Papers

We investigate the propagation profile of positive solutions to \begin{equation*} u_t-du_{xx}=f(t,u) \mbox{ for } t>0,\ x\in(g(t),h(t)), \end{equation*} where $f(t,u)$ is monostable in $u$ and $T$-periodic in $t$, and the free boundaries $x=g(t), \ x=h(t)$ are determined by the Stefan condition $g'(t)=-\mu u_x(t, g(t)),\ h'(t)=-\mu u_x(t,h(t))$, coupled with $u(t, g(t))=u(t, h(t))=0$. For a special nonlinearity satisfying the strong KPP condition, the long-time behavior and asymptotic spreading speed of this problem were considered by Du, Guo and Peng \cite{DGP}. In this paper, by employing new techniques, we extend the results of \cite{DGP} to general monostable nonlinearities beyond the KPP framework and at the same time we obtain more precise description of the propagation profile: we prove the existence and uniqueness of a semi-wave and show that the spreading solution converges to this semi-wave as time goes to infinity.

We show that $id:\mathbb{S}^1 \to \mathbb{S}^1$ is not a minimizing $W^{s,\frac{1}{s}}$-harmonic map for $s \in (0,\frac{1}{8}$). On the other hand, for $s \in (\frac{1}{3},1)$ it is a local minimizing map, and for $s\in [\frac{1}{2}-\varepsilon,\frac{1}{2}+\varepsilon]$ it is a global minimizer. The usual extension or Fourier techniques being unavailable, our argument relies instead of stability analysis in $s$.

It is well known that the reaction-diffusion equation $u_t=du_{xx}+f(u)$ with compactly supported nonnegative initial functions exhibits trichotomy dynamics for bistable and combustion type $f(u)$ \cite{DM, zlatos}. The same is true for the corresponding Stefan type free boundary problem \cite{DL}. In this paper, we reveal a rather different type of trichotomy for this reaction-diffusion equation under a new set of (free) boundary conditions, arising as a model for biological invasion with $u(t,x)$ representing the density of an invading species over the one dimensional spatial regin $[0, h(t)]$. The evolution of the invading front $x=h(t)$ is governed by $h'(t)=-\frac d\delta u_x(t, h(t))$ and $u(t, h(t))=\delta\in (\hat\theta_f, 1)$, with $\hat\theta_f \in [0, 1)$ uniquely determined by $f$; they allow $h(t)$ to advance as well as to retreat when time increases. At the fixed boundary $x=0$, the density is controlled by $u(t,0)=\delta_0\geq 0$. We completely classify the long-time dynamics of the model when $f(u)$ is a monostable, or bistable, or combustion type nonlinear function. In the biologically interesting case that $\delta_0<\delta$, we show that there are exactly three scenarios: (i) successful spreading, (ii) finite-time vanishing, (iii) a transition state characterized by $h(t)\to l_*\in (0, \infty)$ and $u(t,x)\to w_*(x)$ as $t\to\infty$, where $(u(t,x), h(t))\equiv (w_*(x), l_*)$ is the unique stationary solution of the free boundary problem. The model here does not have the usual order-preserving property enjoyed by those considered in \cite{DM, zlatos, DL} and elsewhere (i.e., $u(0,x)\leq v(0,x)$ implies $u(t,x)\leq v(t,x)$ for all $t>0$ if $u$ and $v$ are two solutions of the problem), which is intrinsically linked to the many novel features of the model.

A nonlinear Vlasov–Fokker–Planck equation is considered in a bounded domain, with specular reflection imposed on the boundary. It is proven that if the initial datum is sufficiently close, in a weighted $L^\infty$-space, to a radial and spatially homogeneous function that is regular enough, then a corresponding global-in-time weak solution exists, and this one decays with exponential rate to the global Maxwellian. The proof is split into two steps. In the first, the stability of the manifold of spatially homogeneous solutions to this equation is established. In the second part, the close-to-Maxwellian regime is considered. A careful regularization scheme is designed so that this mollified equation can be well-approximated by a linear system. Inspired by the recent preprint [Carrapatoso and Mischler, arxiv:2407.09031], for this linear system, $L^2$-hypocoercivity along with ultracontractivity properties are derived in order to change the space of functional decay to the $L^\infty$-framework. The properties of the original model are then recovered by means of fixed point arguments and carefully passing to the limit of the regularization parameter.

This paper focuses on a class of fully nonlinear elliptic equations with variable double phase type degeneracy law and Hamiltonian terms. We obtain improved gradient Hölder regularity results at points where the Hamiltonian coefficients and source terms vanish. Furthermore, we establish a Schauder-type estimate at local extrema, which is sharp with respect to the vanishing rate of the Hamiltonian coefficient and source term. Our approach adapts compactness and dichotomy arguments to capture the interplay between the degeneracy rate and the growth of the Hamiltonian term.

This paper focuses on a class of fully nonlinear elliptic equations with general double phase degeneracy/singularity law and Hamiltonian terms of the form $$\Phi(|Du|,x)F(D^2 u, x)+H(Du,x) =f(x) \quad \text{in} \quad B_{1},$$ where $\Phi$ takes one of two typical forms: $$\Phi(|Du|,x)=\sigma_{1}(|Du|)+a(x)\sigma_{2}(|Du|)\quad {\rm or}\quad \Phi(|Du|,x)=\frac{\sigma_{1}(|Du|)}{|Du|}+a(x)\frac{\sigma_{2}(|Du|)}{|Du|}.$$ Under suitable assumptions on the operator $F$, Hamiltonian term $H$, source term $f$ and modulating coefficient $a$, we establish $C^{1}$ regularity for viscosity solutions, provided that $\sigma_{1},\sigma_{2}$ are moduli of continuity and their inverses are Dini continuous. Our argument is based on a tangential analysis via approximating hyperplanes combined with a new recursive renormalization algorithm adapted to the present framework. It is noteworthy that our results are new even for the case $a(x)\equiv 0$.

In this paper, we prove that the 2D Boussinesq equations are strongly ill-posed in the supercritical Besov spaces $B^s_{p,q}$ and Sobolev spaces $W^{s,p}$ with $(p,q)\in(1,\infty)\times [1,\infty]$ and $s\in(0,1+2/p)$ by constructing an initial data with arbitrarily small norm for which the solution of the system exhibits norm inflation almost instantaneously. As a further application, we prove the instability of perturbations near the hydrostatic equilibrium for the 2D Boussinesq equations in the same $B^s_{p,q}$ and $W^{s,p}$.

In part I, \cite{Frauenfelder:2026c}, we showed that collisional periodic orbits of twisted Zeeman systems can be detected variationally by a non-local Hamiltonian action functional. In this part II we show that the linearized gradient flow of this non-local functional is a Fredholm operator and prove a non-local elliptic regularity result. These results are obtained with the theory of almost extendability of weak Hessian fields introduced in \cite{Frauenfelder:2025g}.

We prove a finite-window anti-phantom principle for scale-critical Navier–Stokes defect packages and develop a conditional localized transfer framework around it. In a fixed clean quotient, the main compactness theorem shows that if the only defect that is simultaneously invisible, exactly reproducible, and tax-free is a gauge artifact, then observation, reproduction failure, and tax control the clean quotient distance by a positive finite-window gap. We then isolate the additional localized inputs needed to use this clean gap for Navier–Stokes packages: pressure-source observability, enhanced pressure-tail geometry, chart visibility, and residual sub-budgets for localization, reproduction, and gate/tax mismatch. The localized results are conditional finite-window reductions with explicit error constants, including a comparison theorem between the enhanced-tail geometry and the original intrinsic geometry under stated projection and harmonic tail approximation assumptions. The paper should be read as a rigorous finite-window obstruction framework, not as a proof of Navier–Stokes regularity, a construction of a singular solution, or a scale-uniform regularity criterion.

We prove that the map sending the diffusion profile to the solution of a time-changed Parisi PDE evaluated at time-space $(0,0)$ is concave. This result strengthens the raywise concavity result proven by Auffinger and Chen (2016). As an application, for the balanced multispecies Ising spin glasses, the lower bound of Bates and Sohn (2025) matches the Hopf-type upper bound given by the Hamilton–Jacobi framework developed by Mourrat, Chen and Xia.

We study solutions of the differential inequality $$ \Delta^{m / 2} u \ge f (x) g (u) \quad \mbox{in } B_1 \setminus \{ 0 \}, $$ where $m \ge 2$ is an even integer, $f$ and $g$ are some functions, and $B_1$ is an open unit ball in $R^n$, $n \ge 2$, centered at zero. Our aim is to obtain a necessary condition for a singularity at zero to be removable for any solution of this inequality.

We study the Boussinesq approximation of the Rayleigh-Taylor instability with local energy inequality, within the convex integration plus variational selection criterion framework of arXiv:2505.03278. We show that, under an appropriate rescaling, the energy dissipation functionals obtained in said paper $\Gamma$-converge, and that their limit has a unique maximizer, corresponding to the profile also obtained in arXiv:2008.08853.

We prove Harnack inequality with an optimal tail for linear mixed local-nonlocal parabolic equation on the Heisenberg group. For this purpose, we make use of the expansion of positivity method and a clustering lemma, thereby bypassing Moser iteration and logarithmic estimates.

We consider the Vlasov–Poisson equation on $\mathbb{R}^d \times \mathbb{R}^d$ for any dimension. For initial distribution $f_{0}$ having compact support in $v$ and belonging to $H^{s,p}(\mathbb{R}^d \times \mathbb{R}^d)$, we prove local well-posedness for $s>\frac{d}{p}-\frac{1}{2p}$ and $p\in[2, \infty)$. We proved this by using two main ingredients, which is the averaging property for the density $\rho$ and the Schauder-Tychonoff fixed point theorem. It seems like this is the first application of the Schauder-Tychonoff fixed point theorem in showing existence of solution for evolutionary PDEs.

We discuss the existence of $C^1$-solutions for two related double phase inequalities: \begin{equation*} {\mathcal L}_g u\pm \Delta_s u\geq (|x|^{-\alpha}*u^p)u^q \quad\mbox{ in }\mathbb R^N\setminus \overline B_1, N\geq 1,\tag{$P^\pm$} \end{equation*} in which $\Delta_s u:={\rm div}\big(|\nabla u|^{s-2}\nabla u\big)$ is the $s$-Laplace operator, $s>1$, and $$ {\mathcal L}_g u:= -{\rm div}\Big(|\nabla u|^{m-2}g(|\nabla u|)\nabla u\Big),\quad m>s>1, $$ where $g:[0, \infty)\to (0, \infty)$ is a $C^1(0, \infty)\cap C[0, \infty)$ non-increasing function with some specific behaviour near the origin. In the above context, the general form of ${\mathcal L}_g u$ includes the case of $m$-Laplace and $m$-mean curvature operator. Our study reveals a sharp distinction between $(P^+)$ and $(P^-)$. Precisely, we show that the inequality $(P^+)$ has solutions for all $m>s>1$ and $q>s-1$. In contrast, $(P^-)$ has solutions if and only if $p$ and $q$ are sufficiently large. We also link the solvability of $(P^-)$ with that of the corresponding equation ${\mathcal L}_g u- \Delta_s u= (|x|^{-\alpha}*u^p)u^q$ in $\mathbb R^N\setminus \overline B_1$, for which we derive optimal conditions in terms of $p, q, \alpha, s$ and $N$. The approach combines integral estimates with a new sub and supersolution method that accounts for the presence of the convolution term.

We consider the dynamics of an idealised cross-diffusion model of biological invasion in which the diffusion of an invading population is inhibited by a resident population, which is in turn degraded by the former. We formulate and numerically solve the problem that describes the self-similar smoothing of an initially piecewise-constant invading population. In the limit the height ahead of the initial discontinuity vanishes, the speed of propagation also vanishes, but only logarithmically slowly. This result is confirmed by a matched asymptotic analysis. The singular nature of this limit indicates that the system does not permit the existence of compactly supported solutions that exhibit a moving front or interface, which has important implications for the simulation of more complex models that also feature this degenerate diffusive mechanism.

This paper investigates spreading speeds for nonautonomous Fisher–KPP equations with nonlocal diffusion. The dispersal kernel depends on time and may be asymmetric in space. We assume exponential boundedness of the kernel and the existence of uniform mean values. Under these assumptions, the rightward and leftward spreading speeds are proved to be given by the linearly determined formula. The proof is based on comparison arguments, a refined maximum principle and proper sub- and super-solutions. The result includes, in particular, periodic, almost periodic, and uniquely ergodic time dependence. And it allows kernels whose support does not contain the neighborhood of the origin.

We study the problem of reconstructing the scattering and absorption coefficients in the radiative transport equation using internal data in quantitative photo-acoustic tomography (QPAT). In practical settings, however, this internal data is only partially available near the boundary, owing to medium's strong absorption and limitations of the measurement equipment. Our main contribution is the development of a method to recover these coefficients within a subregion where the internal data can be obtained with sufficient reliability.

We prove an exact inverse theorem for damped waves on $X=\mathbb{T}\times Y$ with arbitrary bounded measurable transverse damping $a=a(x)\geq 0$. Let $\Theta_a(r)$ be the least average of $a$ on radius-$r$ intervals. Then $\liminf_{r\downarrow 0}\Theta_a(r)>0$ characterizes exponential stability, while for every $L(S)=\ell(\log(eS))$ with $\ell:[1,\infty)\to[1,\infty)$ nondecreasing, unbounded, and eventually doubling, $\Theta_a(r)\gtrsim L(1/r)^{-1}$ is equivalent to stationary- and generator-resolvent bounds $O(|s|^{-1}L(|s|))$ and $O(L(|s|))$. One resonant scalar block per dyadic octave already recovers this multiscale mass bound. Every such critical profile occurs for indicator damping on an open dense set of arbitrarily small measure, whereas the stationary resolvent-to-mass implication fails at every sublinear power gauge. For the cusp $a(x)=(\log(e/|x|))^{-A}$, $A>0$, near its isolated damping well, we prove the sharp obstruction is genuine spectrum. Blow-up yields $-\partial_y^2+i\log|y|$, whose spectrum is a simple interlaced vertical ladder. Every fixed finite ladder portion transfers to damped-wave eigenvalues with complete blockwise algebraic count and two-term asymptotics. Combined with the resolvent theorem, these eigenvalues give the sharp regularized decay scale $\exp[-t^{1/(A+1)}]$.

We study the hyperbolic $O(N)$ linear sigma model, i.e. a system of $N$ interacting stochastic damped nonlinear wave equations (SdNLW) with coupled cubic nonlinearities, posed on the two-dimensional torus and indexed by a parameter $\varepsilon > 0$. We show that as $\varepsilon$ goes to zero (Smoluchowski-Kramers approximation) and $N$ goes to infinity (mean-field limit), each component of the solution to the SdNLW system converges to the solution to the stochastic nonlinear heat equation (SNLH) with a mean-field nonlinearity. We prove such convergence via two regimes: first with $\varepsilon$ going to zero to obtain the parabolic $O(N)$ linear sigma model, i.e. a system of $N$ coupled SNLH, and then with $N$ going to infinity; or first with $N$ going to infinity for each component to obtain the mean-field SdNLW and then with $\eps$ going to zero. As a result, we obtain a commutative diagram regarding the convergence from the hyperbolic $O(N)$ linear sigma model to the mean-field SNLH.

We prove the existence of forces and smooth initial data such that the associated Leray-Hopf solution of the 3D Navier-Stokes equations is unique, global-in-time, satisfies the energy equality, and has infinitely many (countable) blow-up instants. Different classes of forces and blow-up strengths are considered. All of them are sharp compared to the boundedness statements in literature. Our method also enables us to construct examples of blow-up for the forced 3D Euler equations.

We consider the two-dimensional incompressible Navier-Stokes equations with supercritical fractional dissipation in the vorticity formulation. In self-similar variables, we analyze the linearized operator in weighted spaces, prove a spectral gap, and construct a finite-dimensional local slow invariant manifold for small solutions. As a consequence, solutions are attracted to this manifold and admit an explicit long-time asymptotic expansion determined by the leading eigenmodes; additional moment conditions yield faster decay. We also show Lipschitz dependence of the manifold on the dissipation exponent, and recover the classical Navier-Stokes dynamics in the limit as the exponent approaches one.

We study the semiclassical wave front set of vector-valued microlocal solutions to the pseudo-differential system with a non-Hermitian matrix-valued symbol. The main result establishes the propagation of the semiclassical wave front sets under a generalized microhyperbolicity condition for the real part of the symbol and the non-negativity for the imaginary part. The proof is based on Hörmander's positive commutator method.

We develop a finite-window audit and transfer framework for the local regularity problem of the three-dimensional incompressible Navier–Stokes equations. The paper has two logically separate components. First, a finite-scale critical ledger estimate shows that a persistent non-CKN branch cannot survive without cumulative untaxed critical supply or accumulated localization leakage. Second, a local-to-clean transfer theorem shows how a clean finite-window anti-phantom gap gives a localized coercive estimate for Navier–Stokes defect packages, provided three structural inputs hold: quotient-lifting stability, componentwise detection comparison, and a normalized residual budget. The technical contribution is an explicit bookkeeping of the residuals that can obstruct this transfer. Pressure tails, cutoff leakage, truncation loss, nonlinear cutoff mismatch, reproduction drift, gauge mismatch, and profit discrepancy are separated as finite-window ledger entries and then assembled into a concrete absorption criterion. The main transfer theorem is algebraic once these entries satisfy the stated normalized bounds; the PDE-facing work left open is precisely to prove the corresponding component compatibility estimates against the localized quotient distance. Thus the article is a conditional reduction. Its conclusion is structural: any surviving obstruction must either defeat the stated compatibility/growth controls or appear as an NS-realizable, cleaned, scale-critical, combined-invisible, profitable, reproducible defect cascade.

In this paper, we develop an intrinsic consensus-based optimization framework on Riemannian manifolds with bounded sectional curvature. In contrast to extrinsic approaches based on an ambient Euclidean embedding, our model is formulated directly in terms of the Riemannian structure, using logarithmic and exponential maps induced by the intrinsic geodesic distance. We prove the global well-posedness of the proposed particle system and its associated McKean–Vlasov dynamics. We also establish the global convergence of the mean-field equation toward a global minimizer of the objective function under suitable conditions. Numerical experiments on the sphere, hyperbolic space, and the special orthogonal group demonstrate the effectiveness of the intrinsic CBO dynamics for nonconvex optimization problems on manifolds.

We develop a regularity theory for fully nonlinear oblique transmission problems. Our main result establishes the optimal regularity of viscosity solutions for flat interfaces. For constant-coefficient equations, we prove that viscosity solutions are piecewise $C^{1,\alpha}$ up to the interface. The same regularity continues to hold for variable-coefficient problems, with sufficiently regular coefficients, in a more restrictive class of viscosity solutions. This is the first regularity result for variable-coefficient transmission conditions.

We introduce elliptic transmission problems involving fully nonlinear equations in both the interior bulk equations and the interface transmission condition. In contrast to all previous works, our transmission condition depends on both the normal and tangential derivatives of the solution from each side of the interface. Our main result establishes the existence and uniqueness of viscosity solutions for constant-coefficient oblique transmission problems with flat interfaces.

We prove rigidity and classification results for the quasilinear Liouville equation associated with the $n$-Laplacian on complete noncompact Riemannian manifolds with nonnegative Ricci curvature. Our first result shows that, under a sharp logarithmic lower bound, the ambient manifold must be isometric to the Euclidean space and the solution must be one of the standard bubbles. We also prove a finite-mass rigidity theorem under the corresponding sharp asymptotic lower bound. We show that any logarithmic lower bound forces positive asymptotic volume ratio and one-endedness of the manifold. Finally, we construct solutions on nonflat manifolds with nonnegative Ricci curvature showing the sharpness of our hypotheses.

We establish boundary Hölder regularity for weak solutions to a class of nonlinear elliptic Dirichlet problems with nonstandard growth posed on Reifenberg flat domains. More precisely, for any prescribed exponent \(\alpha\in(0,1)\), we show that weak solutions are \(\alpha\)-Hölder continuous up to the boundary provided that the Reifenberg flatness parameter is sufficiently small. The proof combines an iterative boundary decay argument with an ABP-type maximum principle in the Orlicz–Sobolev setting.

We establish a strong comparison principle for viscosity solutions of fully nonlinear elliptic equations driven by second-order Hamiltonians that may be discontinuous with respect to all variables. The analysis relies only on a controlled superlinear growth in the gradient variable together with a structural monotonicity in the unknown, and requires neither ellipticity nor continuity of the coefficients. The proof is based on the construction of tailored viscosity test functions obtained as solutions of auxiliary eikonal-type equations. These functions compensate for the lack of regularity and allow for a fully local comparison argument despite the complete discontinuity of the Hamiltonian. This yields a robust comparison principle in open subsets of $\mathbb{R}^N$. As a consequence, we prove the existence of continuous viscosity solutions via a Perron's method adapted to discontinuous frameworks, combined with Ishii's semicontinuous envelope technique. The class of Hamiltonians covered includes linear and quasilinear equations with merely Borel measurable coefficients, as well as Hamilton-Jacobi-Bellman equations arising in stochastic control and differential games.