Papers

We present an expository excursus on modulation spaces, from Feichtinger's original construction on locally compact abelian groups to recent symplectic formulations. We trace the same local-global principle through bounded uniform partitions of unity, the short-time Fourier transform, coorbit methods, and Gabor coefficients, while distinguishing equivalent descriptions from the weighted, quasi-Banach, and Gelfand-Shilov extensions of the underlying functional setting. We also discuss Gabor matrices, convolution and embedding estimates, and an application to a nonlinear dispersive equation. The final part is devoted to metaplectic Wigner distributions $W_{\mathcal A}$. In the shift-invertible case these representations are, up to chirps and a linear change of phase-space variables, rescaled short-time Fourier transforms. This structural fact explains both their characterization of modulation and Wiener amalgam spaces and their discretization by metaplectic Gabor frames, and places the recent symplectic theory naturally within Feichtinger's modulation-space framework.

We establish Bourgain–Brezis selection maps for Hodge systems in new ranges of Besov and Triebel–Lizorkin spaces. Our approach combines the trace-free Beurling–Ahlfors transform with a nonlinear duality method. To control concentrations, we use tools developed by Stolyarov in the study of Maz'ya's $\Phi$-inequalities. These methods yield selection maps in Besov spaces $\dot B^{n/p}_{p,q}(\mathbb{R}^n)$ for every $1\leq q\leq p\leq 2$, and in Triebel–Lizorkin spaces $\dot F^{n/p}_{p,q}(\mathbb{R}^n)$ for every $1<p\leq q\leq 2$. The theorems hold for every dimension $n\geq 2$ and every form degree $1\leq l\leq n-1$. Although our results overlap with some earlier work, they considerably extend the known range of solutions to the Bourgain–Brezis problem and underscore the importance of the trace-free Beurling–Ahlfors transform in its resolution.

In this paper, we study the Fisher–KPP equation \[ \partial_t u = \alpha\Delta_{\TT_{q+1}}u+\beta u(1-u), \] on the homogeneous \((q+1)\)-regular tree \(\TT_{q+1}\), with \(q\geq2\), for nontrivial, compactly supported radial initial data. We first obtain an explicit representation of the fundamental solution of the corresponding linearized problem, together with sharp two-sided pointwise estimates. These estimates identify the propagation threshold \[ \beta>\alpha(\sqrt q-1)^2 \] and determine the associated critical speed \(c_\ast\) and decay rate \(\lambda_\ast\). In this regime, we prove that, for every fixed \(\theta\in(0,1)\), the outermost \(\theta\)-level set satisfies \[ \kappa_\theta(t) = c_\ast t-\frac{3}{2\lambda_\ast}\log t+O(1), \qquad t\to\infty. \] Therefore, while the branching geometry affects the threshold for propagation and the values of the critical parameters, the classical Bramson factor \(3/2\) persists. Our results therefore extend classical propagation and logarithmic-delay phenomena, previously known for one-dimensional continuous and discrete Fisher–KPP models, to homogeneous trees.

We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, uniformly equi-integrable subset of $L^{q_c}$. The main difficulty is the possible concentration of the critical gradient energy. We overcome it through a two-scale blow-up argument: the first scale produces a small Hamiltonian coefficient, while energy normalization at the second scale yields strong endpoint compactness, allowing a parabolic Liouville theorem to rule out concentration. The resulting uniform little-Hölder estimate, combined with critical Gagliardo–Nirenberg interpolation, permits absorption of the nonlinear term in the parabolic Calderón–Zygmund estimate. As an application, we construct bounded strong solutions for $f\in L^{q_c}$ and continuous initial data $u_0$, and prove subsequential strong interior convergence of smooth approximations in $W^{2,1}_{q_c}$.

We study scattering for the defocusing inhomogeneous generalized Choquard equation with data in the conformal energy space. In the full short-range mass-subcritical regime, a key estimate for the nonlinear term yields scattering in \(L^2\). A pseudoconformal argument in the compactified variables then gives scattering in \(H^1\). In the homogeneous case, estimates based on the Galilean vector field further imply scattering in \(\Sigma\) above the Strauss threshold. The method avoids the additional algebraic restrictions arising from intermediate Strichartz-Sobolev estimates.

We study linearized perturbations of the subextremal Kerr-Newman black hole solution of the Einstein-Maxwell equations. We show how to overcome what Chandrasekhar called the "apparent indissolubility of the coupling between the spin-1 and spin-2 fields" by using a matrix integrating factor for suitably transformed first-order systems that describe the radiative degrees of freedom. For the radial ODEs obtained after angular separation, we prove mode stability on the real axis, i.e., the absence of quasinormal modes with real frequencies, using a Whiting-type transform.

Let $1<m<n$ and $0<q<m-1$. We classify positive, locally bounded weak solutions of \[-\Delta_m u=u^p|Du|^q\quad\text{in }\mathbb R^n,\qquad p=\frac{m-q}{n-m}\left(n+\frac{q}{m-1-q}\right)-1.\] Every solution is constant or belongs to an explicit family of radial solutions, up to translation and scaling. No global bound, decay, or finite-energy assumption is required. The proof is based on a scalar differential identity and a maximum principle that remains valid at critical points. These lead to a sharp gradient bound, classification of normalized solutions, and a first-contact argument for arbitrary entire solutions. Local regularity and the behavior at critical points are treated throughout in the weak solution class; global $C^2$ regularity is not assumed.

For the 2D incompressible Euler equations on the half-plane, we establish Lyapunov stability of the Lamb dipole without sign condition on the vorticity. The main challenge arises from growth of the $x_{2}$ weighted norm for mixed sign vorticities, whereas this norm is conserved and responsible for dipole stability in the single signed vorticity case. This difficulty is handled by a Lagrangian bootstrap argument, based on carefully dividing the solution into several pieces and analyzing interactions among them, with the most delicate interaction involving vorticity drifting to infinity in the vertical direction. We expect that the strategy developed here will provide a starting point for the stability analysis of more general vortex configurations, including interactions among multiple vortex dipoles and among multiple vortex rings in settings involving mixed-sign vorticity.

This paper investigates the boundary pointwise regularity of solutions to the p-Laplace equation. By treating curved boundaries as perturbations of flat ones, we establish the boundary pointwise Lipschitz regularity under the exterior $C^{1,\mathrm{Dini}}$ condition and the Hopf lemma under the interior $C^{1,\mathrm{Dini}}$ condition. We also establish the boundary pointwise $C^1$ regularity under the $C^{1,\mathrm{Dini}}$ condition by a compactness argument.

For $3\le n\le6$, we determine the sharp isoperimetric bound for the mass-normalized first weighted Neumann eigenvalue on bounded Euclidean domains. For every finite nonnegative nonatomic measure $\mu$ on $\overline\Omega$, \begin{equation*} \overline\lambda_1^N(\Omega,\mu)\le\frac{n(n-1)}{n-2}\omega_n\sin^2\vartheta_n\left(\frac{|\Omega|}{\omega_n}\right)^{(n-2)/n}, \end{equation*} where $\omega_n$ is the volume of the unit ball and $\vartheta_n\in(\pi/2,\pi)$ is the angle at the first stationary radius of the regular rotational harmonic-map profile. The bound is attained on balls by a smooth positive radial density. In fact, we affirmatively resolve Question~1.12 posed by Vinokurov \cite{Vinokurov2026}. Furthermore, for admissible domains with finite boundary measure, we also prove the sharp strict Steklov bound \begin{equation*} \sigma_1(\Omega)|\partial\Omega|\,|\Omega|^{(2-n)/n}<\frac{n(n-1)}{n-2}\omega_n^{2/n}\sin^2\vartheta_n, \end{equation*} whose constant is approached by $C^1$ perforated domains.

We develop an analytical framework for studying a family of interacting particle systems that we refer to as the multi-particle Oja flow. The common interaction structure underlying this family arises in a variety of applications, including higher-order synchronization models, opinion dynamics, and self-attention dynamics derived from Transformer architectures, motivating a unified mathematical framework for their analysis. We first establish well-posedness for both the finite-particle and the mean-field systems. We also prove convergence of the finite-particle dynamics to the mean-field equation as the number of particles tends to infinity. We then investigate low-dimensional structures underlying the dynamics. In particular, we show that the mean-field solution admits a representation through a time-dependent normalized linear transformation acting on the initial distribution, with the transformation governed by a matrix ODE. Building on this representation, we identify a broad class of parametric families that are invariant under the mean-field dynamics. When the initial distribution belongs to one of these families, the evolution remains within the same family for all time and can therefore be characterized through the dynamics of the corresponding family parameters. As a concrete example, we develop this reduction within the angular central Gaussian family and illustrate how the resulting parameter dynamics provide reduced descriptions of the mean-field evolution for several representative systems.

We study vector-valued local minimizers of double phase functionals with two modulating coefficients and logarithmic-type growth. The energy density is $a(x)|Du|^p+b(x)|Du|^p\log(e+|Du|)$, where the nonnegative coefficients $a(\cdot)$ and $b(\cdot)$ may vanish but their sum is bounded away from zero. Assuming that $a(\cdot)$ is uniformly continuous and that $b(\cdot)$ satisfies a vanishing log-Hölder continuity condition, we prove that minimizers are locally Hölder continuous for every exponent in $(0,1)$. If both coefficients are Hölder continuous, then the gradient of a minimizer is locally Hölder continuous. The proof combines Gehring-type higher integrability with a comparison argument.

In this contribution we consider the Fokker-Planck equation for fermions, commonly known as the Fermi-Dirac-Fokker-Planck equation. This equation is a correction of the classical Fokker-Planck model that incorporates the Pauli exclusion principle. We present a novel approach that combines three ingredients: (1) stability of solutions, (2) quantitative convergence for a dense set of initial data via a robust entropy method, and (3) a De Giorgi level-set iteration, to establish quantitative time-asymptotic thermalisation for general initial data. In particular, we prove quantitative convergence for arbitrarily cold initial data.

This article is devoted to the inverse problem of uniquely determining different classes of coefficients in a hyperbolic equation with space-time nonlocal attenuation from a local source-to-solution map. Our objective is to exploit the nonlocal attenuation mechanism to address inverse coefficient problems that remain open, or are even fundamentally intractable, for linear and nonlinear hyperbolic equations without attenuation. More precisely, by leveraging the interactions induced by nonlocal attenuation, we simultaneously determine general classes of leading and lower order coefficients depending on both space and time. In addition, we formulate our results using data collected on disjoint sets, thereby extending the existing theory and introducing the notion of data on sets disjoint in both space and time for hyperbolic equations. Our approach builds on key features of nonlocal operators, such as memory effects and the unique continuation principle, while integrating the interplay between local and nonlocal operators along with structural properties of hyperbolic equations.

We consider the focusing, $L^2$-supercritical and $\dot{H}^2$-subcritical nonlinear fourth-order Schrödinger equation. The scattering of radially symmetric solutions below the ground state was proved by Guo [Comm. Partial Differential Equations (2016)] and Dinh [Nonlinearity (2021)]. In this paper, we extend the scattering results to group-invariant solutions. In Komada–Masaki [Nonlinearity (2024)], the scattering of group-invariant solutions below the ground state was proved under a certain hypothesis. To remove the hypothesis, we establish the non-optimal scattering result for general solutions, where the threshold of action is less than certain fraction of the action of the ground state. This result is analogous to that in Pausader–Shao [J. Hyperbolic Differ. Equ. (2010)] for the $L^2$-critical nonlinear fourth-order Schrödinger equation.

We establish a local well-posedness theory for the $n-$dimensional Boussinesq equation with a linear restoring force in a modified Sobolev spaces $H_\omega^s$ and Bourgain-type spaces $X_\omega^{s,b}$, which are adapted to its modified dispersion relation. The restoring force introduces a new phase function whose behavior in the low-frequencies is different from ones of the classical Boussinesq equation, that in turn, affects the associated energy functional. We prove bilinear and trilinear estimates for quadratic and cubic nonlinearities, respectively, which in turn imply local well-posedness under suitable conditions on $s$ and $\omega$. To our knowledge, this is the first well-posedness result for this model.

We investigate forward self-similar solutions to the incompressible Navier–Stokes equations with homogeneous initial data $u_0(x)=\sigma(x/|x|)/|x|$, $x\in\mathbb R^n$, focusing on the critical four-dimensional case. For the existence theory in dimensions $3\leq n\leq5$, we develop a Galerkin scheme for the Leray profile problem, featuring a decomposition of the self-similar data and nonlocal heat kernel effects across spatial scales. Exploiting the structural properties of the data and the coercivity of the Leray operator, we derive uniform a priori estimates, pass to the limit, and obtain weak forward self-similar solutions. In the 4D case, we prove new Stokes-type estimates for the linear Leray system. The inverse Leray operator admits an integral representation as a singular integral operator; its $L^p$-boundedness follows from Calderón–Zygmund theory. Combined with a 4D compactness estimate for the convective term, this closes the regularity iteration. For spatial decay, we reformulate the problem in weighted spaces. The non-integrability of the low-frequency kernel causes a logarithmic loss under $\sigma\in W^{1,\infty}(\mathbb S^3)$. With $\sigma\in C^{1,\gamma}(\mathbb S^3)$, $0<\gamma<1$, frequency-localized estimates remove this loss and yield sharp decay.

We investigate the global solvability and the large-time asymptotic behavior of solutions to the Navier–Stokes–Maxwell system in two and three space dimensions. The system describes the interaction between a viscous incompressible fluid and an electromagnetic field through the Lorenz force and Ohm's law. The system exhibits a parabolic–hyperbolic coupling in which the dissipative Navier–Stokes system interacts with the hyperbolic Maxwell system. The Maxwell system is damped through Ohm's law with the Maxwell correction. For sufficiently small initial data with suitable Sobolev regularity, we establish the global well-posedness of strong solutions and derive optimal decay rates for the solution and its higher-order spatial derivatives. In addition, we show that the electric field decays faster by the extra factor $(1+t)^{-1/2}$ compared to the velocity and magnetic components. One of the key ingredients of our analysis is a detailed study of the linearized system, which reveals refined decay properties of the electromagnetic components. These linear estimates play a crucial role in the analysis of the two-dimensional case, allowing us to overcome the logarithmic loss arising from the borderline decay estimates. For the nonlinear system, our approach is based on a time-weighted energy method, specifically designed to capture the distinct dissipative properties of the fluid and electromagnetic components. A suitable compensating functional is introduced to exploit the coupling between the electric and magnetic fields and thereby recover the missing dissipation of the magnetic field. Combined with careful analysis of the nonlinear terms, these estimates yield our desired result.

We construct a class of global naked singularity solutions for the $(3+1)$-dimensional spherically symmetric Einstein–Maxwell-charged scalar field equations. The solutions arise from $C^{1,\alpha}$ asymptotically flat one-ended characteristic initial data with $0<\alpha\ll1$ and develop a curvature singularity that is not enclosed by an event horizon. The resulting spacetimes are genuinely charged, with a nonzero Maxwell field that satisfies a quantitative positive lower bound up to the singularity and admits an explicit asymptotic profile. As part of the construction, we develop a general framework for constructing naked singularity exteriors, namely, the causal future region of the past light cone emanating from the singularity. The framework accommodates a broad range of singular characteristic data and, in particular, allows for singular behavior of the Maxwell field near the singularity.

This paper investigates the fractional Laplacian equation in half-spaces, focusing on cases with nonlocal Neumann boundary conditions. We consider both single equations and systems, and establish two key principles: the decay at infinity principle and the narrow region principle. These foundational results, combined with the direct method of moving planes, allow us to derive symmetry and monotonicity properties for decaying solutions. To the best of our knowledge, these are the first results concerning the symmetry of solutions to fractional Laplacian equations with nonlocal Neumann conditions.

For complex Hessian equations on compact Kähler manifolds, we give an alternative proof of the gradient estimate without using blow-up argument, and clarify its dependence on the right-hand side. In addition, using the similar approach, the case of Kähler manifolds with boundary is also investigated.

We formalize in Lean 4 the Galerkin construction of global Leray–Hopf weak solutions for the unforced two-dimensional Navier–Stokes equations on arbitrary bounded open domains with no-slip boundary conditions. For every initial datum in the solenoidal \(L^2\) velocity space, the solution has a weakly continuous velocity path, satisfies the weak equation locally in time, and obeys the energy inequality at every time. For rectangles, we also formalize finite-horizon solutions with continuous forcing in the dual energy space. The development includes divergence-free graph spaces, compact spectral coordinates, transport cancellation, Ladyzhenskaya estimates, and simultaneous space–time compactness. The abstract Hilbert-space results apply to evolution equations with a compact energy-to-velocity embedding and a skew trilinear nonlinearity.

We study a non-isothermal model for incompressible magnetoviscoelastic fluids in $\mathbb{R}^N$, $N=2,3$, coupling the incompressible Navier-Stokes equations with the evolution equations for the deformation tensor, temperature, and magnetization. The material coefficients are allowed to depend on the temperature, and no artificial diffusion is imposed on the deformation tensor, leading to a coupled hyperbolic-parabolic system. We establish local well-posedness of strong solutions for sufficiently regular initial data and prove global existence for small perturbations of a constant equilibrium. The global analysis relies on a suitable auxiliary variable combining the velocity and the inverse deformation tensor, which reveals a hidden dissipative structure and allows us to derive uniform higher-order energy estimates for the coupled system.

We study stable solutions of the Allen-Cahn equation with homogeneous Neumann boundary condition on Riemannian manifolds with boundary in the regime where the transition layers meet the boundary orthogonally. Under an a priori bound for the enhanced second fundamental form in the transition region, we establish uniform second-order Hölder estimates and quantitative mean curvature decay for the nodal hypersurfaces up to the boundary in ambient dimensions at most 10. The proof adapts the technique pioneered by Wang-Wei, adding a further correction determined by the geometry of the ambient boundary. Its leading contribution to mean curvature cancels on the nodal set but survives on non-zero level sets in the transition region, whose mean curvature need not decay uniformly. This phenomenon has no interior analogue and is exhibited by stable solutions constructed by Kowalczyk.

We consider singular Liouville systems on closed Riemann surfaces with a common sign-changing coefficient. Direct continuation from a positive coefficient is obstructed by the unavoidable appearance of degenerate zeroes along such a homotopy. We instead identify an intrinsic ray in the parameter space on which the system admits synchronized solutions and reduces exactly to a scalar sign-changing mean-field equation. Combining this reduction with the variational existence theory of De Marchis–López-Soriano–Ruiz and the nodal singularity compactness theorem of Wu, we obtain existence for every noncritical parameter on this ray. Positive singular sources are allowed on the nodal curve. In particular, when the surface has nonpositive Euler characteristic and every negative component of the coefficient is a disk, the topological hypothesis required by the scalar min–max construction is automatic.

We study the three-dimensional incompressible Navier–Stokes equations with a componentwise nonlinear damping term of the form $\alpha\sum_{k=1}^3 u_k^{2m+1}e_k$ in Fourier and Fourier– Gevrey spaces based on $\mathcal{X}^0(\mathbb{R}^3)$. We first establish local-in-time existence and uniqueness of solutions and obtain the corresponding unique maximal solutions. We then derive blow-up criteria in the Fourier–Gevrey framework, including an integral blow-up criterion in a weaker exponential weight. Moreover, we establish a quantitative lower bound for the solution near a possible finite maximal existence time. By iterating the loss of exponential weight, we finally obtain a corresponding lower bound in the unweighted Fourier space $\mathcal{X}^0(\mathbb{R}^3)$.

We determine the small-diffusion limit of the principal eigenvalue for a one-dimensional cooperative elliptic system with component-dependent drifts and Neumann boundary conditions. For scalar equations, this limit is determined by local data at the drift equilibria and boundary points. We show that such localization fails for systems: switching between components can make some intervals spectrally relevant. We first prove that the logarithmic transforms of all component eigenfunctions converge to the same function, which solves a scalar Hamilton-Jacobi equation. The associated Aubry set determines the asymptotic behavior of the principal eigenvalue. Under nondegeneracy assumptions, this set decomposes into isolated points, regular intervals, and composite intervals containing common drift zeros. We then identify the effective value of each class through local Ornstein-Uhlenbeck spectra and interval transport problems, and establish that the limit of the principal eigenvalue is the minimum of these class values. A unique minimizing class also determines the common logarithmic eigenfunction profile.

We study two-dimensional steady incompressible Euler flows with finite total curvature. A distinguished family of such flows arises from finite-Morse-index solutions to semilinear elliptic equations in two dimensions. We establish three main properties of finite-total-curvature flows in the $L^\infty\cap H^1_{\rm loc}$ class. First, in the plane and the half-plane, every such flow without stagnation points is a parallel shear flow. Second, in the plane, the half-plane, and the infinite strip, the pressure converges uniformly to a constant at each end at infinity. Third, in the plane, the half-plane, the infinite strip, and the periodic strip, we establish sharp lower bounds for the total curvature in terms of the pressure oscillation. The rigidity result in fact extends to a broader class of flows: in the plane and the half-plane, any steady Euler flow without stagnation points is a parallel shear flow provided that $\nabla P\in L^q$ for some $1\le q\le 2$. Combined with standard elliptic estimates for the pressure, this result yields a concise new proof of Hamel and Nadirashvili's rigidity theorems, with the regularity assumption sharpened optimally to $H^1_{\rm loc}$. The pressure serves as a unifying quantity throughout the analysis.

We establish Morrey-type conditions ensuring that a finite signed Radon measure belongs to the dual of the energy space of solutions to nonlinear parabolic equations of $p$-Laplacian type. More precisely, for the parabolic cylinders $Q_{r,r^p}(z):=B_r(x)\times(t-r^p,t+r^p)$, we prove that \[ |\mu|\bigl(Q_{r,r^p}(z)\cap(\Omega\times (0,T))\bigr) \leq Mr^{n+p-\vartheta}, \qquad \vartheta<p, \] implies duality, and this threshold is sharp in general. More generally, for cylinders with time length proportional to $r^q$, $q>1$, the sufficient threshold is $\vartheta<\min\{p,q\}$. Finally, we discuss the resulting variational, energy, and renormalized solution theories for nonlinear parabolic equations with measure data, relating our conclusions to the existing literature.

In this paper we investigate existence and multiplicity of solutions for a critical growth anisotropic quasilinear elliptic system, depending on two real parameters $\lambda,\,\mu$, that is coupled through a subcritical perturbation term and involves a negative critical part. We identify a specific scaling for the system and introduce a parameter $\gamma$ associated with this scaling, which governs the geometry of the corresponding variational functional. We establish three distinct types of multiplicity results, depending on the regimes $\gamma = 1$, $\gamma > 1$, and $\gamma < 1$. The multiple results of multiplicity of solutions are based on a recent abstract critical point theorem for symmetric functionals on product spaces when $\gamma\geq1$ and on a truncation argument for $\gamma<1$.