Papers

We prove global in time regularity of solutions of the Navier-Stokes equations defined in the complex space.

The paper is concerned with $\alpha$-stable SDEs with singular time-inhomogeneous general drift. Our drifts satisfy a condition that is close to the minimal possible scaling-invariance and can introduce local blow-ups of the type arising in some particle systems with strong attracting interactions.

We study the flow of an isothermal compressible Newtonian fluid around a body that performs a (time-independent) rigid motion. We derive a weak-strong uniqueness principle, and show that in the low Mach number limit, the governing equation is well approximated by the Navier-Stokes equations for incompressible rotating flow. Both results are based on the derivation of a relative energy inequality for weak solutions to this exterior-domain problem.

For the family of Dirichlet stadia $S_t$ whose rectangular part has height $\pi$ and half-length $\pi t/2$, $t \in [1,2]$, we show that for Lebesgue almost every $t$ there exist real eigenfunctions $u_j$ and a smooth mean-zero physical observable $Q$ for which $\langle Q u_j,u_j\rangle$ has a non-zero subsequential limit. Consequently, along the same subsequence, the eigenfunction mass fails to equidistribute on a fixed region whose relative boundary in the interior of the stadium is smooth. This proves a physical-space strengthening of Hassell's non-QUE theorem for generic stadia, and thus gives an affirmative answer to Tao's question in Hassell's generic setting. The proof uses the classification of generic stadia in Hassell's argument. In each of the resulting cases, we construct an appropriate physical observable $Q$ that converts Hassell's phase-space obstruction to QUE into physical-space non-equidistribution.

We show that any three-ball with mean convex boundary contains an embedded free boundary minimal disk. Moreover, when the three-ball is a strictly convex domain with nonnegative Ricci curvature (for instance, a compact convex domain in Euclidean three-space), we prove the existence of at least three embedded free boundary minimal disks. Our approach is based on a multiplicity-one theorem for the free boundary Simon-Smith min-max theory.

The deterministic incompressible Navier-Stokes equations are physically incomplete: any viscous fluid at finite temperature must exhibit thermal fluctuations whose form is dictated by the fluctuation-dissipation relation. We formulate the stochastic Navier-Stokes equations with the kinematically selected deformation Laplacian on compact Riemannian manifolds with strictly negative Ricci curvature. The fluctuation-dissipation relation, derived from a topological (Poincaré lemma) argument, uniquely determines the noise from the viscous operator. For the spectrally truncated system, we prove that the unique stationary distribution is the Gibbs measure (Gaussian in the mode amplitudes, because the nonlinear convective terms preserve energy), and that convergence to equilibrium is exponentially fast with rate at least $2\nu\lambda_\Def$, where $\nu$ is the kinematic viscosity and $\lambda_\Def$ is the spectral gap of the deformation Laplacian. The spectral gap satisfies $\lambda_\Def \geq \kappa^2$ when $\Ric \leq -\kappa^2 g$, and is independent of the volume of the domain. On flat space, the analogous thermalisation rate vanishes in the infinite-volume limit. The equilibrium velocity-velocity correlation function decays exponentially in geodesic distance, in contrast to the algebraic decay on flat space. These results provide a rigorous statistical-mechanical foundation for viscous fluids on negatively curved manifolds and illustrate how the geometry of the domain controls not only the deterministic dynamics but also the approach to thermal equilibrium.

This paper investigates a class of Schrödinger-Poisson systems in $\mathbb R^3$ featuring the (p,q)-Laplacian operator and a combination of critical and subcritical nonlinearities in the Schrödinger equation while the m-Laplacian and a power type nonlinearity in the Poisson's one. We consider both the attractive and repulsive cases, which correspond to different signs in front of the nonlocal term. While most existing literature relies on auxiliary functionals or specialized techniques to overcome the lack of compactness and ensure the boundedness of Palais-Smale sequences, we employ a direct variational approach. By applying the Mountain Pass Theorem and concentration compactness principles, we establish the existence of positive solutions. A careful analysis is conducted to identify the parameter ranges for which the Mountain Pass level falls within the compactness threshold, despite the technical challenges posed by the unbalanced growth of the operator and the nonlocal interaction.

In this article, we study the existence of normalized solutions to the following mixed nonlinear Choquard equation with exponential growth \begin{align*} \left\{ \begin{aligned} \mathcal{L}u+\lambda u \; &=\; \Lambda(I_{\alpha}\ast F(u))F'(u), \quad \text{in }\mathbb{R}^{2}, \int_{\mathbb{R}^{2}}|u|^{2}\,dx \; &=\; a^{2}, \end{aligned} \right. \end{align*} where $\mathcal{L}= -\Delta+(-\Delta)^s$, $0<s<1$, $a>0$, $I_{\alpha}$ is the Riesz potential of order $\alpha\in (0,2)$, $\Lambda>0$ is a parameter and $\lambda\in \mathbb{R}$ appears as a Lagrange multiplier. Here, the nonlinearity $F$ has exponential growth in $\mathbb{R}^{2}$. Using variational methods, we prove the existence of normalized solution in the Pohožaev manifold. Moreover, we discuss the regularity result and the construction of the Pohožaev identity, essential for the existence. \keywords{Normalized solutions; Nonlinear Schrödinger equations; Choquard nonlinearity; Critical exponential growth; Trudinger-Moser inequality}

We study a system of nonlinear Schrodinger equations under the mass-resonance condition and provide a complete description of the asymptotic behavior of small solutions in two space dimensions. Our analysis is based on a detailed study of an associated system of ordinary differential equations governing the asymptotic profile. We establish a phase-amplitude representation for this ODE with arbitrary initial data, where the amplitude is expressed explicitly in terms of Jacobi elliptic functions and the phase is given by elliptic integrals of the third kind. As a consequence, we obtain a fully explicit characterization of the asymptotic profile for the original PDE. In particular, the long-time behavior is described by a modified-scattering-type profile whose profile evolves according to the above integrable structure. While elliptic-function-type asymptotics were previously constructed for special solutions in final-state problems, the present work provides the first complete characterization, in two space dimensions, of asymptotic dynamics associated with arbitrarily small initial data through elliptic-function-type profiles.

In this paper, we investigate critical mass threshold for the Patlak-Keller-Segel-Navier-Stokes system on the two-dimensional whole space and obtain global existence of strong solutions if the initial mass is less than or equal to $8\pi$, regardless of the initial norm of the velocity. One new observation is that the local mass of the density function rearrangement satisfies a good inequality that is independent of velocity; and then an improved maximum principle is applied by choosing a nice auxiliary function.

We derive a fully 3-dimensional Summation-By-Parts scheme for a class of linear wave equations on hyperboloidal slices that meet future null infinity on a Minkowski background. The scheme is derived in spherical polar coordinates, with a major strength being that it is provably stable and allows having grid points at the origin and on the $z$-axis, despite coordinate singularities, and at infinity, by introducing compactification followed by rescaling. Reducing it to the standard Cauchy problem, or on finite spacelike slices with an outer boundary, will follow a similar procedure. Interesting relations are obtained between the rescaling and compactification factors that simplify the equations, and the conditions on constraint addition terms are discovered to maintain symmetric hyperbolicity. Numerical implementation is achieved using finite-difference methods at second-order accuracy, which can be generalized to higher-order or spectral accuracies as well. Dissipation operators are given a more abstract treatment, which makes it possible to define them everywhere in the domain, including at the boundary points, in curvilinear coordinates, such that they satisfy the dissipative property (DP) in our energy norms. These generalizations reduce to the well-known Kreiss-Oliger dissipation operators whenever defined on a Cartesian grid in the bulk and satisfy the DP in the standard $L^2$-norms. We also propose new norm convergence tests that produce more accurate outputs. Promising results are obtained, giving hope for application to fully nonlinear systems, like the Einstein Field Equations, and extracting the resulting gravitational waves free of systematic errors or gauge ambiguities.

We consider singular problems for a general class of quasilinear hyperbolic systems that involve two a priori independent stiff parameters. We argue that such situations may lead to the rapid development of small-amplitude spatial oscillations of small wavelength starting from arbitrarily smooth initial data. Despite this phenomenon we provide sufficient conditions on initial data that secure the uniform control of solutions and show strong convergence in the singular limit of small stiff parameters. We apply our general theory to the rapidly rotating shallow-water system with bottom topography, and to hyperbolic systems stemming from a constraint-relaxation strategy applied to dispersive models for the propagation of water waves, specifically the Benjamin-Bona-Mahony, Boussinesq-Peregrine and Serre-Green-Naghdi equations.

This paper studies an inverse coefficient problem for a time-fractional damped wave equation on a finite time interval. The aim is to determine two spatially varying coefficients, namely the fractional damping coefficient and the zeroth-order potential, from the associated Dirichlet-to-Neumann (DtN) map. We first prove the well-posedness of the forward problem for boundary data with sufficient regularity to admit a suitable lifting. The main result is a uniqueness theorem showing that if two coefficient pairs give rise to the same DtN map, then the corresponding coefficients coincide almost everywhere in the domain. The proof is based on a convolution-type integral identity that eliminates the unknown boundary traces, the construction of high-frequency beam solutions adapted to the singular kernel of the Caputo derivative, and a detailed asymptotic analysis of the resulting fractional convolution terms. This result extends classical uniqueness results for hyperbolic inverse problems to a fractional-order model with damping and provides a theoretical basis for related applications involving memory and viscoelastic effects.

This paper investigates the long time well-posedness of the Benjamin-Ono (BO) equation with quasi-periodic initial data in analytic space. We demonstrate that the Lax operator of the BO equation exhibits a trivial spectrum, and elucidate the fundamental mathematical obstacles to establishing global well-posedness. Furthermore, by employing Tao's gauge transformation in conjunction with the Birkhoff Normal Form method, we extend the lifespan of the solutions to $O(\epsilon^{-3})$.

We study a high-velocity inverse scattering problem for nonlinear Schrödinger equations with spatially dependent nonlinearities in dimensions $d\ge3$. We consider the whole mass-supercritical and energy-subcritical range, including the endpoint cases. By introducing a moving frame adapted to highly boosted initial data, we construct the scattering operator for a class of large incoming states generated by Galilean boosts. The key observation is that, although the boosted data become large in Sobolev norms, the nonlinear interaction becomes effectively weak at high velocity due to rapid spatial separation. Using the resulting high-velocity asymptotics, we derive a reconstruction formula for the X-ray transform of the coefficient. As a consequence, we prove that the scattering operator uniquely determines both the nonlinearity exponent and the spatial coefficient. Our results extend previous work of Watanabe to all dimensions $d \ge 3$, include the endpoint nonlinearities, and replace the repulsiveness and radial monotonicity assumptions on the coefficient by suitable decay conditions.

In this article we study periodic orbits of an electron attracted by a proton subject to Lorentz, electric, and Euler forces where each of them is allowed to depend periodically on time. This setup is motivated by the elliptic restricted three-body-problem where the Lorentz force corresponds to Coriolis force, the Coulomb force is replaced by the gravitational force, and the electric force of an external source is a combination of centrifugal forces and gravitational forces of other bodies. This is a singular version of a Euler-Hamilton system as discussed in [FW26b]. The singularity is due to collisions of the electron with the proton, respectively of two masses. Due to the possibility of collisions this problem has to be regularized. We show how periodic collisional solutions of this problem can be detected variationally in a non-local Lagrangian setup as well as in a non-local Hamiltonian setup.

We show that any entire, capillary minimal graph in a half-space must be linear in low-dimensions or, more generally, when some tangent cone at infinity does not split off a vertical line. We also show that the regular set of any entire, capillary-minimizing hypersurface must be connected, and we discuss connections with the one-phase Bernoulli problem.

Aubin–Talenti bubbles describe the decaying positive solutions of the zero-frequency critical Lane–Emden equation in Euclidean space. By appealing to bifurcation methods, Dancer constructed in his seminar paper \cite{DancerSolution} positive-frequency solutions to the Lane–Emden equation which decay in the noncompact directions and are periodic in one direction. Alternatively, we give in this paper an energy-based variational construction of such Dancer-type solutions via the semivirial-vanishing geometry developed in author's recent work for studying focusing NLS on waveguide manifolds. The main new ingredient is a strict sub-bubbling estimate below the Euclidean Sobolev threshold. Unlike the usual Brezis–Nirenberg mechanism, no lower-order focusing perturbation is available in our model. Instead, the energy drop is produced by the bounded periodic direction: truncating a Euclidean bubble to one period removes a leading-order part of the gradient tail, while the nonlinear tail is of lower order. This restores compactness of minimizing sequences and yields normalized ground states for every prescribed mass, thereby answering an open question from \cite{Luo_energy_crit}.

In this work, we address the stability of $N$-soliton solutions to the completely integrable modified Camassa–Holm (mCH) equation. Recently, Li, Liu, and Zhu (Math. Ann. 392 (2025), 899–932) established the orbital stability of 2-soliton solutions in $H^4(\mathbb{R})$ with respect to the solution $u$ and highlighted the stability of mCH $N$-soliton solutions remains an urgent challenge. Motivated by their work, we systematically investigate the stability of mCH $N$-solitons. We first employ the bi-Hamiltonian structure of mCH to construct a novel hierarchy of explicit conservation laws with well-defined regularity domains. Then by formulating an appropriate Lyapunov functional, we apply the Inverse Scattering Transform to conduct a rigorous spectral analysis on the recursion operators. Finally, we demonstrate that the mCH $N$-solitons are non-isolated constrained minimizers of a variational problem. Our analysis proves that the $N$-soliton solutions of the mCH equation are both dynamically and orbitally stable in $H^{N+1}(\mathbb{R})$. Notably, when reduced to the 2-soliton case, our framework establishes stability in $H^3(\mathbb{R})$, which improves upon the existing regularity threshold.

This paper establishes gradient estimates and Liouville-type theorems for the \(\Phi\)-Laplacian equation \(\Delta_{\Phi}(u) = G(|\nabla u|^2)\) on complete Riemannian manifolds and its parabolic counterpart \(\partial_t u = \Lambda_{\Phi}(u)\) on compact Riemannian manifolds. Using a nonlinear \(\Phi\)-Bochner formula and the Nash-Moser iteration technique, we prove local gradient bounds under the lower bound assumption of Ricci curvature and suitable conditions on \(\Phi\) and \(G\), which leads to Liouville theorems for global solutions. For the parabolic case, we employ the maximum principle to derive gradient estimates on compact Riemannian manifolds, and subsequently obtain Liouville-type results. Our work provides a unified framework that generalizes prior results for \(p\)-harmonic functions and other quasilinear equations.

We are interested in the asymptotic behavior of solutions toward strong rarefaction waves for a parabolic-hyperbolic system arising from chemotaxis. Suppose that the Riemann problem to the corresponding inviscid system admits rarefaction waves. We show that if the initial data is a small perturbation of an approximate rarefaction wave, then the Cauchy problem has a unique global solution that converges to the rarefaction wave asymptotically in time. The waves can be either a single rarefaction wave or a superposition of two rarefaction waves. Furthermore, the stability results hold regardless of the wave strengths. The proofs are based on the energy method, where the key observations are the monotonicity of the approximate rarefaction waves with respect to both space and time, and the appropriate replacement of spatial derivative of the hyperbolic component with the parabolic component in proper forms.

For a properly embedded Willmore surface $\Sigma$ in $\mathbb R^3$, we prove that if the scale-invariant second fundamental form is sufficiently small near infinity, the surface has finitely many ends. Moreover, if this scale-invariant quantity vanishes at infinity, or if there is only one end, the total $L^2$-norm of the second fundamental form is finite.

Suppose $\mathcal{V}$ is a class of stationary integral $n$-varifolds in $B^{n+k}_2(0)\subset\mathbb{R}^{n+k}$ which is closed under weak limits, homotheties, rotations, and disjoint decomposition, and suppose that $\mathcal {V}$ satisfies an $\epsilon$-regularity property near planes of (integer) multiplicity $\leq Q\in \{2,3,\dotsc\}$. This last condition, more precisely, requires that there be a constant $\epsilon = \epsilon({\mathcal V}, Q) \in (0, 1)$ such that if $V\in \mathcal{V}$ is, in the unit cylinder ${\mathbb R}^{k} \times B_{1}^{n}(0)$, $\epsilon$-close as varifolds to the plane $\{0\} \times {\mathbb R}^{n}$ taken with multiplicity $\leq Q$ then, in the half-cylinder ${\mathbb R}^{k} \times B_{1/2}^{n}(0)$, $V$ is represented by the graph of a Lipschitz multi-valued function over $B_{1/2}^{n}(0)$ with uniform quantitative estimates of a $C^{1,\alpha}$ nature. For any varifold in such a class $\mathcal{V}$, we prove that the set of branch points with density $\leq Q$ has Hausdorff dimension $\leq n-2$. By choosing suitable $\mathcal{V}$, a direct consequence of this result and the recently established regularity theorems of the second and third authors (one of which being joint with Becker-Kahn) is that if $V$ is a stationary integral $n$-varifold which is either: (a) represented by the graph of a $2$-valued Lipschitz function; or (b) codimension one, stable, and with no classical singularities of density $<Q$, then the Hausdorff dimension of the density $Q$ branch set ($Q=2$ in (a)) is at most $n-2$. Our proof utilises the planar frequency function introduced by the first and third authors in their work on area minimising currents, and thus does not require the Almgren center manifold for the analysis of branch points except in a single, geometrically canonical case where the center manifold satisfies additional simplifying properties.

The main aim of this paper is to establish the Hölder continuity and the Harnack inequality for weak solutions to Dirichlet problems associated with superposition operators of mixed fractional order, thereby complementing our previous work \cite{BGKL2026}. To achieve this, we extend the De Giorgi–Nash–Moser theory to the framework of superposition operators by introducing a novel nonlocal superposition tail, which appears to be the first contribution of its kind in the literature. The obtained results are new even in the classical linear case $p=2$, thereby illustrating the broader applicability of the analytical techniques developed in this work. As intermediate steps toward the proof of the main results, we also establish a logarithmic estimate for weak supersolutions, local boundedness for weak subsolutions, a weak Harnack inequality for weak supersolutions, an expansion of positivity for weak supersolutions, and tail estimates for weak solutions.

The Boltzmann equation describing the transport of electrons in semiconductor devices with an external electrostatic potential is considered when the spatial variable is in a torus and the wave vector is in the Brillouin zone. We prove the exponential time decay of solutions towards the global equilibrium in a weighted $L^2$ space. Our result holds for wide classes of energy functions of electrons and external electrostatic potentials, and the estimates on the rate of convergence are explicit and constructive. We remove the close-to-equilibrium assumption on the initial datum and the parabolic band approximation assumption that were required in previous works. The technique is based on the construction of a suitable highly non-linear Lyapunov functional by modifying the relative entropy. The analysis benefits from uniform bounds for the solution in terms of the global equilibrium.

We present a variational approach to ferronematics in a three dimensional setting. The ferronematic energy functional is described by two established theories: the Landau-de Gennes energy to explain the nematic part, the micromagnetic energy to explain the magnetic part, and coupling energies between them. We explicitly include the nonlocal stray field energy in a bulk setting and the coupling energy accounting for the nematic and stray field interaction. We prove the existence of an energy minimizer for the introduced ferronematic energy functional in a bulk setting. We then provide a reduced local ferronematic energy in a two-dimensional setting via $\Gamma$-convergence.

Let $\Delta$ be the Dirichlet Laplacian on a bounded domain $\Omega \subset \mathbb{R}^{N}$, and let $(-\Delta)^\alpha$ be the associated spectral fractional Laplacian with $\alpha \leq 1, \ \rho <2$. For general bounded domains with $C^2$ boundary, we prove a symmetry formula for $\alpha <1/2$, extending a result previously proven on rectangles for $\alpha <1$. As a consequence of this formula, well-posedness results are proven for the structurally damped plate equation $$u_{tt}+\Delta^2u+(-\Delta)^\alpha u_t=0$$ subject to Dirichlet or moment boundary control. For rectangular domains with $\alpha <1$, we prove boundary null-controllability results. For $\alpha <1/2, \ \rho \leq 2$, Dirichlet null controllability is proved for the unit disk in $\mathbb{R}^2$. This analysis then extended to the classical case, $\alpha =1$, on rectangles, where higher regularity is required for Dirichlet control.

We develop a nonlinear potential theoretic framework for Schauder estimates for vector-valued solutions of a broad class of nonautonomous variational problems at nearly linear growth. Our approach naturally embraces the variable exponent as well as the Double and Multi phase setting, yielding new regularity results in basic models and recovering optimal regularity recently established in specific cases.

We review the kinetic theory of one-dimensional nonlinear oscillator chains, of which the most famous example is the Fermi-Pasta-Ulam-Tsingou equation. We provide detailed, though not rigorous, accounts of the microscopic to mesoscopic, and mesoscopic to macroscopic limits: derivation of the kinetic wave equation and hydrodynamic limit. We also present the state of the art of the mathematical theory, including proofs. We discuss the connection to two famous problems of Mathematical Physics: the Fermi-Pasta-Ulam-Tsingou paradox, and the derivation of Fourier's law. Finally, many open problems and possible directions for future research are proposed.

We investigate the ill-posedness mechanism of the hard-sphere Boltzmann equation in $H_{x}^{s}$ Sobolev space. Via a direct construction, we prove a strong-weak type ill-posedness result in the low-regularity regime $s<1$, establishing a sharp threshold in connection to the local $s>1$ well-posedness result [11]. Instead of originating from the large-velocity growth of the collision kernel, this illposedness is generated by the loss term and dispersive effects. Consequently, we prove a dispersion-driven nonlinear instability mechanism for the hard-sphere Boltzmann equation, and provide a capstone of the ill-posedness series [18,20].