Papers

We consider the Cauchy problem of the viscous $\beta$-plane equations. We first establish the global well-posedness of the system for the initial data sufficiently small compared to the Rossby parameter. The smoothing effect of the flow is then proved, and it is shown that the obtained global solution satisfies the equation in the classical sense. We also reveal that under some additional assumptions, the solution decays as fast as the corresponding linear solution and asymptotically behaves like a constant multiple of the integral kernel of the linearized equation. In particular, the decay rate of the solution is faster than expected from the flow of the heat equation.

In this paper, we consider the existence of mean curvature type hypersurfaces with prescribed gradient image. Let $\Omega$ and $\tilde{\Omega}$ be uniformly convex bounded domains in $\mathbb{R}^n$ with smooth boundary. We show that there exists unique convex solutions for the second boundary value problem of mean curvature type equations.

This article is concerned with the second boundary value problem of the Lagrangian mean curvature type equation arising from special Lagrangian geometry. By the parabolic method, we consider a fully nonlinear parabolic equation with oblique derivative boundary condition, and show the long time existence and convergence of the flow. It follows that the existence and uniqueness of the smooth uniformly convex solution are obtained, which generalizes the Brendle–Warren's theorem about minimal Lagrangian diffeomorphism in Euclidean metric space.

We classify the symmetry and multiplicity of the second Neumann eigenvalue on isosceles trapezoids and kites as the height varies. Precisely, away from a unique transition height, the second Neumann eigenvalue is simple, and its eigenfunction is either symmetric or antisymmetric. At a transition height, it has multiplicity two, with one symmetric and one antisymmetric basis eigenfunction. We prove that every nonzero real second Neumann eigenfunction has no interior critical points, including arbitrary linear combinations at a transition height, thereby verifying the hot spots conjecture for both classes of domains. In the simple-eigenvalue case, we also classify the nonvertex boundary critical points for trapezoids and kites. Reflection reduces the spectral classification to a comparison of a Neumann branch and a mixed Dirichlet–Neumann branch. Energy estimates establish strict transversality at every crossing and hence uniqueness wherever a transition occurs. To treat the two-dimensional eigenspace, we use Jacobian estimates and directional and radial comparison arguments to exclude cancellation of the two basis gradients.

In this paper, we prove Anderson localization for a hierarchical Anderson-Bernoulli model on lattice with arbitrary dimension, where the potential is characterized by a geometric hierarchical structure combined with fluctuations induced by independent and identically distributed (i.i.d.) Bernoulli random variables. Our method is also applicable to proving a probabilistic unique continuation result on $\mathbb{Z}^d$.

These are lecture notes from the Austral Winter School on Microlocal Analysis and Non-elliptic Fredholm Theory, held at the Australian National University, Canberra, June 30 – July 11, 2025.

In this paper, we establish nonlinear Landau damping and asymptotic stability of a large class of translation-invariant steady solutions to the time-dependent Hartree–Fock equations in the presence of an off-diagonal exchange operator, which arises naturally in the meanfield theory of a large fermionic system, in the whole space $\mathbb{R}^d$, $d\ge 3$. Despite being a sub-order operator, the inclusion of the exchange term disturbs the classical Schrödinger dispersion and causes a complex linear response from the background electrons to the space density whose dispersion relation is no longer a Fourier multiplier as in the classical Vlasov and Hartree theory. In addition, the group velocity of each elementary waves involves a mixture of all other Fourier modes, leading to delicate momentum-dependent echo resonances. To overcome the issues, we develop a nonlinear iterative scheme that relies on a detailed resolvent analysis, makes use of a transport type dispersion in Fourier spaces, and propagates phase mixing and Landau damping in weighted $L^\infty_{k,p}$ norms.

This paper is concerned with a fluid-particle system given by the incompressible Navier-Stokes equations coupled with the Vlasov(-Fokker-Planck) equation through a drag force. Such a model arises naturally in the study of aerosols, sprays, and more generally two-phase flows. In dimensions $d\in \{2,3\}$, we establish a rate of convergence for a system of $N$ interacting stochastic particles coupled with a fluid, towards the Vlasov(-Fokker-Planck)-Navier-Stokes system, as $N\to \infty$. The case of particles with a noise that vanishes as $N\to \infty$ is considered and leads specifically to the Vlasov-Navier-Stokes system. More precisely, we prove that the empirical measure associated with the particle system converges to the Vlasov(-Fokker-Planck) component, while the fluid velocity converges to the Navier-Stokes component of the coupled system. The proofs combine stochastic calculus and PDE techniques to establish energy estimates and commutator estimates for both the discrete and continuous systems.

Within many bacterial colonies, persister cells exist as a subpopulation that is tolerant to antibiotics and other stressors, yet not genetically distinct from the rest of the colony. A recent study has proposed epigenetic inheritance as a mechanism that leads to the presence of persister cells. We analyze a nonlocal PDE–ODE model introduced in that study to describe the epigenetic inheritance process and establish its mathematical well-posedness, including existence, uniqueness, and nonnegativity of solutions. We identify a sharp parameter threshold delineating extinction from persistence of the colony: below this threshold the washout equilibrium is globally asymptotically stable, while above it a unique positive equilibrium exists and the population is weakly persistent. Notably, this threshold is independent of the internal community structure.

We provide a simple and direct proof of a strong-type unique continuation principle for the fractional $p$-Laplacian $(-\Delta_p)^s$ for a range of $s$ and $p$. The result extends to strong solutions of the fractional nonlinear Schrödinger equation. We adapt the recent proofs of the weak UCP by Berger, Schilling and Prasad.

Adversarial training of binary classifiers can be reformulated as regularized risk minimization involving a nonlocal total variation. Building on this perspective, we establish a characterization of the subdifferential of this total variation using duality techniques. To achieve this, we derive a dual representation of the nonlocal total variation and a related integration of parts formula, involving a nonlocal gradient and divergence. We provide such duality statements both in the space of continuous functions vanishing at infinity on proper metric spaces and for the space of essentially bounded functions on Euclidean domains. Furthermore, under some additional conditions we provide characterizations of the subdifferential in these settings.

We present a rigorous spectral analysis of plasmonic resonances in the nonlocal regime of spatially dispersive media. We adopt the quasi-static approximation of the hydrodynamic Drude model, which provides an analytically tractable setting to account for nonlocal effects. By reformulating the governing equations as a boundary integral system, we obtain an analytic Fredholm operator pencil that characterizes resonant behaviour. This framework enables the study of nonlocal plasmonic eigenvalues in general bounded Lipschitz domains, together with a corresponding resonant expansion of the scattered field. Our main result reveals a fundamental change in spectral structure: in contrast to the local theory – which exhibits infinitely many surface plasmon modes and field singularities in domains with corners – the nonlocal model admits a discrete real spectrum with a single accumulation point at a strictly positive value. Consequently, only finitely many surface plasmon modes exist, and the singular behaviour associated with sharp geometries is regularized. As such, our results show that nonlocality does not merely shift plasmonic resonances but completely reshapes the spectral landscape in a way that provides a mathematically transparent explanation for several nonlocal phenomena previously observed in numerical studies.

In this article, we study certain transcendental function spaces arising in potential theory within the framework of Orlicz spaces. Specifically, we generalize Bessel and Lizorkin-Triebel spaces to the nonstandard setting of Orlicz spaces. We recover classical results from potential theory, such as the fact that Bessel-Orlicz spaces of integer order coincide with Orlicz-Sobolev spaces (Calderón type theorem), and we establish inclusion results for fractional orders. Moreover, we prove a Strauss-type lemma for potential spaces. In the last sections, we show that certain Orlicz-Lizorkin-Triebel spaces coincide with Bessel-Orlicz spaces, and we provide a useful atomic decomposition for these spaces.

We study the spectral properties of minimizers of the Müller functional for atoms and molecules with $N$ electrons and total nuclear charge $Z$. We prove that under some suitable assumptions on $Z$ and $N$, the $k$-th eigenvalue of a Müller minimizer $\gamma_*$ behaves as $A_* k^{-8/3}$ when $k\to \infty$, with a constant $A_*>0$ determined explicitly by the density of $\gamma_*$. In particular, in the atomic case $V=Z|x|^{-1}$ our assumption holds if $Z$ is sufficiently large and $N\le Z- C_0 Z^{1/3}$. While our proof is inspired by Sobolev's work on the asymptotic behavior of the one-particle density matrix of Schrödinger ground states, the analysis in Müller theory requires several new ingredients concerning both the singular behavior of the integral kernel of the minimizers near the diagonal and the decay properties at infinity.

This paper is concerned with the global existence and uniqueness of classical solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients in three-dimensional bounded domains or in the whole space $\mathbb{R}^N$ $(N=2,3)$ with non-vacuum far-field density. Specifically, we assume that the shear viscosity coefficient $\mu(\rho)=\rho^\alpha$ and the bulk viscosity coefficient $\lambda(\rho)=(\alpha-1)\rho^\alpha$, which satisfy the BD entropy relation. For arbitrarily large spherically symmetric initial data, we establish the global existence and uniqueness of spherically symmetric classical solutions under the following conditions: for $N=2$, $\alpha \in (0.54369,1)$ and $\gamma \in (1,\infty)$; for $N=3$ (both bounded domains and the whole space), $\alpha \in (0.67661,1)$ and $\gamma \in (1,6\alpha-3)$. In the two-dimensional case $\mathbb{R}^2$, the restriction on $\alpha$ can be further relaxed to $\alpha \in (9-6\sqrt{2},1)$ provided that the initial data satisfy additional weighted integrability conditions. Moreover, we show that the solution will not exhibit vacuum in any finite time provided that no vacuum is present initially.

We introduce an extended exterior $(K,K^{\prime},\alpha_0)$–quasiconformal mapping method to study the asymptotic behavior at infinity of solutions to the supercritical phase Lagrangian mean curvature equation \[ \sum_{i=1}^{n} \arctan \lambda_i(D^2u) = \theta + f(x) \] on exterior domains in $\mathbb{R}^n$, where the constant $|\theta|\in((n-2)\pi/2,n\pi/2)$, $n\geq 2$, and $f=O(|x|^{-\beta})$ is a perturbation term with the sharp decay condition $\beta>2$ at infinity. Our work generalizes the classical exterior Bernstein-type theorem for the special Lagrangian equation ($f\equiv0$) established by Li–Li–Yuan [Adv. Math. (2020)]. Via Perron's method, we solve the corresponding Dirichlet problem outside a bounded, uniformly convex domain, prescribing asymptotic behavior at infinity. For $n \geq 3$, we establish existence and uniqueness of viscosity solutions in both the supercritical phase case with $f \not\equiv 0$ and the subcritical phase case with $f \equiv 0$. This extends earlier work by Li [Trans. Amer. Math. Soc. (2019)] on the exterior Dirichlet problem for the special Lagrangian equation ($f \equiv 0$) under weaker regularity assumptions on the interior boundary and boundary data.

The multi-scale and non-linear nature of phase-field models of solidification requires fine spatial and temporal discretization, leading to long computation times. This could be overcome with artificial-intelligence approaches. Surrogate models based on neural operators could have a lower computational cost than conventional numerical discretization methods. We propose a new neural operator approach that bridges classical convex-concave splitting schemes with physics-informed learning to accelerate the simulation of phase-field models. It consists of a Deep Ritz method, where a neural operator is trained to approximate a variational formulation of the phase-field model. By training the neural operator with an energy-splitting variational formulation, we enforce the energy dissipation property of the underlying models. We further introduce a custom Reaction-Diffusion Neural Operator (RDNO) architecture, adapted to the operators of the model equations. We successfully apply the deep learning approach to the isotropic Allen-Cahn equation and to anisotropic dendritic growth simulation. We demonstrate that our physically-informed training provides better generalization in out-of-distribution evaluations than data-driven training, while achieving faster inference than traditional Fourier spectral methods.

We prove existence of weak solutions and weak-strong uniqueness for a mathematical model which couples the evolution of a phase-parameter $\varphi$ satisfying a Cahn-Hilliard type relation with the one of an additional variable $\sigma$ influencing the phase separation process. The main application of the model refers to cancer growth processes, where $\sigma$ may represent the concentration of a chemical substance affecting the evolution of the tumor, and is governed by a nonlinear parabolic equation characterized by a cross-diffusion term alike that occurring in the Keller-Segel model for chemotaxis. This term is also responsible for the most relevant difficulties in the mathematical analysis of the system. Complementing previous results on the model, we prove here global in time existence for a very weak notion of solution to which a suitable energy imbalance and a logarithmic inequality for the nutrient are added. Noting that the system also admits local in time "strong" solutions, we can also exhibit a weak-strong uniqueness result whose proof exploits in an essential way the entropy-type inequality satisfied by weak solutions.

This work analyzes a predator-prey cross-diffusion system coupled with two chemical substances under homogeneous Neumann boundary conditions in a bounded domain Omega subset of R^n (n >= 2) with smooth boundary dOmega. Under appropriate conditions on the model parameters, the global existence of classical solutions is established. Furthermore, by constructing a suitable Lyapunov functional, the asymptotic stability of the spatially homogeneous steady state is proved. The emergence of spatial patterns induced by diffusion-driven instability is also investigated. Owing to the complexity of the resulting four-equation system, the criteria for Turing bifurcation are derived numerically rather than analytically. Numerical simulations are performed to generate Turing bifurcation diagrams, illustrating the dynamical responses of the system to variations in the predation rate. These results provide new insights into the role of predation intensity in the formation of spatial patterns in predator-prey systems mediated by two chemical substances.

This paper studies critical fractional Sobolev inequalities with lower-order terms on the standard CR sphere $\mathbb S^{2n+1}$. Let $Q=2n+2$, let $s\in(0,1)$, let $1<p<Q$, and let $p_s^*=\frac{Qp}{Q-sp}$. For the inequality $\|u\|_{L^{p_s^*}(\mathbb S^{2n+1})}\le A[u]_{s,p}+B\|u\|_{L^p(\mathbb S^{2n+1})}$, we prove that the admissible lower-order coefficients are exactly $\left[|\mathbb S^{2n+1}|^{-s/Q},\infty\right)$. For the power-type inequality $\|u\|_{L^{p_s^*}(\mathbb S^{2n+1})}^p\le A[u]_{s,p}^p+B\|u\|_{L^p(\mathbb S^{2n+1})}^p$, we show that the admissible set is $\left[|\mathbb S^{2n+1}|^{-sp/Q},\infty\right)$ when $1<p\le 2$, and $\left(|\mathbb S^{2n+1}|^{-sp/Q},\infty\right)$ when $2<p<Q$. Via the Cayley transform, we derive the exact weighted counterpart on the Heisenberg group and prove that the corresponding admissible sets coincide with those on the sphere. We also show that nonlinear first-moment constraints do not improve the optimal lower-order coefficient, whereas finite-codimensional linear constraints excluding nonzero constants yield coercive inequalities.

For the Newtonian \(N\)-body problem at nonnegative energy, we study solution sets selected by the Jacobi–Maupertuis variational principle and by the associated stationary Hamilton–Jacobi equation. We prove a compactness/stability theorem for classical initial data generating geodesic rays: limits in the ambient phase space remain collision-free, generate geodesic rays, and carry locally relatively compact normalized Busemann functions. The limiting horofunction of normalized Busemann functions yields a viscosity solution of the limiting stationary equation. For a fixed collision-free hyperbolic limit shape \(a\), we also prove closedness of the corresponding slice of geodesic-ray data. Finally, after passing to the reduced configuration space \(X\), we show that this fixed-shape slice is countably \(d(N-1)\)-rectifiable in phase space and has Hausdorff dimension exactly \(d(N-1)\). Thus the paper combines phase-space compactness of calibrated minimizing motions with a geometric-measure description of a fixed-shape Hamilton–Jacobi calibrated slice.

We study nonlinear stationary Kolmogorov equations with degenerate diffusion matrices and discontinuous coefficients. The existence of a solution is proved. We propose a new approach based on an integral condition with Lyapunov functions and a regularity of projections of solutions in the partially degenerate case. Examples are given to illustrate the results.

We study resonances for the semiclassical magnetic Laplacian in the full plane with a compactly supported magnetic field in the framework of semiclassical complex scaling and black box scattering theory. Assuming that the magnetic field is locally constant, we prove the existence of semiclassical resonances near the Landau levels with exponentially small imaginary parts. We also prove that resonances emerge from a magnetic step discontinuity along a curved interface or a non-degenerate magnetic well, and in the vicinity of anharmonic Landau levels if the field has an isolated zero.

For the 3D cubic quasilinear wave system $\square_{c_i} u^i=G^i(u,\partial u,\partial^2u)=\displaystyle\sum_{\substack{0\le|\alpha|,|\beta|,|\gamma|\le1 \\ 1\le j,k,l \le m}}g_{\alpha\beta\gamma}^{ijkl}\partial^{\alpha}u^j\partial^{\beta}u^k\partial^{\gamma}u^l$, it is well known that global solution $u$ exists when the small smooth initial data $(u,\partial_tu)|_{t=0}$ $=(u_0(x), u_1(x))$ are compactly supported or decay rapidly at spatial infinity. However, when $(u_0, u_1)\in (H^{s+1}, H^s)$ with $s>\frac{5}{2}$ are small, it remains unknown whether $u$ exists globally or not. In this paper, we show that if $\|u_{0}\|_{H^{N+1}}+\|u_{1}\|_{H^N}\le\varepsilon$ ($N\ge 6$) is small, then the almost global solution $u$ exists in $[0, T_{\varepsilon}]$ with $T_{\varepsilon}\ge e^{C\varepsilon^{-1}}$ for the general $G(u,\partial u,\partial^2u)$ depending on $u$ and $T_{\varepsilon}\ge e^{C\varepsilon^{-2}}$ for the nonlinearity $G(\partial u,\partial^2u)$ independent of $u$, respectively. In addition, if $\displaystyle\sum_{|a|\le 5}\|\langle x\rangle^{\mu}\partial^a_x(u _0,u_1)\|_{L^2}\le\varepsilon$ holds for any fixed constant $\mu\in (0,1)$, then the solution $u$ exists globally and meanwhile the scattering property of $u$ is derived. Our main ingredients consist in establishing a series of new weighted $L^\infty-L^2$ estimates and Strichartz estimates based on the strong Huygens' principle for 3D linear wave equations.

The special unitary group SU(2) plays a fundamental role in the description of symmetries in quantum mechanics, theoretical physics, and spherical signal processing. In this paper, we address the computational challenges of performing spectral analysis on this non-abelian compact Lie group. We present the Fourier Transform (FT) on SU(2) and develop a Fast Fourier Transform (FFT) algorithm inspired by the classical Cooley-Tukey divide-and-conquer scheme. Our approach efficiently discretizes the group using Euler angles, applying a two-dimensional FFT on the angular variables and exploiting the recursive properties of Jacobi polynomials. We provide an analysis of the computational complexity, demonstrating that our FFT-based method significantly outperforms the direct computation of the FT. This algorithm serves as a foundational tool for understanding the implementation of the FFT on SU(2), a key component in numerical simulations and advanced data analysis for high-performance computing applications on curved manifolds and quantum systems.

We consider discounted infinite-horizon potential mean-field games (MFGs) on the $d$-dimensional torus. Without imposing monotonicity assumptions, we prove that every weak limit point of a time-dependent equilibrium, as time tends to infinity, is a stationary equilibrium. As a consequence, equilibria converge whenever the stationary solution is unique. The short proof is based on a novel Lyapunov functional for the time-dependent MFG system. We also provide a new uniqueness criterion for stationary equilibria. Finally, we apply our results to the subcritical Kuramoto MFG studied by Carmona, Cormier, and Soner, showing that every equilibrium converges to the incoherent solution.

We present a novel numerical framework for studying nonlinear dispersive equations in higher-dimensional settings, specifically designed for solutions featuring traveling waves along a preferred axis (or field-aligned traveling waves). Using the three-dimensional generalized Zakharov-Kuznetsov (gZK) equation as a model, we convert it into cylindrical coordinates and implement a domain decomposition strategy. By partitioning the computational domain into distinct regions based on expected solution behavior, we significantly reduce computational complexity while maintaining the high resolution necessary for capturing small-scale dynamics. Another key innovation of our method is the ability to efficiently handle fractional nonlinearities, specifically, the critical power $p = 7/3$ in 3D, which typically introduces significant computational overhead and numerical instabilities that compromise simulation accuracy. Using this framework, we are able to investigate the dynamics of solutions (with cylindrical symmetry) close to the ground state soliton and show that for the 3D critical ZK equation, the ground state serves as the sharp threshold for global vs. finite time existence of solutions. Our method successfully tracks the profiles of these singular solutions, providing new insights into the dynamics of wave collapse in three-dimensional magnetized media.

For dimensions $n\geq8$, we are concerned with the quotient functional of the biharmonic Brézis-Nirenberg problem under the Navier boundary condition $$ S(\varepsilon V):=\inf_{0\not\equiv u\in H^2(\Omega)\cap H_0^1(\Omega)}\frac{\int_{\Omega}|\Delta u|^2dx+\varepsilon\int_{\Omega}V|u|^2dx}{\big(\int_{\Omega}|u|^{2^\star}dx\big)^{2/2^\star}}, $$ where $2^\star=\frac{2n}{n-4}$ is the critical Sobolev exponent of the embedding $H^2(\Omega)\cap H_0^1(\Omega)\hookrightarrow L^{2^\star}(\Omega)$, $\Omega\subset\mathbb{R}^n$ is a bounded open set and $V:\overline{\Omega}\rightarrow\mathbb{R}$ is a continuous function. Under certain assumptions on $V$, we establish sharp asymptotics for the energy difference $S(0)-S(\varepsilon V)$, as $\varepsilon\rightarrow0^+$, by means of matching upper and lower bound estimates. Moreover, we give a precise description of the blow-up profile of (almost) minimizing sequences and characterize the blow-up rate and the location of concentration points.

We study the stability of the extended Morse index, defined as the number of negative and zero eigenvalues of the Jacobi operator, for sequences of harmonic maps on degenerating Riemann surfaces. As the conformal structure approaches the boundary of moduli space, collar collapse creates major analytical challenges. We analyze the second variation of the energy under these degenerations and identify conditions ensuring upper semicontinuity of the extended Morse index. Refining earlier results of the first and second authors in [7], we obtain sharper control of the spectrum of the Jacobi operator on degenerating domains. A key new aspect is the explicit contribution of geodesics arising as limits of the images of degenerating collars. We show that these neck regions converge to geodesic segments whose Morse index contributes nontrivially to the limiting extended index.

We study ground states of the fermionic nonlinear Schrödinger system $J_2(p)$ in $\R$, where $p>1$ denotes a polynomial exponent of the nonlinear term. It is known that the system $J_2(p)$ admits ground states for any $1<p<2$, while there is no ground state for $J_2(2)$. We prove that there is no ground state of $J_2(p)$ as $p\searrow 2$, which addresses the special case of Conjecture 5 in [D. Gontier, M. Lewin and F. Q. Nazar, ARMA, 2021]. The refined limiting profile of ground states for $J_2(p)$ is also analyzed as $p\nearrow 2$, which shows that the corresponding density admits exactly two bumps whose distance goes up to infinity as $p\nearrow 2$.