Papers

Let $(X,\mathbf{q},\mu)$ be an ultra-RD-space with upper dimension $n\in(0,\infty)$; i.e., it is a quasi-ultrametric space of homogeneous type whose measure $\mu$ satisfies an additional reverse doubling property. Let $\mathrm{ind\,}(X,\mathbf{q})\in(0,\infty]$ denote its lower smoothness index, as introduced by Mitrea et al. In this monograph, the authors first construct a new approximation of the identity on quasi-ultrametric spaces of homogeneous type, achieving a maximal degree of smoothness $0<\varepsilon\preceq\mathrm{ind\,}(X,\mathbf{q})$. This fundamental tool is then used to derive sharp homogeneous (as well as inhomogeneous) continuous/discrete Calderón reproducing formulae on ultra-RD-spaces. As applications, the authors establish Littlewood–Paley function characterizations for both Hardy spaces and Triebel–Lizorkin spaces on ultra-RD-spaces. The authors further introduce Hardy–Lorentz spaces $H^{p,q}_\ast(X)$ via the grand maximal function, with the sharp range $p\in(\frac{n}{n+\mathrm{ind\,}(X,\mathbf{q})},\infty)$ and $q\in(0,\infty]$, and provide their real-variable characterizations using radial/non-tangential maximal functions, (finite) atoms, molecules, and various Littlewood–Paley functions. Based on these characterizations, the authors prove a duality theorem between Hardy–Lorentz spaces and Campanato–Lorentz spaces, establish a real interpolation theorem for Hardy–Lorentz spaces, and derive boundedness results for Calderón–Zygmund operators on them. It should be emphasized that many of the main results in this monograph are indeed established in the more general setting of quasi-ultrametric spaces of homogeneous type.

We characterize possible pairs $(u_\varepsilon,c)\in C(\mathbb{R}^n\backslash\varepsilon\mathbb{Z}^n,\mathbb{R})\times\mathbb{R}$ addressing the homogenization problem for Hamilton–Jacobi equations $$ H\left(\frac{x}{\varepsilon}, d u_\varepsilon, u_\varepsilon\right)=c, \quad \left({\mathrm resp.} \quad H\left(\frac{x}{\varepsilon}, d u_\varepsilon, u_\varepsilon\right)=\varepsilon\Delta u_\varepsilon+c \right) $$ for all $\varepsilon>0$. Under a (not necessarily strict) monotonicity assumption on the Hamiltonian, we proposed certain criteria (based on the structure of Mather measures), under which all possible solutions $u_\varepsilon$ converge to a uniquely identified limit $u\in C(\mathbb{R}^n,\mathbb{R})$ solving the effective equation \[ \overline H( du,u)=c,\quad ({\mathrm resp.}\quad \overline H(du,u)=\Delta u+c) \] as $\varepsilon\rightarrow 0_+$ with a uniform rate $\mathcal{O}(\varepsilon)$.

This paper is Part II of a series on boundedness, global existence, and asymptotic behavior of positive classical solutions to the chemotaxis model $$\begin{cases} u_t=\Delta u-\chi_0\nabla\cdot\left(\frac{u^m}{(1+v)^\beta}\nabla v\right)+au-bu^{1+\alpha}, & x\in\Omega,\\ 0=\Delta v-\mu v+\nu u^\gamma, & x\in\Omega,\\ \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0, & x\in\partial\Omega, \end{cases} \tag{CM}$$ where $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain, $\alpha,\gamma,m,\mu,\nu>0$, $\chi_0\in\mathbb{R}$, and $a,b,\beta\ge0$. Part I established biologically relevant parameter regimes ensuring boundedness and global existence of positive classical solutions. Here we study persistence and stabilization of globally defined bounded positive solutions, quantifying how $\beta$ in the sensitivity $\chi(v)=\chi_0(1+v)^{-\beta}$ influences long-time dynamics. The factor $(1+v)^{-\beta}$ models signal-dependent desensitization, so large $\beta$ should promote stabilization, though it considerably complicates the analysis. We take $\mu=\nu=1$ with either $a=b=1$ (logistic source) or $a=b=0$ (minimal model). We show that when $m\ge1$, every globally defined bounded positive solution stays uniformly away from zero in space. We determine exact $\beta$-dependent critical sensitivity thresholds for local stability of constant solutions and establish $\beta$-dependent thresholds for their global stability; both tend to $\infty$ as $\beta\to\infty$. Thus signal saturation (large $\beta$) or repulsion ($\chi_0<0$) can prevent aggregation and promote relaxation to spatially homogeneous states. For $\beta>0$ and $m,\alpha,\gamma$ not all 1, we develop new techniques, including a Lyapunov function, pointwise estimates for $\nabla v$, and a nontrivial extension of the rectangle/ODE method from $\beta=0$ to $\beta>0$.

We investigate several instances of the Hadamard inequality in the mean in two dimensions. As a consequence, we prove the uniqueness of minimizers of an integral functional with a polyconvex integrand, subject to mixed Dirichlet and Neumann boundary conditions. The theoretical findings are complemented by computational experiments that illustrate the behavior of the minimizers.

Machine learning, especially physics-informed neural networks (PINNs) and their neural network variants, has been widely used to solve problems involving partial differential equations (PDEs). The successful deployment of such methods beyond academic research remains limited. For example, PINN methods primarily consider discrete point-to-point fitting and fail to account for the potential properties of real solutions. The adoption of continuous activation functions in these approaches leads to local characteristics that align with the equation solutions while resulting in poor extensibility and robustness. A general explicit network (GEN) that implements point-to-function PDE solving is proposed in this paper. The "function" component can be constructed based on our prior knowledge of the original PDEs through corresponding basis functions for fitting. The experimental results demonstrate that this approach enables solutions with high robustness and strong extensibility to be obtained.

This note is an addendum to the work initiated by Promyslov on the integro-differential equation arising in the ruin problem for annuity payment models. First, the existence of viscosity solutions is proved. Then the regularity of these solutions is established, showing that they are indeed classical solutions.

On rank one Riemannian symmetric spaces of compact type (of dimension $\ge 2$), we first obtain a quantitative characterization of Hölder continuity in terms of Cesàro means. In addition to some approximation theoretic applications, we also apply it to study the celebrated physical phenomenon known as `Talbot effect' arising from diffraction theory. More precisely, for almost every fixed time instance, we study the Hölder continuity and the fractal profile of the Schrödinger propagation in terms of the decay of the Littlewood-Paley projections of the initial data. In the process, we also obtain oscillatory expansions of zonal spherical functions uniformly near the origin and near the cut locus respectively, which may be of independent interest.

We investigate the semiclassical regularity of thermal equilibria in the presence of a harmonic potential at low temperature; that is, we obtain the asymptotic behavior of the Schatten norms of commutators of the one-body operators associated with these equilibria and the position and momentum operators. We also obtain upper bounds in the magnetic field case for the Fock-Darwin Hamiltonian. Our estimates, in particular, allow us to observe several regimes depending on the joint behavior of the Planck constant, the temperature, and the strength of the magnetic field.

This article is concerned with Kronecker flows on the infinite torus. The work is partly motivated by the fact that many Hamiltonian PDEs and systems on infinite lattices admit invariant tori, of possibly infinite dimension, on which the dynamics is linearizable. Finite-dimensional Kronecker flows are well understood: the dynamics can be reduced to a non-resonant flow on a subtorus, which is equivalent to being topologically transitive, to minimality, and to unique ergodicity in the projection. We prove that these properties still hold when the dimension of the torus is infinite if and only if the integer (finite) linear combinations of the frequencies form a free abelian group. Next, we construct a class of orbits whose closure is locally homeomorphic to the product of a ball and a Cantor set, extending a recent result by Sakbaev and Volovich. We also show that the Benjamin-Ono equation admits this type of solutions. Finally, we prove the equivalence between a classification problem for Kronecker flows and that for countable abelian groups without torsion.

We prove a localized big bang formation result, which does not require proximity of the initial data to any background solution. Suppose that we are given initial data for the Einstein–nonlinear scalar field equations on an open set $U \subset \mathbb{R}^3$. We identify a general condition on the initial data such that if the condition is satisfied in a large enough neighborhood of $x \in U$, then the corresponding maximal globally hyperbolic development has a local quiescent big bang singularity with curvature blow up to the past of $x$. We achieve the localization by introducing a new kind of foliation by spacelike hypersurfaces, given by the level sets of a time function satisfying a certain second order differential equation. This time function allows us to synchronize the singularity while at the same time yielding a symmetric hyperbolic formulation of Einstein's equations. Our new formulation also has two key advantages over previous localized big bang stability results. First, it is independent of the matter model, so it is possible that it could be used to prove big bang formation with matter models different from a scalar field. And second, it allows us to conclude that our solutions induce geometric initial data on the singularity, thus giving a complete description of the asymptotics towards the big bang.

This paper proposes a novel Mean-Field Game (MFG) framework for large-scale attacker-defender systems aimed at protecting one or multiple High-Value Units (HVUs). Motivated by classical agent-wise attrition models, we introduce a population-wise attrition mechanism formulated by statistical distance between populations, enabling a macroscopic description of weapon-based interactions between large populations. Leveraging this and Lions derivative on the space of probability measures, we derive the associated MFG system, which characterizes optimal strategies and the evolution of population distributions in attacker-defender interactions. We analyze the model by establishing upper and lower bounds on the defender density, ensuring physical consistency by preventing concentration and depletion. For numerical investigation, we develop a numerical scheme combining physics-informed neural networks with Sinkhorn method to solve attacker-defender MFG system. Simulations confirm the effectiveness of the framework and reveal key insights, including sensitivity to weapon strengths and population dispersion.

We study the conformal wave equation $\square_g \psi + \frac{2}{l^2} \psi = 0$ on 4-dimensional Schwarzschild–Anti de Sitter spacetimes under dissipative boundary conditions. We prove boundedness and decay of the non-degenerate energy of $\psi$ at an arbitrary polynomial rate of $(1+v)^{-n}$ provided that we control the (up to) $n$-times $T$-commuted energy. This contrasts with the inverse logarithmic decay obtained under Dirichlet boundary conditions and is in line with the result obtained in the pure Anti-de Sitter case under dissipative boundary conditions. In particular, the decay is not affected by the additional trapping at the photon sphere.

We demonstrate the stray-field-mediated skyrmion stabilizing capabilities of ultrathin exchange-decoupled antisymmetric ferromagnetic bilayers based on conventional transition metal materials. Using an asymptotically exact micromagnetic model valid in the ultrathin film limit, we show that the antisymmetric tailoring of the bilayer allows the Dzyaloshinskii-Moriya interaction and the dipolar interaction to act synergistically to stabilize skyrmions, in contrast to the monolayer case, in which these energies compete. To obtain optimal stability of these skyrmions against collapse and bursting – the two fundamental processes determining skyrmion lifetime, we carry out an asymptotic analysis of the saddle point solution that separates the skyrmion from the demagnetized state. The result is an optimal stability line for compact skyrmions in the non-dimensional parameter space of the effective Dzyaloshinskii-Moriya interaction strength and the effective film thickness. Our predictions are confirmed by extensive micromagnetic simulations of antisymmetric bilayers, using magnetic parameters of the conventional Pt/Co/AlO$_x$ systems. Our results provide a new pathway for experimental observations of 10 nm radius zero-field skyrmions with lifetimes compatible with information technology applications.

The curve shortening flow is a geometric heat equation for curves and provides an accessible setting to illustrate many important concepts from nonlinear partial differential equations, including maximum principle estimates, monotonicity formulas, Harnack inequalities and blowup analysis. All these techniques will be combined to give an exposition of Huisken's proof of Grayson's beautiful theorem that the curve shortening flow shrinks any closed embedded curve in the plane to a round point.

We discuss solutions of the one dimensional scalar conservation law with the flux function $y\longmapsto G_{c,\rho}\left(y\right)=((1-\rho)c-y)\mathbb{1}_{\{y>c\}}-\rho y\mathbb{1}_{\{y\leqslant c\}}$ for two specific initial conditions $u(\cdot,0)=u_0$. This equation arises as the limit of a specific conceptual hydrological model. For initial data strictly below (resp. above) the threshold level $c$, the equation reduces to a constant-speed transport equation with velocity $p$ (resp. $1$). Our goal is to understand precisely what happens when the initial condition crosses the threshold $c$, which corresponds to a generalisation of the Riemann problem, and to provide, in such cases, quasi-closed-form expressions for the corresponding solutions.

We derive a sharp spectral estimate for a superlinear free boundary problem arising in plasma physics. The semilinear equation is coupled with a constraint, which forces the analysis of a non-local eigenvalue equation. Consequently the corresponding first eigenvalue, say $\sigma_1$, is not a standard one and it is shown that it cannot satisfy a general isoperimetric property of Faber-Krahn type. This motivates a careful analysis of the problem on balls in any dimension $N\geq 2$, where we prove that in fact $\sigma_1$ is always positive. The implications about the uniqueness problem for the Emden equation are also discussed.

We present novel self-similar finite-time blowup scenarios for the 1D Hou–Luo model. We numerically demonstrate that solutions that initially satisfy certain derivative degeneracy condition can develop asymptotically self-similar finite-time blowups with singular self-similar profiles that are unbounded at some point. Moreover, this blowup phenomenon exhibits a two-stage feature: the solution first undergoes a local $L^{\infty}$ blowup at some time $\tilde{T}$, then continues in the weak sense beyond $\tilde{T}$ and develops a local $L^p$ blowup at a later time $T>\tilde{T}$ for some $p>0$. A further numerical investigation indicates that both stages are asymptotically self-similar. Finally, we extend our numerical study to the 2D Boussinesq equations and discover similar self-similar finite-time blowups with singular profiles that also exhibit a two-stage feature.

Volume-filling cross-diffusion equations for the components of a tissue structure are formally derived from mass conservation laws and force balances for the interphase pressures and viscous drag forces in a multiphase approach. The equations include Maxwell-Stefan, tumor-growth, thin-film solar cell models as well as novel volume-filling population systems. The Boltzmann and Rao entropy structures are explored. If the drag coefficients are all equal to one, the global-in-time existence of bounded weak solutions, their long-time behavior, and the weak-strong uniqueness of solutions to a regularized system are proved using entropy methods. In the general case, the resulting diffusion matrix is positively stable, ensuring local-in-time existence of solutions. Global-in-time existence of weak solutions is proved if the drag coefficients are sufficiently close to each other. This restriction is explained by the fact that the pressure forces are of degenerate type, while the drag forces are nondegenerate in the volume fractions. Numerical simulations are presented in one space dimension to illustrate the solution behavior beyond the entropy regime.

A comprehensive methodology for establishing the existence of gradient flows for cross-diffusion systems with respect to suitable energies is proposed. The approach is based on the construction of piecewise-in-time constant approximations via the Jordan-Kinderlehrer-Otto scheme. Compactness of the approximate sequence is obtained using either the flow interchange technique or the five gradient inequality. These methods are illustrated for both parabolic and hyperbolic-parabolic Busenberg-Travis systems, as well as for several of their variants. This paper reviews the results from the literature and discusses additional properties.

A model for morphoelastic growth, that is, growth influenced by elastic stress, driven by the absorption of nutrients is considered. The model features a multiplicative decomposition of the deformation gradient into an elastic contribution and a growth tensor. While the evolution of the system is governed by an ordinary differential equation for the growth tensor on a suitable Banach space, which depends on the elastic stresses and the concentration of a nutrient field, the total deformation is given by the solution of a quasi-static equilibrium equation arising from the formal Euler-Lagrange equations of a hyperelastic variational integral. The nutrient concentration is determined by a linear elliptic reaction-diffusion equation which is formulated in Lagrangian coordinates and whose coefficients depend on the growth tensor as well as the deformation gradient accounting for the change of material properties due to elastic deformation and growth. Existence and uniqueness of solutions of this fully coupled system of differential equations is proven via a fixed-point argument.

This paper introduces a new coupled p-Laplacian system with logarithmic nonlinearities, on locally finite graphs for p>4 and, in the critical case p=N>4, on R^N via a regularised formulation. The logarithmic coupling renders the energy functional ill-defined on the natural Sobolev space. To address this non-separable singularity, we develop an exponent calibration technique that converts the logarithmic terms into strictly lower-order power estimates. This technique underpins the existence proofs in two distinct settings and is essential in the continuous critical setting, where the failure of the L-infinity embedding renders classical methods inapplicable. The same technique is also used to recover compactness, thereby completing the passage from the regularised problem to the original ill-posed equation. In the discrete setting, we further develop a rigidity theory for ground states, which yields an explicit Hessian factorisation dictated by the Nehari constraint. The compactness restoration and limit theorem, the convergence rates, and the existence results in both settings together form a discrete-continuous double loop, from ill-posedness through compactness and rigidity to asymptotics, providing a unified variational framework for logarithmically coupled systems.

The aim of the course is to lead to an understanding of homogenisation processes in an operator-theoretic sense. In fact, using solely operator-theoretic means not referring to the particular form of the coefficients, we will identify an operator topology on the level of coefficients that will fully capture the convergence involved in the context of homogenisation. One upshot of this perspective will be that we will obtain homogenisation results for time-dependent partial differential equations (almost) for free.

We introduce and study a one-parameter family of curve diffusion flows with a scale-critical cubic curvature term for closed immersed planar curves. We first classify all closed stationary solutions, showing that they are precisely circles or a unique family of “super-lemniscates”. We then analyse the dynamical stability of homothetic circles. Under a sharp spectral condition, we establish, by purely variational methods, that any small perturbation of an $\omega$-fold circle monotonically approaches the unit $\omega$-circle after rescaling, translation, and reparametrisation. As a corollary, we determine the sharp ranges of the parameter for the stability of an embedded circle, and of all $\omega$-circles. We also uncover a striking arithmetic structure in the stability landscape, where the stability of $\omega$-circles depends non-monotonically on $\omega$.

The spin-diffusion Landau–Lifshitz–Bloch (SDLLB) system is a nonlinearly coupled system of quasilinear vector-valued PDEs which models the interaction between spin-polarised currents and magnetisation at high temperatures. The aim of this paper is twofold. Firstly, assuming the initial data is sufficiently small, we show the existence of a unique global strong solution to the SDLLB equation in a bounded domain $\Omega\subset \mathbb{R}^d$, where $d\leq 3$, thus ensuring well-posedness of the model. Secondly, we propose a decoupled linearised fully-discrete finite element scheme to solve the problem. Despite the strong nonlinearity of the system, the proposed scheme only requires the solution of two completely decoupled linear systems per time-step. Assuming adequate regularity of the exact solution and a certain time-step constraint, we rigorously show that the numerical scheme converges at an optimal rate. Several numerical experiments corroborate our theoretical results.

We address analytic regularity for the divergence equation $\text{div}\, u = f$ in $\Omega$, with $u=0$ on $\partial\Omega$, where $\Omega$ is an arbitrary bounded analytic domain and $\int_{\Omega} f\,dx=0$. If $f$ is analytic on the closure of $\Omega$, then we prove that there exists a solution that is analytic on the closure of $\Omega$.

We demonstrate that the failure of $L^1$ regularity in Calderón-Zygmund theory is a universal phenomenon: every non-constant holomorphic function in $\C^n$ generates a counterexample to the Poisson equation. In order to achieve this goal, we shall establish sharp level-set estimates that link harmonic analysis to the geometry of complex structure through Hironaka's resolution of singularities and the Łojasiewicz gradient inequality.

In this paper we discuss the solvability of the Neumann and Regularity boundary value problem of elliptic Schrödinger-type equation $-\DIV(A(x)\nabla u(x,t))+V(x)u(x,t)=0$ with bounded measurable uniformly elliptic coefficinets $A(x)$ independent of $t$ and $V$ in Reverse Hölder class $\mathcal{B}_q$, and Neumann boundary data $\partial_{\nu_A}u(x,0)=f(x)\in H^p_{\mathcal{L}}(\rn)$, or Regularity data $u(x,0)=g\in H^{1,p}_V(\rn)$, utilizing the method of layer potential. We prove the solvability when $A$ is a small $L^\infty$ perturbation of a matrix satisfying De Giorgi-Nash-Moser bounds. Besides we also give the Campanato norm estimate of the double layer potential related to the Dirichlet problem with boundary data in certain Campanato-type spaces.

The Aviles-Giga energy provides sequences of maps converging to weak solutions $m\colon\Omega \subset\mathbb R^2\to\mathbb R^2$ of the eikonal equation \begin{align*} \mathrm{div}\, m=0\text{ in }\mathcal D'(\Omega),\quad |m|=1\text{ a.e. in }\Omega\,, \end{align*} whose entropy productions $\mathrm{div}\,\Phi(m)$ are Radon measures in $\Omega$, controlled by the energy. Here, the entropies are all $C^2$ vector fields $\Phi\colon\mathbb S^1\to\mathbb R^2$ such that $\mathrm{div}\,\Phi(m_*)=0$ for any smooth solution $m_*$. It is conjectured that the entropy production measures are concentrated on the one-dimensional jump set of $m$, as follows from the chain rule if $m$ has bounded variation. In particular, the entropy production measures should vanish if they coincide with $L^p$ functions: this is what we establish in this note if $p$ is not too small and under natural boundary conditions.

This paper develops boundary feedback controls to stabilize traffic congestion toward a predefined shock equilibrium in the Aw-Rascle-Zhang (ARZ) traffic flow model. We transform the corresponding moving-boundary $2\times2$ hyperbolic system, covering free and congested flow regimes, respectively, into a shock-free $4\times4$ augmented system on a fixed domain via shock-location-based moving coordinates. By applying the modified Lyapunov functionals concerning shock perturbation, we show that the shock position and the state of the system in $H^2$-norm can be stabilized with an arbitrary exponential decay rate via the given feedback controls. Finally, the stabilization results are demonstrated by numerical simulations.

In this paper we consider the Schrödinger operator $\mathcal L_V= -\Delta + V$ in $\mathbb R^d$ with a non negative potential $V$, and $V\not\equiv 0$. We define the logarithmic Schrödinger operator $\log \mathcal L_V$ proving its main properties. We obtain a pointwise representation of $\log \mathcal L_V$ when $V$ satisfies a reverse Hölder inequality of exponent $q> \frac{d}{2}$ by using the semigroup of operators $\{T_t^V\}_{t>0}$ generated by $\mathcal L_V$. We consider the Lipschitz function space adapted to the Schrödinger setting to solve the initial value problem \[ \begin{cases} \frac{\partial u}{\partial t}=-(\log \mathcal{L}_V)u, & \text{in } \mathbb{R}^n \times (0,\infty), \\ u(x,0)=f(x), & x \in \mathbb{R}^d \end{cases} \] in terms of the fractional integral associated with $\mathcal L_V$.