Papers

We construct a nonconstant weakly harmonic map from a three-dimensional ball into the two-sphere with constant trace. This gives an affirmative answer to a question raised by Rivière in 1995.

We investigate the compressible Navier-Stokes equations with heat conduction subject to inflow/outflow boundary conditions, establishing the existence of a strong solution. The primary contribution of this work lies in the derivation of a new regularity criterion for three-dimensional compressible flows, which serves as an analogue to the Beale-Kato-Majda criterion for incompressible fluids. Notably, our findings refine both the strong solution results established by Valli and Zajaczkowski and the conditional regularity framework proposed by Basaric, Feireisl, and Mizerova. Our analysis extends the framework of Basaric, Feireisl, and Mizerova, allowing us to successfully derive the required uniform estimates.

Let $X$ be a compact Kähler manifold and $(\alpha,\beta)$ be a pair of Kähler classes. We show that when $(\alpha,\beta)$ is $J$-semistable, the collection of positive-dimensional irreducible $J$-null subvarieties is finite. We also obtain a uniform positive lower bound for the normalized $J$-slope of every irreducible non-$J$-null subvariety. This confirms the finiteness and uniform-gap expectation of Khalid-Sjöström Dyrefelt \cite[Section 1.2, Theorem 1.5; Remark 1.8]{KSD26}. As a corollary, we show that the collection of optimal destabilizers is finite when the stability threshold is nonpositive.

In this paper, we prove the $C^{1,1}$ regularity of Green functions associated with the complex $k$-Hessian operator for $1\leq k<n$, and give a new proof of the corresponding regularity of pluricomplex Green functions. We establish real Hessian estimates for approximating problems associated with homogeneous complex Hessian equations on punctured domains. The main idea is to introduce new auxiliary functions generated by complex linear vector fields to reduce the global real Hessian estimate to the boundary. We also treat the complex Monge-Ampère case. Similar real Hessian estimates also works for the exterior problem.

We prove uniqueness of solutions to the Navier-Stokes equations in the space \[ L^p\bigl([0,T],\widetilde{L}^{3,\infty}(\mathbb{R}^3)\bigr) \cap C\bigl([0,T],B^{-1,\infty}_{\infty}(\mathbb{R}^3)\bigr), \qquad p>2, \] where $\widetilde{L}^{3,\infty}(\mathbb{R}^3)$ denotes the closure of the Schwartz space $\mathcal{S}(\mathbb{R}^3)$ in the Lorentz space $L^{3,\infty}(\mathbb{R}^3)$. Moreover, we prove that any solution $u$ in this class, if it exists, is smooth for positive time, namely, $u\in C^\infty((0,T]\times \mathbb{R}^3)$. In addition, we establish precise estimates for the behavior of $u(t)$ as $t\to0^+ $ in various Banach spaces.

We study a free boundary problem for a diffusive Lotka–Volterra competition system describing the invasion of a new species into the habitat of a native competitor, where habitat suitability shifts from unfavourable to favourable at a constant speed $c>0$ due to climate change. Only the invader is affected by the shifting environment and only its range is governed by a Stefan-type free boundary, while the native species occupies the whole half line. We work throughout in the weak competition regime, in which the two species may coexist. We prove a spreading–vanishing dichotomy: either the invader spreads and the pair converges to the coexistence steady state $(u^*,v^*)$, or the invader vanishes and the native species recovers its carrying capacity. In the vanishing case we obtain the explicit bound $\lim_{t\to\infty}h(t)\le\frac{\pi}{2}\sqrt{d_1c_2/(a_1c_2-a_2c_1)}$, and we give criteria guaranteeing each alternative. When spreading occurs, we determine the exact asymptotic spreading speed: $\lim_{t\to\infty}h(t)/t=\min\{c,c_0\}$, where $c_0$ is the spreading speed of the corresponding homogeneous weak competition system. In particular the invasion is slowed down both by the competitor and by the climate shift, and the slower of the two mechanisms is the one that determines the speed. Numerical simulations illustrate the results.

We consider the Nernst–Planck–Darcy system describing the electrodiffusion of $N$ ionic species with arbitrary valences and different diffusivities, transported by a Darcy flow driven by the electric force, in the whole space $\mathbb{R}^d$, $d \ge 3$. We prove the existence of global solutions for initial ionic concentrations that are small in the critical Lebesgue space $L^{d/2}(\mathbb{R}^d)$. Moreover, we establish uniqueness for initial data in $L^{d+\epsilon}(\mathbb{R}^d)$ when $3 \le d \le 4$ and in $L^{2d-1}(\mathbb{R}^d)$ when $d \ge 5$, improving on results in the literature, which require Sobolev regularity of the data. The proofs rely on energy estimates exploiting cancellations of the highest-order nonlinear terms, and on the resulting dissipative structures, which yield an instantaneous gain of regularity used to control the doubly nonlinear velocity term in the uniqueness argument.

We establish a quantitative Gidas–Ni–Nirenberg theorem for the pure fractional Dirichlet problem \[ (-\Delta)^s u=k(x)g(u)\quad\text{in }B_1, \qquad u=0\quad\text{in }\mathbb R^n\setminus B_1, \qquad 0<s<1. \] For positive bounded solutions under a two-sided $L^\infty$ normalization, we prove \[ \mathcal D(u)\le C\,\mathcal D(k)^\gamma,\qquad 0<\gamma<1, \] where $\mathcal D$ measures spherical oscillation and outward radial increase. To the best of our knowledge, this provides the first quantitative Gidas–Ni–Nirenberg estimate for the pure fractional Dirichlet problem. The key new ingredient is a shift-uniform, weak Harnack inequality for shifted antisymmetric supersolutions, which overcomes the exterior-tail obstruction in the quantitative moving-plane argument. This mechanism extends to nonseparable nonlinearities.

We establish an exponential turnpike property for optimal boundary control of multi-dimensional first-order linear hyperbolic systems in which both the control and observation operators are unbounded and the cost functions may be nonquadratic. The main assumptions are internal and boundary dissipation conditions. These conditions ensure that the associated input-to-state and input-to-output maps are bounded, with operator norms independent of the time horizon T. This uniform boundedness is essential to the proof, which combines weighted energy estimates for the forward and adjoint equations, input-output duality, and strong convexity. Finally, we apply the result to the linearized two-dimensional Saint-Venant equations with sluice-gate control and illustrate the turnpike behavior numerically.

This monograph develops an introduction to global analysis centered on the interaction between differential geometry, functional analysis, partial differential equations, and variational methods on Riemannian manifolds. Beginning with smooth and Riemannian geometry, it develops Sobolev spaces, distributions, interpolation and fractional regularity, differential and pseudodifferential operators on vector bundles, elliptic theory, heat methods, bounded geometry, and trace theorems. It then treats Fredholm and index theory, culminating in the Atiyah–Singer index theorem, followed by geometric evolution equations and Ricci flow, infinite-dimensional geometry on Banach and Hilbert manifolds, and variational methods including the direct method, Palais–Smale theory, deformation arguments, the mountain pass theorem, and the Nehari method. Particular emphasis is placed on explicit proofs, the passage from local Euclidean estimates to intrinsic global statements, and the precise geometric hypotheses required in compact, noncompact, and boundary settings. The text is intended for advanced undergraduate and graduate students, as well as readers approaching global analysis from geometry or differential equations.

In this paper we consider a free boundary problem for the spreading of an invasive species in the habitat of a native competitor. We assume that the invasive species benefits from a climate shift that turns the environment from unfavorable to favorable at a constant speed $c$. We show that if the invasive species is an inferior one, then it must vanish, while the native competitor always persists. However, if the invasive species is a superior one, then a dichotomy occurs: either the invasive species vanishes and the native one persists, or the invasive one spreads and the native one vanishes. We also show that in the latter case the asymptotic spreading speed equals $\min\{c, c_0\}$, where $c_0$ is the spreading speed in the corresponding homogeneous environment. Numerical simulations are provided to illustrate our theoretical results.

We consider ground states of a two-component logarithmic fermionic system in $\mathbb{R}^d$, where $d\ge 1$ is arbitrary. We prove that up to translations and scalings, ground states of the logarithmic system are the $L^\infty $-limits of ground states for a two-component $2p-1$ power-law fermionic system as $p\searrow 1$. As a byproduct, we also establish a sharp logarithmic Sobolev inequality for orthonormal functions in \(\mathbb{R}^d\), whose optimizers are, up to scalings, the minimizers of a constraint variational problem associated with the logarithmic system.

We study the conformal parametrisation of smooth star-shaped planar domains and its dependence on perturbations of the boundary. The boundary correspondence of the normalized Riemann map is described by the Theodorsen equation, which we formulate in weighted Wiener algebras. We derive an explicit representation of the inverse of the linearized Theodorsen operator through a canonical Wiener–Hopf factorisation and obtain quantitative a posteriori estimates that are stable under perturbations of the boundary function. For a collection of nonperturbative reference domains, rigorous computer-assisted estimates yield explicit neighborhoods in a weighted Wiener algebra such that every boundary function in these neighborhoods admits a uniquely determined conformal parametrisation in the validated ball, with quantitative control of the corresponding Riemann map. Thus the validation applies to open, infinite-dimensional families of star-shaped domains. As an application, we consider the semilinear Dirichlet problem \[ -\Delta v=v^3\quad\text{in }\Omega, \qquad v=0\quad\text{on }\partial\Omega. \] Pulling the equation back by the validated conformal maps yields a problem on the unit disk. Uniform a posteriori estimates then imply persistence of the elliptic solutions throughout explicit neighborhoods of the reference domains.

We prove linear ill-posedness in tangential Sobolev spaces for the three-dimensional resistive MHD boundary layer equations. We assume that a suitable linear combination of the initial tangential velocity components has a non-degenerate critical point and that both initial tangential magnetic components have a double zero at the same point. Compared with the corresponding two-dimensional MHD result, our construction requires only this double-zero condition, rather than a higher-order degeneracy of the magnetic shear. The proof combines the three-dimensional Prandtl critical layer construction with a vector correction to the two-dimensional magnetic quotient. This correction cancels the leading terms in both induction equations while preserving the magnetic divergence constraint. The resulting approximate solutions grow like $\exp(\sigma t/\sqrt{\varepsilon})$, and their residuals have a prefactor of order $O(\varepsilon^{-3/8})$, up to logarithmic factors. A Duhamel argument then yields linear ill-posedness for every tangential derivative loss $\mu<\frac18$. Thus the three-dimensional Prandtl instability persists when the stabilizing tangential magnetic field degenerates.

We show that the fractional perimeter of any order $0<s<1$ is maximal, among all planar sets of fixed perimeter, for the disk.

We start from a simple problem in geometric optics. A particle moves between two homothetic ellipses, with refraction at the inner interface and reflection at the outer one. In a suitable nearly circular regime, the small geometric anisotropy of the ellipses is amplified by the strong refraction, and the corresponding return map develops a transverse heteroclinic connection between the two axial motions. We turn this geometric mechanism into a construction for a two-mode Kirchhoff system while keeping the standard quadratic elastic variable throughout. The singular optical model is first reduced to an explicit kick–drift map, for which the heteroclinic connection is obtained by a contraction argument. The same argument also gives exponential convergence along the two tails and transversality. We then show that the connection persists for the genuine elliptic billiard and, subsequently, through a smooth regularization of the reflecting and refracting interfaces. Finally, a small positive background is added to the Kirchhoff coefficient, making it uniformly positive without destroying the hyperbolicity of the axial modes or the transverse connection. This produces a smooth and uniformly positive Kirchhoff coefficient admitting a transverse heteroclinic orbit between two simple modes. Combined with the road-map theorem developed in our previous work, the construction yields a finite-time blow-up example for a forced abstract Kirchhoff equation with a regular forcing term.

This paper is concerned with the parabolic-elliptic chemotaxis system involving weakly singular sensitivity and a nonlocal source: $u_t=\Delta u-\chi\nabla\cdot\left(\frac{u}{v^k}\nabla v\right) +u^\alpha\left(r-\mu\int_\Omega u^\beta\,dx\right)$ and $0=\Delta v-v+u^\gamma$ under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega\subset\mathbb{R}^N\) with \(N\ge1\), where \(\chi,r,\mu,\gamma>0\), \(k\in(0,1)\) and \(\alpha,\beta\ge1\). In view of the enhanced aggregation induced by singular sensitivity and the suppression of excessive population growth by the nonlocal damping, we identify sufficient conditions under which solutions remain globally bounded. We show that classical solutions are globally bounded in the following cases: \par\smallskip \begin{center} \small \setlength{\tabcolsep}{5pt} \setlength{\arrayrulewidth}{0.3pt} \renewcommand{\arraystretch}{1.25} \begin{tabular}{c|c|c|c} \multirow{2}{*}{$\beta>1$} & \multirow{2}{*}{$\gamma=1$} & $N=1$ & $1\leq\alpha<1+2\beta$ \\ \cline{3-4} & & $N\geq2$ & $\begin{gathered} 1\leq\alpha<2,\quad \alpha+\beta>2+\tfrac{N}{2},\\[-0.5mm] \text{or}\quad 2\leq\alpha<1+\tfrac{2\beta}{N} \end{gathered}$ \\ \hline \multirow{2}{*}{$\beta=1$} & $\gamma=1$ & $N=1$ & \multirow{2}{*}{ $\begin{array}{l@{\quad}l@{\quad}l} \text{Case 1:} & 1\leq\alpha<1+\tfrac{2}{N} & \text{if}\quad m_0<\tfrac{r}{\mu},\\[4pt] \text{Case 2:} & \alpha\geq1 & \text{if}\quad m_0\geq\tfrac{r}{\mu} \end{array}$ } \\ \cline{2-3} & $0<\gamma<\tfrac{2}{N}$ & $N\geq2$ & \\ \end{tabular} \end{center} \par \smallskip \noindent Here \(m_0:=\int_\Omega u_0\) denotes the initial total mass. It is worth noting that both the uniform-in-time \(L^p\)-estimate and the \(L^\infty\)-bound obtained by Moser iteration are derived without first establishing a uniform positive lower bound for \(v\).

This paper investigates Anderson localization in subwavelength elastic periodic systems with random perturbations. For the unperturbed system, we use layer potential techniques to reformulate the eigenvalue problem as boundary integral equations, and derive asymptotic formulas for the subwavelength eigenvalues. For perturbed systems, we apply the Floquet transform to obtain a periodic formulation and derive equations determining the resonant frequencies under general perturbations. Numerical experiments for perturbed periodic monomers and dimers agree with the theoretical predictions. We further demonstrate Anderson localization by increasing the strength and number of random perturbations. These results provide a mathematical foundation for understanding subwavelength localization in elastic metamaterials.

In this note, we give an explicit construction of global solutions to the prescribed Jacobian equation \[ \det(\nabla u)=f\mbox{ in }\mathbb{R}^2, \] for a class of data. For every $f\in C_c^1(\mathbb{R}^2)$ and $p>1$, we construct a solution $u\in \dot W^{1,2p}(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$. In particular, no sign condition or integral constraint is imposed on the compactly supported data. For $f\in C_c^\infty(\mathbb{R}^2)$, the construction yields a smooth solution. We also consider rapidly decaying data and prove the existence of global solutions with bounded gradient for every $f\in\mathcal{S}(\mathbb{R}^2)$. Finally, we extend the construction to compactly supported data that are measurable in one variable and $C^1$ in the other. Our proof relies on a similar idea as in [Moser, Trans. Amer. Math. Soc. 1965].

We prove quantitative instability results for the light ray transform in the Minkowski setting, and locally in the smooth and Gevrey categories. In the smooth Sobolev setting, we show that no modulus of continuity can be better than $t^\alpha$ for any $\alpha \in(0,1)$, while in the Gevrey setting we obtain an explicit logarithmic lower bound. On the other hand, using the relation between stationary geometries and magnetic(-potential) systems, we obtain a stability result for "time moments" of the light ray transform.

We study spectral and nonlinear instability for some steady states of a two-dimensional inertial magnetohydrodynamic system on the torus. For equilibria with aligned velocity and magnetic fields, the linearized problem reduces to a matrix recurrence of Fourier coefficients, and a stable matrix continued fraction produces a smooth eigenmode. For a purely magnetic family, the square of the linearized operator admits a compact self-adjoint reduction on each transverse Fourier blocks that classifies the entire unstable spectrum. In an invariant symmetry class, we identify the unstable branches and a spectral gap below the leading eigenspace. The spectral gap then permits an approximate-trajectory construction that proves nonlinear instability on the logarithmic time scale. Our study is motivated by the physics literature, where the nonlinear instability of purely magnetic tearing modes is invoked to explain magnetic reconnection.

We consider the Cauchy problems for a class of nonlinear diffusion equations where each initial data is given by a characteristic function of an open set $\Omega$ in the whole Euclidean space. If the boundary $\partial\Omega$ is of class of $C^2$ in a neighborhood of a point $x$ on it, then we extract the mean curvature of $\partial\Omega$ at $x$ from the initial diffusion at $x$. This corresponds to the author's previous result for the two-phase linear diffusion equation. Applications to some overdetermined problems for nonlinear diffusion equations are given.

In this paper, we study the continuous dependence of solutions to the three-dimensional incompressible anisotropic Navier-Stokes equations $(ANS)$ with initial data in the critical space $\dot{B}^{0,\frac12}$. We prove that the data-to-solution map is continuous in this setting. Due to the critical regularity of the problem, the classical Bona-Smith argument \cite{BoSm} (see also the abstract result \cite{ABITZ}) does not appear to apply because a key ingredient (the Lipschitz continuity at lower regularity) does not appear to hold. As a consequence, we develop a new direct Fourier-weighted method within the Besov framework.

We derive computable upper bounds on the maximum wave speed in the Riemann problem for the compressible Euler equations with a composite equation of state, in which the pressure is the sum of a convex hydrodynamical part and a barotropic correction. The correction may destroy the convexity of the equation of state, so that the Riemann solution can contain composite shock-rarefaction waves. The bounds are obtained by comparison with the auxiliary Riemann problem for the hydrodynamical equation of state alone. The bounds hold irrespective of whether the outer waves are elementary or composite and require no case distinction on the wave type.

We prove two complementary Lewy-type theorems for elliptic equations whose coefficients depend only on the gradient. First, let $\Omega\subset\mathbb{R}^3$ and let $u$ solve \[ a^{ij}(Du)u_{ij}=0, \] where $a$ is a smooth, symmetric, positive definite matrix field on an open set containing $Du(\Omega)$. We show that if the gradient map $Du$ is a local homeomorphism, then $\det D^2u$ never vanishes; hence $Du$ is a local $C^\infty$-diffeomorphism. Second, in every dimension, we consider capacitary solutions on convex rings. If the solution has no critical points and its level hypersurfaces are strictly convex, then ellipticity alone forces the Hessian to have one positive and $n-1$ negative eigenvalues, that is, inertia $(1,n-1)$. Consequently, the gradient is a global diffeomorphism onto a radially parametrized ring in gradient space. Both results apply to the $p$-Laplace and minimal surface equations. For the global minimal-surface result, existence of a smooth solution is assumed. Classical convex-ring results supply the noncriticality and strict level-set convexity needed in the global corollaries. Since the two coefficient matrices are real analytic on the relevant gradient ranges, the corresponding local and global gradient diffeomorphisms are real analytic.

We study physics-informed neural networks (PINNs) for the Dirichlet boundary control of a semilinear parabolic equation with Tikhonov regularization. Two approaches are considered. A direct PINN parameterizes the state and control by separate networks and minimizes a penalized form of the tracking objective. An indirect PINN instead represents the state, adjoint, and control by unconstrained networks trained jointly to satisfy the first-order optimality system, with the state-control coupling and the homogeneous adjoint boundary and terminal conditions imposed as soft penalty terms rather than enforced architecturally. For the indirect formulation we develop an error estimation framework that decomposes the total error into approximation, optimization, quadrature, and soft boundary/terminal-constraint contributions. Under standing assumptions on optimal-solution regularity and compatibility, network approximability, uniform Hölder control of the soft-constraint residuals, and a local neighborhood of the reference optimality-system solution, we derive a quantitative linearized stability estimate and a conditional local nonlinear residual-to-error estimate, and construct a computable residual indicator with a conditional reliability bound. Numerical experiments on two manufactured test problems - a cubic reactiondiffusion equation and a linear equation with a nontrivial boundary control and adjoint - illustrate the behavior of the direct and indirect formulations.

We study the linear stability of compactly supported static spherical equilibria of the non-isentropic Euler–Poisson system. In the Schwarzschild-stable case, the linearized equations are realized as a separable Hamiltonian system on weighted spaces adapted to the physical vacuum. We prove stability against non-radial perturbations and show that the algebraic dimension of the radial unstable subspace is the Morse index of a density quadratic form subject only to the mass constraint. Thus the infinitely many linearized entropy constraints reduce, for radial perturbations, to one mass constraint and an explicit entropy reconstruction. We apply this criterion to two entropy prescriptions. No smallness condition is imposed on a fixed entropy–density relation: under the stated effective-pressure and branch hypotheses, a simple mass maximum is not a stability transition. By contrast, for a small entropy distribution fixed as a function of enclosed mass, the turning-point principle holds. In the Schwarzschild-unstable case, we construct the self-adjoint velocity operator from its quadratic form, prove a strictly negative spectral bottom and sharp exponential growth, and reconstruct energy solutions of the original first-order linearized system.

We prove the global well-posedness and nonlinear stability of classical solutions to the three-dimensional compressible viscous, non-resistive MHD system on $\mathbb{T}^3$ near the equilibrium $(1,\mathbf{0},\mathbf{e}_3)$, for small $x_3$-symmetric perturbations and without any Diophantine condition on the background magnetic field. The central obstruction is the $x_3$-independent sector, in which the density and the magnetic field possess no dissipation, no damping, and no decay. We overcome it by exhibiting a hidden wave structure for the exact total pressure $\mathcal{D}=P(1+a)+B_3+\frac12|\mathbf{B}|^2$, the averaged pair $(\overline{\mathbf{u}},\overline{\mathcal{D}})$ obeys a closed system, and $\overline{\mathcal{D}}$ satisfies a strongly damped wave equation whose principal wave part propagates at the fast magnetosonic speed and whose non-parabolic branch damps at a rate that involves no gain of derivatives. This hidden damping substitutes for the missing magnetic dissipation, and simultaneously absorbs the magnetic pressure $\frac12\nabla|\mathbf{B}|^2$, the main obstruction created by compressibility. Together with a damped wave structure for the oscillatory sector and space-time weighted energy functionals with shifted time weights, this yields global existence, uniform stability, and explicit polynomial decay rates.

We study simultaneous recovery of the (variable) mass density, initial displacement, and initial velocity for the three-dimensional isotropic elastic wave equation with known constant Lamé parameters. The data are the complete displacement trace on an enclosing boundary. We first assume that the density-weighted initial displacement and velocity have fixed known profiles in one spatial direction. The $s^0$ and $s^1$ coefficients of the zero-frequency Laplace expansion identify these weighted states. The $s^2$ and $s^3$ coefficients then give static Lamé orthogonality identities for the density difference. The exact difference expansion through order $s^3$ has an $O(s^4)$ remainder, uniformly on bounded spatial sets and bounded density-contrast and weighted-state classes. Under alignment of the two initial states and a nonzero moment of the density-weighted initial velocity, the two density identities reduce to a constant-vector static transform. Two opposite elastic null phases give uniqueness for one or two fixed vertical density profiles when the profile family is two-sided Laplace nondegenerate. When $\lambda+\mu\ne0$, each nonzero normal root has partial multiplicities $2$ and $1$ and admits a length-two Jordan chain. The associated polynomial–exponential Lamé mode produces derivatives of the bilateral profile transforms. The resulting Hermite–Laplace system gives uniqueness for aligned classes with up to four fixed vertical profiles and independent horizontal coefficients, provided that the profile system is Hermite–Laplace nondegenerate. Distinct translations of one compactly supported profile provide an explicit four-profile class. We also prove rigidity of the alignment reduction and exhibit an infinite-dimensional kernel for the reduced transform on unrestricted densities.

We construct a smooth, strictly convex ring that admit a unique solution of Gaussian $p$-Capacity problem whose super-level sets are nonconvex. This result provide a negative answer.