) with reasonably large time step sizes. Theoretically, the ESP method has been shown to be unconditionally energy stable for arbitrary number of iteration steps for the case where two spectral points are used. To demonstrate the advantages of the ESP approach, we consider two applications that have stability difficulties in large-time simulations. One of them is the Allen-Cahn equation with the symmetry breaking problem that most existing time discretizations face, and the second one is about the complex Ginzburg-Landau equation, which also suffers from large-time instabilities.

Chaos and convergence in 3D H\`{e}non maps

Jiayi Wei Wenzhou-Kean University Peoples Rep of China Co-Author(s):    Emanuel Guariglia, Lianglin Li

Abstract: This work concerns the chaotic behavior in H\`{e}non maps. More precisely, we deal with its aperiodic orbits. We thus remove these points from the dynamics of the 3D H\`{e}non map. Our results are based on a perturbation technique so that the perturbed map has the same fixed points of the 3D H\`{e}non map.

Stability of the Caffarelli-Kohn-Nirenberg inequality

Yuanze Wu China University of Mining and Technology Peoples Rep of China Co-Author(s):    Juncheng Wei

Abstract: In this talk, I will report our recent results on the stability of the Caffarelli-Kohn-Nirenberg inequality both in the functional inequality setting and in the critical point setting. In these results, we establish the sharp Bianchi-Egnell stability in the functional inequality setting under the nondegenerate assumption and discuss the existence of minimizers of the variational problem related to the optimal constant. We also establish the sharp Figalli-Glaudo stability in the critical point setting both under the nondegenerate assumption and the degenerate assumption. Rather surprisingly, the optimal power of the stability under the degenerate assumption is an absolute constant which is independent of the power of the nonlinearity and the number of bubbles.

Parabolic-scalings on asymptotic expansion of the incompressible Navier-Stokes flow

Masakazu Yamamoto Niigata University Japan Co-Author(s):

Abstract: In this talk, large-time behavior of the incompressible Navier-Stokes flow in n-dimensional whole space is discussed. For this theme, Carpio (1996) and Fujigaki and Miyakawa (2001) derived the asymptotic expansion up to n-th order by employing the theory via Escobedo and Zuazua (1991). In those expansion, integrability of the moments of solution is required. However, on the next profile, the moment is growing logarithmically in time. Not only that, but there is also a problem with spatial integrability since the nonlinear effect contains the nonlocal operator. More precisely it contains the Riesz transform which is coming from the solenoidal conditions. To omit those difficulties, we employ the renormalization together with Biot-Savart law. By employing this method, asymptotic expansion up to 2n-th order is derived. Any terms on the expansion have their own parabolic-scalings. The parabolic scaling guarantees uniqueness of the expansion. Furthermore the above logarithmic evolution is specified on this expansion.

Optimistic Sample Size Estimate for Deep Neural Networks

Yaoyu Zhang Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):

Abstract: Estimating the sample size required for a deep neural network (DNN) to accurately fit a target function is a crucial issue in deep learning. In this talk, we introduce a novel sample size estimation method inspired by the phenomenon of condensation, which we term the optimistic estimate. This method quantitatively characterizes the best possible performance achievable by neural networks. Our findings suggest that increasing the width and depth of a DNN preserves its sample efficiency. However, increasing the number of unnecessary connections significantly deteriorates sample efficiency. This analysis provides theoretical support for the commonly adopted strategy in practice of expanding network width and depth rather than increasing the number of connections.

Singularity formation for the heat flow of the H-system

Yifu Zhou Wuhan University Peoples Rep of China Co-Author(s):    Yannick Sire, Juncheng Wei, and Youquan Zheng

Abstract: In this talk, we shall briefly report a recent gluing construction of finite-time blow-up for the heat flow of the $H$-system, describing the evolution of surfaces with constant mean curvature. One key observation is a decoupling property of the system at linear level, and as a by-product, non-degeneracy of H-bubbles with higher degree is shown.