Papers

We prove the first boundedness and pointwise decay result for the scalar wave equation on rotating extremal black holes without any symmetry assumptions. The result applies to slowly rotating (equivalently, strongly charged) extremal Kerr-Newman spacetimes. We establish uniform energy boundedness, integrated local energy decay, and a hierarchy of boundary-weighted estimates at the extremal horizon and at null infinity, from which inverse-polynomial pointwise decay follows in the entire exterior region. As a consequence, we also prove the expected Aretakis instability: for generic initial data, suitable transversal derivatives fail to decay along the event horizon, and higher transversal derivatives blow up asymptotically. The proof uses the $b$-structure of the wave operator near the two boundary hypersurfaces, together with a treatment of normally hyperbolic trapping on extremal Kerr–Newman.

We develop a nonlinear potential theory for elliptic equations with Orlicz growth under general monotonicity and growth conditions, without any homogeneity or scaling assumptions. The lack of scaling invariance prevents the use of many classical tools from nonlinear potential theory. To overcome this difficulty, we establish a new framework that includes global Hölder regularity for obstacle problems, a balayage theory, the construction and analysis of Riesz measures associated with superharmonic functions, the identification of capacitary potentials, capacitary estimates for polar sets, and the quasicontinuity of superharmonic functions. As an application of this theory, we prove that the classes of superharmonic functions and renormalized solutions to elliptic measure data problems coincide. This extends the classical equivalence theory from the homogeneous $p$-growth setting to general Orlicz growth and is new even for power-growth operators without homogeneity assumptions.

We establish the global-in-time existence of a codimension $1$ canonical multi-phase free boundary Brakke flow in the upper halfspace which is integer rectifiable up boundary, starting from a countably $n$-rectifiable set. Under a suitable uniform density ratio assumption on the initial datum, we show that the free boundary carries no positive mass for some short time.

Of concern is a class of non-autonomous evolution equations of second order in Hilbert spaces, with a nonnegative self-adjoint operator $A$, time-varying damping and nonlinear source term. We give an upper decay rate of the energy, valid for all solutions and solely based on the damping coefficient and the geometrical index of the source term. Furthermore, we prove under suitable conditions that for all initial data, except for those in the kernel of $A$, the solutions decay (in the energy norm) at most as fast as this decay rate. The result not only shows the optimality of the decay rate, but also reveals an unusual phenomenon: “slow solutions", i.e. those that decay at exactly this rate, are dominant in amount. Moreover, specialized to the case when the nonlinear source is absent, our result improves relevant existing ones to a large extent.

We study qualitative and quantitative homogenization for a Cahn–Hilliard system with a nonconservative source term in a periodically perforated domain. Using the periodic unfolding method, we derive uniform energy estimates and prove convergence to a homogenized Cahn–Hilliard system whose effective diffusion tensor is characterized by scalar Neumann cell problems on the pore cell. For the quantitative analysis, we construct first-order corrector approximations by means of a scale-splitting operator, so that the cell correctors are only required to belong to $H^1_{\mathrm{per}}(Y_p)$. Under $H^2$-regularity of the homogenized solution and well-prepared initial data, we obtain an order $\varepsilon^{1/2}$ corrector estimate: the corrected order-parameter error is controlled in $L^2(0,T;H^1(\Omega_p^\varepsilon))$, while the uncorrected order parameter is controlled in $L^2(0,T;L^2(\Omega_p^\varepsilon))$. This improves the rate $\varepsilon^{1/4}$ previously established for fourth-order phase-field equations in perforated media, and matches the natural rate for second-order elliptic problems in perforated domains. The rate reflects the boundary layer caused by incomplete cells near $\partial\Omega$ and improves to order $\varepsilon$ on the flat torus $\mathbb{T}^d$.

We prove that every entire smooth strictly plurisubharmonic solution of the self-shrinking $J$-equation on $\mathbb{C}^n$ is a quadratic polynomial. This removes the asymptotic lower bound assumption on the complex Hessian in \cite[Theorem 4]{HJ}. The result also recovers the corresponding real rigidity theorem in \cite[Theorem 1.1]{HOW} as a special case. More generally, our method applies to self-shrinking equations associated with a broad class of fully nonlinear elliptic operators satisfying suitable structural conditions, including the adjacent inverse complex Hessian quotient operators $-\sigma_{k-1}/\sigma_{k}$, $1\leq k\leq n$. For the remaining quotients $-\sigma_l/\sigma_k$ with $l<k-1$, we prove the same rigidity conclusion under the additional assumption that the solution is proper.

In this paper, we investigate the multiplicity of normalized solutions to a nonlinear Dirac equation with localized nonlinearities on noncompact metric graphs under the \(L^2\)-constraint, as well as the asymptotic behavior of these solutions in the nonrelativistic limit. First, we establish the existence of multiple normalized bound states. Moreover, we explore the nonrelativistic limit and show that, as the speed of light tends to infinity, the solutions converge to those of a nonlinear Schrödinger equation. Our results including the mass-subcritical, mass-critical and, in particular, mass-supercritical regimes.

Theorems on uniqueness and continuous dependence on the initial condition of solutions to stochastic Navier-Stokes equations driven by Wiener processes and Poisson martingale measures are presented. These theorems generalise some results from \cite{GK2026}.

Under appropiate hypotheses on the external force acting on an incompressible flow diffusing through a porous medium, we show that there is a unique stationary solution to the diffusive porous media equation. Moreover, we show that this solution is asymptotically stable and estimate the decay rate of any perturbation towards this steady state.

This work is concerned with the nonlinear information in the linear sampling method for the inverse medium scattering problem. In addition to the well-known capability in shape characterization, we demonstrate that the imaging indicator represents nonlinear information about the unknown contrast. We further demonstrate additional representations for this nonlinear information using the generalized linear sampling method. We shed light on this new result using the Born model and analytical examples.

In the present note, certain scenarios of potential Type II blowups of solutions to the Navier-Stokes equations are considered on the local level. They generalise particular scenarios described in the previous papers of the author. The main features of the approach, adopted in the note, are a zoom based on the Euler scaling and Liouville type theorems for the Euler equations in classes motived by a particular scenario of the Type II blowup.

This paper investigates a novel quasi-singularity formation phenomenon in the isentropic compressible Euler equations in three dimensions. For any prescribed finite set of points $\{z_j\}_{j=1}^N \subset \mathbb{R}^3$ and an arbitrarily large parameter $M \gg 1$, we explicitly construct real-analytic initial data generated by Herglotz-type wave fields. We prove that the corresponding classical solution $$ (p,v)\in C^1([0,T]\times\mathbb{R}^3;\mathbb{R}\times\mathbb{R}^3), $$ with $$ T = C_0 \, M^{-\frac{3\gamma+1}{\gamma-1}}, $$ where $\gamma > 1$ is the adiabatic index and $C_0 > 0$ is a constant independent of $M$. Throughout the time interval $[0, T]$, the velocity field, its spatial gradient, and the pressure gradient exhibit simultaneous localized amplification at each prescribed point $z_j$. More precisely, each component of the velocity field is bounded below by $M$, each component of the pressure gradient is of size at least $M^{\frac{2\gamma}{\gamma-1}}$, and the velocity gradient exhibits a prescribed anisotropic sign pattern with entries of size at least $M$ at every $z_j$. All implicit constants in these estimates are independent of $M$ and depend solely on the fixed physical parameters and the locations $\{z_j\}_{j=1}^N$. This mechanism generates highly localized, shock-like gradient concentrations and implosion-like spatial profiles within the smooth regime. The underlying mechanism relies on the construction of the Herglotz kernel via Helmholtz far-field patterns, together with delicate quantitative energy estimates for the quasilinear hyperbolic system.

We consider self-maps of a sphere in the critical Sobolev space with a given Brouwer degree. Our main result is that the (directed) distance between maps of different degrees is equal to an explicit constant times the difference in degrees. In the case of the 2-sphere this resolves an open problem by Brezis.

In this paper, we study the initial-boundary value problem of the 1D compressible Navier–Stokes/Cahn–Hilliard system with vacuum. We establish the global existence and uniqueness of strong solutions to this initial-boundary value problem. No any initial compatibility conditions are required via time weighted techniques, which leads to a loss of regularity near the initial time. Therefore, the uniqueness of solutions obtained in this paper is even more challenging. To address this issue, we establish refined growth estimates and singular-in-time weighted energy estimates that induce a Gronwall-type structure, which ultimately allows us to close the uniqueness proof in Eulerian coordinates without passing to Lagrangian coordinates.

In this paper, we study the long-time behaviour of the two-dimensional stochastic damped Euler equation on the torus driven by bounded random forcing. Unlike stochastic Navier-Stokes or fractionally dissipative Euler equations, the model possesses no viscous regularisation, so the classical parabolic smoothing is unavailable. We prove that when the damping coefficient is sufficiently large, the associated Markov semigroup admits a unique invariant measure and converges exponentially fast to equilibrium. The key ingredient is the establishment of a global-in-time $W^{1,\infty}$ estimate for the vorticity. This estimate yields a compact absorbing mechanism in $C(\mathbb{T}^2)$, which enables us to establish the uniqueness of the invariant measure and exponential mixing. To the best of our knowledge, this is the first exponential mixing result for a genuinely inviscid stochastic Euler-type equation. Our approach demonstrates that sufficiently strong linear damping can effectively replace the compactness mechanism usually provided by viscosity and is expected to be applicable to other inviscid or weakly dissipative stochastic partial differential equations driven by bounded random forcing.

This paper addresses the existence of nontrivial solutions to a class of mixed local-nonlocal problems involving a mixed interpolated Hardy potential. We first establish a concentration-compactness principle for mixed local and nonlocal operators. This result is combined with Ricceri's variational principle to obtain an existence result for quasilinear elliptic problems under different growth assumptions on the nonlinearity. Furthermore, we apply the classical mountain pass theorem to obtain a second existence result in the superlinear case.

It is well known that solutions of elliptic equations inherit analyticity from analytic coefficients, while much less is understood about the inheritance of ultra-analytic regularity, especially for nonlocal equations. This paper develops a systematic Fourier-analytic framework to study fractional and more general nonlocal pure-potential equations whose potentials satisfy ultra-analytic derivative bounds. We prove sharp quantitative logarithmic ultra-analytic estimates for normalized solutions, and show that both the logarithmic power and the leading constant involving the fractional exponent are optimal in natural periodic model examples. We also establish a general transfer principle for weighted ultra-analytic scales, which reveals why standard scales are not preserved, and singles out a natural family of invariant ultra-analytic spaces.

We establish four local-to-global propagation results for Dirichlet–to–Neumann maps. Our first two results are proved in the general setting of smooth compact Riemannian manifolds with boundary. The first shows that if two smooth Riemannian metrics coincide in a collar neighborhood of a connected boundary component \(\Gamma\), then equality of the corresponding local Dirichlet–to–Neumann maps on a nonempty open subset of \(\Gamma\) propagates to equality of the associated global Dirichlet–to–Neumann maps on all of \(\Gamma\). The proof combines unique continuation and self-adjointness arguments. The second replaces the geometric collar assumption by an exponential spectral assumption on the difference of the corresponding global Dirichlet–to–Neumann maps. The proof relies on the spectral unique continuation theory of Jerison–Lebeau, through the formulation of Le~Rousseau–Lebeau. Our third and fourth results establish local-to-global propagation principles under Ingham-type quasi–analytic spectral assumptions. Assuming that the boundary manifold is respectively a compact Riemannian symmetric space or a compact quasi–analytic Riemannian manifold, they rely on the propagation theorems of Ganguly–Thangavelu and of Bhowmik–Pradhan. As an application, we consider a class of conformally warped product metrics. In this setting, the local Borg–Marchenko theorem and Weyl–Titchmarsh theory relate the required Ingham-type spectral decay to a suitable quasi–analytic boundary closeness of the conformal factors, yielding new local-to-global uniqueness results for Dirichlet–to–Neumann maps.

We study the two-dimensional incompressible viscous non-resistive magnetohydrodynamics in the periodic strip $\mathbb T\times\mathbb R$, subject to a smooth sheared background magnetic field $(\xi(x_2),0)^{\top}$, where $\xi(x_2)$ is bounded and away from zero. For sufficiently smooth perturbations satisfying even-odd symmetry, we prove global-in-time well-posedness and nonlinear stability in Lagrangian coordinates. The spatial inhomogeneity of the shear profile generates persistent linear contributions, most critically a nontrivial pressure term that precludes the uniform-in-time estimates. We straighten the integral curves of the initial magnetic field and construct a volume-preserving corrector. This geometric reduction transforms the intractable linear pressure into a quadratic nonlinearity. These structures yield the global energy bounds and the anisotropic algebraic decay rate for the system. This mechanism appears to provide the first rigorous framework for establishing global nonlinear stability for viscous non-resistive magnetohydrodynamics near the genuinely nonuniform sheared magnetic profile.

This paper presents a generic approach for establishing mean field game master equations, applicable whenever the mean field equilibrium can be characterized by a McKean-Vlasov forward-backward stochastic differential equation system. The core of our approach is a representation formula for the first-order Lions derivative of the decoupling field of this forward-backward SDE system. We then employ a bootstrap argument to recursively compute its higher-order derivatives. To demonstrate the method's versatility, we establish the local well-posedness for master equations in three distinct models: extended mean field games, mean field games with volatility control, and mean field games with a major player.

In this paper, we establish higher-order convergence rates of the periodic homogenizatio for symmetric Lévy-type operators, encompassing the subcritical $\alpha$-stable regime, critical regime, and supercritical diffusive regime. To this end, we develop a systematic framework to decompose the contributions of the underlying jumping kernel across small, intermediate, and large spatial scales – a strategy tailored to all the aforementioned regimes. To the best of our knowledge, this work represents the first comprehensive study of higher-order convergence rates in the homogenization of non-local operators.

We study an obstacle problem for surfaces minimizing an anisotropic surface energy of ellipsoidal type. Given a convex obstacle and a boundary datum, we seek a surface that minimizes the anisotropic area functional while remaining above the obstacle. The central novelty is the systematic use of the Cahn-Hoffman transform to convert the anisotropic problem into an equivalent isotropic problem with a generalized Robin boundary condition. We prove optimal regularity of the solution ($C^{1,1}$ up to the free boundary) and $C^{1,\alpha}$-regularity of the free boundary itself under a non-degeneracy condition. The singular set of the free boundary is shown to have Hausdorff dimension at most $n-1$, and a logarithmic epiperimetric inequality yields its $(n-1)$-rectifiability. The approach combines Caffarelli's classical theory of obstacle problems with the geometric theory of anisotropic mean curvature and the Alexandrov reflection principle adapted to the anisotropic setting.

We analyze via Evolutionary Gamma-convergence a stratified composite structure consisting of a thin adhesive layer with vanishing thickness and undergoing rate-independent damage, as well as two adjacent elastic adherents. As the width of the intermediate layer tends to zero, we prevent complete degradation of the material by assuming that the damage variable scales minimally like the thickness of the adhesive layer. As a result, we identify a limiting model that combines both a brittle constraint and an adhesive-type energy contribution featuring the jump of the admissible displacements.

An analytic solution has been recently developed for the Maxwell's equation in a medium with zero currents such as vacuum. The solution is attractive in the sense that it is formulated based on the Fourier expansion of the initial value. It has been used to study the properties of solutions like certain conservative laws and construct electromagnetic waves with certain features. In this paper, we study Maxwell's equation in a medium with non-zero currents. The structure of solutions in this setting turns out to be much more complicated than what has been achieved without currents, and a clean structure of analytic solutions as with zero current is no longer available in general. Nevertheless, we can still develop an algorithm to construct the solution effectively. Our efforts in seeking analytic solution focus on two special cases. First, we develop analytic solution under the assumption that Ohm's law is satisfied, i.e. the current density is proportional to electronic density; secondly, we add skew symmetric components under generalized Ohm's law, which is also refereed as Hall effect in literature, and study the properties of solutions. In addition, we consider the case where an independent local electromagnetic field is included and derive the analytical solution accordingly. As an application, we provide an example to use the analytic solution to construct parallel electronic and magnetic waves.

This first part of the series builds the analytic layer of noncommutative anisotropic diffusion in a separable Hilbert space. Let $\mu_0=\mathcal{N}(0,Q)$ be the reference Gaussian measure, with $Q\in L^1(\mathcal{H})$, and let $D(x)$ be a positive, state-dependent anisotropy. We do not assume that $[D(x),Q]=0$. Consequently, for the forward SDE with $\sigma(x)=D(x)^{1/2}Q^{1/2}$, the correct energy form is given not by the expression $\langle D\nabla u,\nabla v\rangle$ but by the consistent form $\Gamma_A(u,v)= \langle Q^{1/2}D(x)^{1/2}\nabla u, Q^{1/2}D(x)^{1/2}\nabla v\rangle$. We prove closability of the form, well-posedness of the forward dynamics, Galerkin convergence, stability of the $A$-LSI under a Mosco limit, the chain rule for relative entropy, and a general weak-bridge theorem. The main result of Part~I is a functional-analytic theorem: if $A$-consistency, a uniform $A$-LSI, and representability of the right-hand side of the backward weak form in the negative energy space all hold, then a backward weak drift $v=\mathsf{A}\nabla\Phi$ exists and the basic entropy dissipation estimate holds. In addition, we single out a three-dimensional tensor class of anisotropies, formulate a condition for the absence of diffusion degeneracy, and obtain a rate estimate for the homogenization limit, first on cylindrical subspaces and then on compact-tail classes, which yields strong resolvent convergence and convergence of the forward SDEs. The statistical closure, an independent isotropic benchmark, and an approximation theorem for $A$-adapted networks are treated in Part~II.

We establish Liouville-type theorems for the stationary fractional Navier-Stokes equations in $\mathbb{R}^n$ under suitable integrability conditions on the velocity field $u$ and a large-scale Morrey-type bound on the fractional energy. As a corollary, these assumptions are automatically satisfied if $u \in \dot{H}^{\frac{\alpha}{2}}(\mathbb{R}^n)$, yielding Liouville-type results under the finite fractional energy condition for $\frac{n}{3} \le \alpha < \frac{n+2}{3}$, where $\alpha$ denotes the order of the fractional Laplacian $(-\Delta)^{\frac{\alpha}{2}}$. This range reflects a scaling-critical correspondence between Liouville-type theorems in the finite-energy setting and the threshold arising in partial regularity theory. The proof relies on direct kernel estimates for the commutator of the fractional Laplacian, based on a dyadic decomposition of the tail term, which remain valid in the hyper-dissipative case. The argument also uses a bootstrap argument that propagates integrability from near the scaling-invariant exponent down to lower exponents, including the Sobolev embedding exponent.

This paper focuses on the regularity of viscosity solutions to normalized $p$-Laplacian equations with variable-exponent double phase type degeneracy/singularity and Hamiltonian terms. Based on a new improved oscillation-type estimate combined with a localized analysis, we establish sharp interior $C^{1,\alpha}$ regularity estimates in a unified way. In addition, in the degenerate case, we obtain improved gradient Hölder regularity results at points where the Hamiltonian coefficient and source term vanish, and establish a Schauder-type estimate at local extrema. Notably, our results are still novel even restricted to single power-type singularity or degeneracy law.

In this paper, we study a quantitative Runge-type global approximation theorem for the linearized magnetohydrodynamic (MHD) system in bounded domains with arbitrary topology. In the context of magnetic relaxation, the interplay between the domain topology and magnetic field structure plays a crucial role. Recent studies illustrate a sharp contrast in the dynamics: while Enciso–Peralta-Salas (2025) highlights that the geometric complexity of magnetic fields acts as an obstruction to relaxation in non-resistive regimes, Kozono-Shimizu-Yanagisawa (2025) proves that in resistive regimes, the flow stably relaxes towards a harmonic equilibrium. Focusing on this resistive scenario, we adopt a control-theoretic viewpoint to quantitatively approximate the relaxation trajectory generated by the linearized initial-boundary value problem. Specifically, after decomposing the bounded-domain solution into the time-evolving part and the stationary part, we approximate it by a global solution on $\mathbb{R}^3$ under a remote forcing. An explicit dependence of the forcing cost on the approximation error is provided.

The hydrodynamic limit to the barotropic Euler equations, including power-law pressure $P(\rho)=\rho^\gamma$, for a kinetic nonlinear Fokker–Planck equation with degenerate diffusion is established. This extends the well-known result of the derivation of isothermal Euler equations via Fokker–Planck equation with linear diffusion. We establish the asymptotic analysis using the relative entropy method by quantifying error estimates for pressures and employing the generalized Log-Sobolev inequality for degenerate diffusion.

This paper establishes a pointwise gradient potential estimate for solutions to linearized Monge-Ampère equations and derives a modulus of continuity estimate for the gradient in terms of the associated section adapted potential. As applications, we obtain a sufficient criterion for the continuity of the gradient, local BMO and VMO estimates for the gradient with respect to Monge-Ampère sections, and a local $C^{1,\gamma}$ regularity criterion.