Special Session 69: Mathematical Models and Analysis of (Partial) Differential Equations in the Applied Sciences

Nonlinear gradient effects in viscoelasticity

Aseel AlNajjar King Abdullah University of Science and Technology(KAUST) Saudi Arabia Co-Author(s):

Abstract: We consider a system of strain-gradient viscoelasticity arising in phase transition models, with a particular focus on nonlinear dispersive effects. In the onedimensional case, we prove the existence of weak solutions under minimal assumptions, highlighting the role of nonlinear strain-gradient terms in the analysis. In the case of constant dispersion and in the multi-dimensional case, we prove global existence results and investigate the zero dispersion limit, including a rate of convergence in two space dimensions. We conclude by discussing some of the main challenges that arise in the presence of physical boundaries.

Nonexistence of multi-dimensional solitary waves for the Euler-Poisson system

Junsik Bae Kyungpook National University Korea Co-Author(s):    Daisuke Kawagoe

Abstract: We study the nonexistence of multi-dimensional solitary waves for the Euler-Poisson system governing ion dynamics. It is well-known that the one-dimensional Euler-Poisson system has solitary waves that travel faster than the ion-sound speed. In contrast, we show that the two-dimensional and three-dimensional models do not admit nontrivial irrotational spatially localized traveling waves for any traveling velocity and for general pressure laws. Our results provide theoretical evidence for the stability of line solitary waves in multi-dimensional Euler-Poisson flows. We derive some Pohozaev type identities associated with the energy and density integrals. This approach is extended to prove the nonexistence of irrotational multi-dimensional solitary waves for the two-species Euler-Poisson system for ions and electrons.

Analytical and Computational Methods for Bifurcation Analysis of Collapsing Solutions to Nonlinear Dispersive PDEs

Efstathios Charalampidis San Diego State University USA Co-Author(s):

Abstract: In this talk, the finite-time blow-up in nonlinear dispersive PDEs will be discussed as a bifurcation problem. First, we will present a universal framework for the identification of self-similar waveforms as stationary solutions in a frame that co-explodes with the solution. This will allow us to perform a spectral analysis of these solutions in the co-exploding frame in order to infer their stability. As prototypical examples, we will consider the 1D Nonlinear Schroedinger (NLS) equation with power-law nonlinearity and generalized Korteweg-de Vries (gKdV) model. Self-similar collapsing branches emanate from the solitary branch at critical nonlinearity exponent therein. However, their stability analysis will reveal the emergence of unstable modes which in turn are intimately connected with symmetries of the systems in the original frame. We will show that these modes do not correspond to true instabilities but rather to neutral eigen-directions. Numerical results both at the existence and stability will be compared with normal forms and WKB results where an excellent agreement will be observed between the two. Then, time-permitting, we will depart from 1D settings and focus our considerations on the 2D NLS model with power-law nonlinearity. The accurate computation of states and their stability requires to use alternative techniques. Indeed, we will present a computational framework that have been developed in the open source software FreeFEM that combines domain-decomposition methods and mesh adaptivity tools. Novel results will be discussed together with the performance of the numerical codes.

Mathematical model for tumor growth and lactate kinetics in glioma

Laurence L. Cherfils La Rochelle University France Co-Author(s):

Abstract: Gliomas are widespread and invasive brain tumors. Except for grade I gliomas, which can be cured after complete surgical resection, the prognosis remains poor for grade II to IV gliomas, despite radiotherapy and chemotherapy treatments. Recent studies have shown that lactate plays an important role in tumor growth, and new therapeutic strategies targeting lactate metabolism have emerged. In this talk, I will present a mathematical model (in the form of partial differential equations) that describe both the temporal evolution of tumor cell density and lactate kinetics within the tumor. Two therapies are incorporated into the model: a chemotherapy treatment and a therapy specifically targeting lactate production or transport. These treatments are considered as control functions, and we seek an optimal therapeutic strategy, i.e., patient-specific dosages that are as low as possible to minimize side effects without reducing treatment efficacy. I will present results related to the mathematical analysis of these models and propose several numerical simulations to illustrate our findings.

A Coupled Surface Diffusion and Mean Curvature Flow problem - Steady States and Stability

Daniel Goldberg Technion - Israel Institute o Technology Israel Co-Author(s):    Daniel Goldberg and Amy Novick-Cohen

Abstract: Patterned thin gold lines when undergoing solid-state dewetting break up into linear grain sections interspersed with occasionally larger ``abacus`` particles. We model this system using coupled geometric flows, where the exterior surfaces of the grains evolve via Surface Diffusion $V = -\Delta_{\Gamma(t)}H$ and the internal grain boundaries evolve via Mean Curvature Flow $V = A H$. In this talk we first motivate the problem using these experimental observations. Next, we describe the steady states for this mixed-order system. By assuming axi-symmetry, we can systematically construct composite equilibria by piecing together Delaunay surfaces --- specifically, combining unduloidal, cylindrical and spherical exterior surfaces partitioned by planar or catenoidal internal grain boundaries. We then introduce a stability analysis by first linearising the coupled (nonlinear) problem about these composite steady states, deriving a Jacobi operator whose strict positivity implies asymptotic stability of the linearised problem. Using this approach and building upon the classical Rayleigh stability criteria, we obtain stability predictions for steady-state configurations consisting of near-cylindrical and near-spherical grains meeting at a single planar grain boundary.

Localization of Self-Similar Solutions in Diffusion-Relaxation Systems

Hoyoun Kim King Abdullah University of Science and Technology (KAUST) Korea Co-Author(s):

Abstract: In this talk, we construct localizing solutions for diffusion-relaxation systems by exploiting their self-similar structure. Diffusion-relaxation models arise in a wide range of applications, from shear-induced motion to biological chemotaxis, and exhibit equilibration and localization depending on parameter regimes. We extend the self-similar formulation to multi-dimensional domains and analyze the resulting asymptotic behavior of solutions. Using geometric singular perturbation theory and the Poincar\'e-Bendixson lemma on a perturbed invariant manifold, we establish the existence of localized self-similar solutions.

Stokes` Phenomenon Within a Small-Time Boundary Layer

Christopher Lustri The University of Sydney Australia Co-Author(s):

Abstract: The 5th-order KdV equation is a PDE that describes shallow water waves with surface tension. The steady version of this problem is a canonical ODE problem in exponential asymptotics, studied by Akylas, Grimshaw & Joshi, and others, who demonstrated that the (symmetric) steady solution is unstable and cannot be reached by the time-varying PDE. Instead, the PDE features a burst of waves that propagate in one direction and cannot be described by the steady-state solution. With the advent of exponential asymptotics for PDEs, it is now possible to understand the evolution of the PDE system: the appearance of these rapidly propagating waves occurs due to Stokes' phenomenon in a small-time boundary layer. I will demonstrate the Stokes structure and show the form of these rapidly propagating waves.

Constructing Steady States Using Delaunay Surfaces

Amy Novick-Cohen Technion - Israel Institute of Technology Israel Co-Author(s):    D. Goldberg, K. Golubkov, Y. Vaknin, A. Zigelman

Abstract: Delaunay surfaces were introduced by Delaunay in 1841, and the motions known as mean curvature flow and surface diffusion were introduced by Mullins in 1956 and 1957, respectively, in modelling materials science processes. Quite recently has it been noted that the Delaunay surfaces may be successfully combined to describe various materials science phenomena, such as hillock and hole formation, and line grating structures. We review some of these constructions and their implications, indicating open questions and future directions. [1] K. Golubkov, A. Novick-Cohen, Y. Vaknin, Coupled surface diffusion and mean curvature motion: an axisymmetric system with two grains and a hole. Quart. Appl. Math. 83 (2025) 97-134. [2] A. Zigelman & A. Novick-Cohen, Critical effective radius for holes in thin films: energetic and dynamic considerations. J. Appl. Phys. 134 (2023) 135302. [3] D. Goldberg & A. Novick-Cohen, Rayleigh type stability criteria for certain coupled surface diffusion and mean curvature flows, in preparation.

Convergence to self-similarity for the additive Smoluchowski coagulation equation with source

Robert Pego Carnegie Mellon University USA Co-Author(s):

Abstract: We study long-time behavior of cluster-size distributions for the Smoluchowski coagulation equation with time-dependent injection of mass, in the critical case when the rate kernel is additive in cluster sizes. Scaled by an expected cluster size, solutions converge to the same self-similar profile for a great variety of injection schedules, corresponding to mass growth histories that are constant, power law, exponential, and double exponential, e.g. Key ingredients in the analysis are: an improved continuity theorem for Bernstein transforms (aka Laplace exponents), and scaling analysis of characteristics for the PDE that transforms satsify.

Sharp front asymptotics in cascading Fisher KPP systems and multitype Branching Brownian Motion

Alexandra Stavrianidi University of Munster Germany Co-Author(s):

Abstract: The Fisher KPP equation is closely connected to Branching Brownian Motion, through the McKean representation, its solution describes the distribution of the maximal particle in the branching system. In this talk, I will discuss the long time behavior of coupled Fisher KPP reaction diffusion systems with interacting components associated with multitype Branching Brownian Motion. For a two component system, we show that the interaction modifies the classical Bramson logarithmic delay. Our PDE approach confirms previously known probabilistic results and extends them to general Fisher KPP nonlinearities. More generally, for cascading systems with k components, we establish sharp front asymptotics and convergence to the minimal speed traveling wave, providing a PDE proof of the conjectured asymptotics for the median position of the maximal particle in a cascading Branching Brownian Motion.

Mullins` nonlinear grooving solutions

Rawan Tarabeh Technion - Israel Institute of Technology Israel Co-Author(s):    Rawan Tarabeh

Abstract: The problem of thermal grooving was first proposed by Mullins in 1957. Mullins considered the morphological evolution of thin solid surfaces, namely solid films, and focused primarily on the exterior surface motion with surface diffusion as the dominant mass transport process. By assuming a small slope approximation, the problem can be reduced to a linear surface diffusion problem. Mullins showed that there exist self-similar solutions for the linearized problem. In the present study, we prove the existence and uniqueness of a self-similar solution for the original nonlinear Mullins` problem for sufficiently small values of the contact angle $\beta$. We do so by studying a reduced linear ODE system, which allows us to understand the asymptotic behavior of the solutions for the corresponding linear homogeneous operator, and then we prove existence of a unique solution to the nonlinear problem in closed form by iterations in an appropriate Banach space. Our methodology should be possible to implement in other problems of interest, such as the Wang`s nonlinear problem for surface evolution of wedges.

Nonlinear waves in metamaterials: from the left-handed regime to the frequency band gaps

Nikolaos L. Tsitsas Aristotle University of Thessaloniki Greece Co-Author(s):

Abstract: 1

Conservation Laws Constrained by Linear PDEs

Andreas Vikelis University of Vienna Austria Co-Author(s):

Abstract: Linear PDE constraints, expressed through constant rank operators, arise naturally in continuum mechanics and impose a specific analytical structure on the admissible states of conservation laws. This structure leads to a natural notion of convexity, acting precisely along the directions in which the operator loses ellipticity. In the talk, it will be discussed how the convexity determined by the PDE constraint, together with the constraint itself, yields robust stability and uniqueness properties for the associated general systems of conservation laws.

Stochastic homogenization of a droplet model in liquid-liquid phase transitions

Konstantinos Zemas University of Bonn Germany Co-Author(s):    Adriana Garroni, Caterina Zeppieri

Abstract: We rigorously derive, by means of $\Gamma$-convergence, a sharp-interface model for the coexistence of different liquid phases in a domain with a random distribution of droplets, in which a certain phase is prescribed. Starting from a diffuse-interface model of Modica-Mortola type, we show that under very broad assumptions on the droplet geometry (modelled probabilistically via a stationary marked point process), and at a critical scaling, a stochastic bulk term of capacitary type attributed to each other phase appears in the limit. This is joint work with Adriana Garroni (Sapienza University of Rome) and Caterina Zeppieri (Uni-Muenster)

On the density of the supremum of nonlinear SPDEs

Pavlos Zoubouloglou University of Muenster Germany Co-Author(s):

Abstract: We study the one-dimensional stochastic partial differential equation \begin{equation*} \frac{\partial u}{\partial t}(t,x) = -\kappa \frac{\partial^4 u}{\partial x^4}(t,x) + \rho \frac{\partial^2 u}{\partial x^2}(t,x) + b(u(t,x)) + \sigma(u(t,x))\, \dot W(t,x), \end{equation*} posed on a bounded spatial domain, where $u$ is understood in the random field sense. Depending on the value of $\kappa$, this equation includes the nonlinear stochastic heat equation with Dirichlet or Neumann boundary conditions, as well as the linearized stochastic Cahn-Hilliard equation with Neumann boundary conditions. We prove that the supremum of the solution admits a density with respect to the the Lebesgue measure. Our approach is based on Malliavin calculus, and in particular on the version of the Bouleau--Hirsch criterion for suprema developed by Nualart and Vives. One of the main difficulties lies in the analysis of the argmax set of the solution and in showing that the Malliavin derivative is almost surely nondegenerate on this set. As part of our arguments, we establish H\"older continuity properties for the Malliavin derivative of the solution as an $L^2-$valued process in the regimes considered in this work.