Papers

We study an inverse problem for the time-dependent nonlinear fourth-order Schrödinger equation on both compact Euclidean domains and compact Riemannian manifolds, (say) $M$. This model arises in nonlinear fiber optics and the theory of optical solitons in gyrotropic media. Our main objective is the identification of unknown coefficients from the associated source-to-solution map, which assigns to each source term $f$, supported in $(0, T)\times \Gamma$, the corresponding solution $u$ restricted to the same set, where $\Gamma \subset M$ is a neighborhood of $\partial M$. We prove that the zeroth-order term, the second-order coefficient, and the nonlinear coefficient are uniquely determined by this map. Moreover, the recovery of the symmetric second-order tensor reduces to the inversion of a divergent beam transform.

In this paper, we establish Riesz–Fejér type inequalities for the $\alpha$-harmonic functions $f=P_{\alpha}[f^*]$ in $\mathbb B^n$, where $f^*\in L^{p}(\mathbb{S}^{n-1})$ and $1<p<\infty$. More precisely, for $n\geq2$ and $\alpha>-1$, we prove the existence of a constant $\mathcal{C}_{n,p,\alpha}$ such that $\int_{-1}^{1} |f(r\eta)|^p(1-r^2)^{n-2}\,dr \leq \mathcal{C}_{n,p,\alpha} \int_{\mathbb S^{n-1}}|f^*(\xi)|^p\,d\sigma(\xi)$. Moreover, in the range $\alpha>\max\left\{ -\frac{n-1}{p},\,n-2-\frac{2(n-1)}{p} \right\}$, we determine the sharp constant explicitly. The result generalize and extend the corresponding results of Ahmed et al. (J. Math. Anal. Appl., 563:13, 2026), Hu et al. (Anal. Math. 51:15, 2025) and Long (arXiv: 2410.12137).

We consider a hydrodynamic model of flocking-type with pressure on the torus, with integrable interaction kernel and density bounded away from zero. We prove that, if an entropy weak solution exists, then its $L^2$ norm decays exponentially fast in time towards the mean values on the period. The proof relies on the study of a suitable energy functional that combines a strictly convex entropy for the system and a potential term, and this allows us to treat the nonlocal source term for a class of strictly positive convolution kernels in $L^1$.

We study the logarithmic Schrödinger equation \[ -\Delta u+V(x)u=u\log u^2,\qquad x\in\mathbb R^N,\ N\ge3. \] Since the logarithmic energy is not \(C^1\) on the natural space \(H_V^1(\mathbb R^N)\), direct invariant-set minimax arguments for sign-changing solutions are not available. We introduce an \(L^p\)-regularization perturbation, which restores a \(C^1\) variational structure while preserving the logarithmic nonlinearity, and prove via a limiting argument that the original equation admits infinitely many sign-changing weak solutions.

We prove polynomial bounds on the growth of high Sobolev norms of solutions to the defocusing fractional nonlinear Schrödinger equation with cubic and Hartree-type nonlinearities in two and three dimensions. Our result is based on the upside-down I-method and the method of higher modified energies. The analysis of the fractional case in higher dimensions is possible due to a higher-dimensional analogue of a resonance inequality, which allows us to control the nonresonant frequency contributions.

In recent years, the technique of convex integration has demonstrated that for some initial data (also referred to as 'wild initial data'), many PDE models of mathematical fluid mechanics allow for a multitude of admissible (weak) solutions. This paper is concerned with the question regarding how large the set of such wild initial data is for the isentropic Euler equations. We prove that wild initial data form a dense set in $L^r$ for any $r \in [1,\infty)$. In contrast to existing results in the literature, in this paper 'wild initial data' are data which give rise to infinitely many global-in-time weak solutions which are admissible in the sense that the local energy inequality holds. In other words, the set of initial data with infinitely many admissible weak solutions (independent of the choice of time interval) is dense. A novel part of the construction is that we use a measure-valued (dissipative) solution as the ansatz for the subsolution. This requires several new ideas, in order to ensure the required regularity of the subsolution and to obtain a lower bound for the density. Another crucial ingredient of the proof is that the (local) energy density and the energy flux are constructed as part of the convex integration scheme, in order to obtain solutions which adhere to the local energy inequality.

In this article we propose a novel thermodynamical model which couples the evolution of the thickness of a grounded shallow ice sheet with the evolution of its internal temperature. Both the grounded shallow ice sheet surface elevation and the ice internal temperature are subjected to physical constraints. The equations governing the evolution of the grounded shallow ice sheet thickness are degenerate, and the ice internal temperature evolves in a moving domain. First, we formally model the phenomenon under consideration by adopting strategies akin to those employed in the construction of diffuse-interface models. Second, we establish the existence of solutions for one such formal model by means of the penalty method, and we observe that the low regularity of the problem under consideration prevents us from obtaining a standard concept of solution.

We study the time-asymptotic stability of viscous-dispersive shock waves for the outflow problem of the barotropic Navier–Stokes–Korteweg equations, which describe viscous fluids with internal capillarity. Assuming that the far-field state is subsonic or transonic and that the velocity at the boundary is larger than the far-field velocity, we prove that the solution converges to the corresponding viscous-dispersive shock wave as $t \to +\infty$, provided that the shock amplitude and the initial perturbation are sufficiently small. The proof is based on the method of $a$-contraction with shifts (for viscous equations) introduced in \cite{KV17,KV21,KVW23}. A main difficulty comes from controlling the boundary effect of the viscous-dispersive shock wave, as well as the influence of capillarity near the boundary.

In this article we give regularity results for the geodesic X-ray transform on nonsmooth simple manifolds. As an application, we improve previous injectivity results for the geodesic X-ray transform acting on $L^p$ functions. The results are based on symbol smoothing arguments for the normal operator and mapping properties of pseudodifferential operators with low regularity symbols.

We give a unified analytic account of correlation emergence and the Epps effect in two coupled limit order books. The Epps effect is the empirical reduction in measured cross-asset log-return correlation at short aggregation scales, or equivalently the recovery of measured correlation as the aggregation interval increases. The model starts from a fixed-grid discrete random walk for order flow in operational time, with creation, cancellation and diffusion. A pair-trader coupling between the books is introduced at the level of order creation, and calendar time is imposed through separate observation clocks. We clarify how the operational-time model reduces to coupled reaction–diffusion equations with a moving reaction boundary defining the model log-mid-price. Using a regularised local-response representation of the coupling, we derive approximate closed-form expressions for realised correlations as a function of aggregation time. Here the Epps effect is shown to arise from two distinct mechanisms: asynchronous observation clocks (subordination), finite coupling response times, and their combination.

This paper investigates the nonlinear asymptotic stability and optimal decay rates of entropy waves for the Landau equation with physically realistic Coulomb interactions under general perturbations. We consider the infinite channel domain $\mathbb{R} \times \mathbb{T}^2$ in three dimensions, which possesses both one-dimensional and high-dimensional characteristics, thereby posing two primary analytical challenges: (i) for the one-dimensional Landau equation with Coulomb potentials, the absence of a spectral gap in the linearized operator has obstructed the derivation of wave pattern stability results with explicit time decay rates; (ii) in the study of contact discontinuities, the multidimensional case fundamentally differs from the one-dimensional setting due to lack of a key structural condition. We develop effective analytical approaches to treat those difficulties. To overcome the weak dissipation caused by the spectral gap deficiency, we implement a time-velocity interpolation technique to enhance dissipation and simultaneously construct coupled diffusion waves to compensate for the loss of time decay. To address the missing structural condition in higher dimensions, a novel transformation is introduced to recover the two-sided structural condition within the perturbation system. By developing a derivative-level transformation and a refined energy framework, we restore the necessary structural condition for derivatives, establish the optimal decay of the solution, and prove the stretched exponential decay of its non-zero modes. In contrast to previous methods that rely on artificial viscosity or the Navier–Stokes approximation, our approach directly leverages the intrinsic physical dissipation of the equation and its coupling with the microscopic kinetic component, ensuring broader applicability.

We consider the existence of vortex solutions to the complex sine-Gordon II (CSG2) equation, which can be viewed as an analogy of the Ginzburg-Landau (GL) equation. Using the nontrivial kernels $\eta_{\pm}$ of the linearized CSG2 equation at the standard degree-2 vortex solution $\Psi_2$, we show that it bifurcating to a one-parameter family of symmetry-breaking solutions $\Psi_{2,\alpha}$. Explicit formulas of these solutions are also available, from which we propose a new bilinear system for this equation. Our method can be generalized to higher degree case. Nondegenracy and stability of degree-1 solution are also proved. Finally, we formally discuss the Lyapunov-Schmidt reduction procedure for the multivortex solutions of the CSG2 and GL equation.

This note is concerned with the spreading speed of the predator component in a class of prey–predator reaction–diffusion systems with spatiotemporal heterogeneity depending on a moving variable. The main difficulty is that the full system lacks a direct comparison principle. We establish a pointwise estimate showing that the prey density is close to its carrying capacity wherever the predator density is sufficiently small. This allows us to compare solutions of the system with those of scalar Fisher–KPP equations in shifting environments. Consequently, we obtain an explicit formula for the spreading speed of the predator. In particular, for nonmonotone shifting environments, locking and nonlocal pulling phenomena may occur.

In this paper, we establish some geometric and analytic foundations for free boundary Brakke flows. Specifically, we (i) introduce unit-regular and cyclic free boundary flows and show that they are preserved under reflections and weak limits, (ii) prove that the support of free boundary Brakke flows satisfies an avoidance principle, and (iii) introduce free boundary inner and outer flows and prove the existence of matching free boundary Brakke flows. These results serve as general tools to analyze free boundary flows through singularities, and in particular will be applied in forthcoming work with Haslhofer, where we address the mean-convex neighborhood conjecture and uniqueness conjecture for free boundary flows through (half) cylindrical singularities.

In the paper [M. Keel, H. Smith, C.D. Sogge, Almost global existence for quasilinear wave equations in three space dimensions. J. Amer. Math. Soc. 17 (2004), no. 1, 109-153], the authors prove that for the 3-D quadratic quasilinear wave equation in exterior domains with homogenous Dirichlet boundary value and small initial data of size $\varepsilon$, the lifespan ${\bar T}_{\varepsilon}$ of the smooth solution fulfills ${\bar T}_{\varepsilon}\ge e^{C/\varepsilon}$. However, for the corresponding 2-D quadratic quasilinear wave equation in exterior domains with homogenous Dirichlet or Neumann boundary value, so far it is still open whether the expected sharp lifespan $T_{\varepsilon}\ge\frac{C}{\varepsilon^2}$ holds or not. In this paper, we will solve this open question. Our main ingredients include: introducing the suitable Friedlander radiation field for the 2-D linear wave equation in exterior domains with homogenous Dirichlet or Neumann boundary value, constructing the delicate approximate solution, and establishing some crucial space-time decay estimates for the solutions of 2-D quasilinear wave equation in exterior domains. On the other hand, for the radial symmetric solutions to a class of 2-D quadratic quasilinear wave equation in exterior domains, the upper bound of the lifespan $T_{\varepsilon}\le\frac{C}{\varepsilon^2}$ is derived and the sharp constant $C$ is also determined explicitly.

We study inverse boundary value problems for the fully nonlinear Monge-Ampère equation $ \det D^2u=a(x,u,\nabla u) $ in a bounded domain. Motivated by Calderón-type problems, we introduce nonlinear Cauchy data and investigate the recovery of $a$ from boundary measurements. Around a strictly convex background solution, under a sign condition ensuring invertibility of the linearized Dirichlet problem, we establish local well-posedness and differentiable dependence on boundary data. The first variation yields a linear elliptic operator with principal coefficient given by the cofactor of the background Hessian and lower-order coefficients determined by derivatives of $a$. An Alessandrini-type identity reduces equality of nonlinear measurements to an anisotropic Calderón problem. Conditional on uniqueness for this linear problem, we recover the first derivatives of $a$ along the background jet. Higher-order linearization yields integral identities whose inversion depends on a product-density property. We also establish unconditional uniqueness results in special settings. For $\det D^2u=f(x)$, local nonlinear Cauchy data determine $f$ in dimension two, and in higher dimensions under real-analytic anisotropic hypotheses. We compare this result with a planar small-data method and an all-dimensional large-data approach based on the Euclidean X-ray transform. For semilinear nonlinearities around $\Phi(x)=|x|^2/2$, Schrödinger inverse results recover the accessible $z$-jet. For equilibrium optimal-transport nonlinearities, linearization yields an isotropic conductivity equation and uniquely determines the transport density.

Our aim is to investigate the existence of local maximizers for the singular Trudinger-Moser functional $$ F_{\alpha}(u) := \int_{\Omega} \frac{e^{\alpha u^2}-1}{|x|^{a}} \mathrm{d}x,\;\;\; u \in W_0^{1,2}(\Omega), $$ restricted to the manifold $ \Sigma=:\left\{u\in W_0^{1,2}(\Omega):\|\nabla u\|_2=1\right\},$ where $a\in [0, 2)$, $\alpha\ge 0$ and $\Omega$ denotes a smooth bounded domain $\mathbb{R}^2$ containing the origin. Adimurthi and Sandeep (Nonlinear. Differ. Equ. Appl. 13, 2007) showed the following singular Trudinger-Moser type estimate \begin{equation}\nonumber \sup_{u \in W_0^{1,2}(\Omega), \, \|\nabla u\|_{L^2} \le 1} F_\alpha(u)<\infty\;\;\;\mbox{iff}\;\;\alpha \leq \alpha_a:=2\pi(2-a). \end{equation} In particular, the functional $F_{\alpha}$ is bounded on $ \Sigma$ whenever $\alpha \le \alpha_a$. In addition, Csató and Roy (Calc. Var. Partial Differ. Equ., 54, 2015) were able to ensure the existence of maximizers for $F_{\alpha}$ on $\Sigma$ when $\alpha\le \alpha_a$. In the supercritical regime $\alpha > \alpha_a$, the functional $F_{\alpha}$ becomes unbounded on $\Sigma$. Nevertheless, we prove that $F_{\alpha}$ still possesses local maximizers on $\Sigma$ beyond the critical threshold, at least for $\alpha > \alpha_a$ sufficiently close to $\alpha_a$. Our approach relies on a variational analysis near the set of maximizers associated with the critical parameter $\alpha_a$, together with a suitable local compactness argument. Our result improves and complements related findings due to Struwe (Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 5, 1988).

Resolution spaces play a central role in constructing solutions for nonlinear partial differential equations. One of the main goals in the area of nonlinear dispersive PDEs has been to construct effective resolution spaces which capture the known bilinear restrictions estimates for free solutions. In this paper we propose a new structure for the Schrödinger equation which effectively replicates the classical bilinear $L^2_{t,x}$ estimate. In addition, the new structure has the property that its "dual" is an effective candidate for a space for the forcing in the linear inhomogeneous Schrödinger equation, a feature that has been elusive so far in the literature. As an application, we show how these structures can recover the known global well-posedness results for derivative NLS with null structure, with Schrödinger Maps being one such model.

We prove a finite-scale supply–tax reduction for suitable weak solutions of the three-dimensional incompressible Navier–Stokes equations near the local regularity threshold. Along an admissible chain of parabolic windows, let B_k = A_k + C_k + D_k denote the scale-critical reservoir badness. The main PDE theorem says that if B_k stays outside a decay basin for many consecutive scales, then either cutoff/window leakage accumulates or the solution generates a positive cumulative amount of untaxed critical supply. The supply terms come from nonlinear flux, pressure transport, interpolation amplification, and pressure regeneration; the tax terms come from viscous dissipation, expected decay of old reservoir coordinates, and harmonic pressure decay. The theorem is deliberately finite-scale. It is not a proof of global regularity and it is not a construction of a singular solution. Its role is to turn persistent scale-critical badness into an explicit accounting alternative: badness can survive only by paying through untaxed supply or through non-negligible leakage. The second part of the paper interprets the ledger coordinates as a PDE-realizable finite-scale defect package. This leads to a dynamic obstruction language in which a dangerous mechanism must be profitable, reproducible, Navier–Stokes-realizable, and visible or invisible through specified pressure, flux, energy, and trace channels. The final part records finite-window quotient tests, a clean positive-cone anti-phantom theorem, and a conditional localized transfer framework. These results are finite-window or conditional tools built around the main ledger theorem, not an unconditional Navier–Stokes regularity theorem.

We present a proof of Nash's smooth embedding theorem with an emphasis on accessibility and rigor.

We prove Hadamard ill-posedness for the non-autonomous linearised Prandtl equations around time-dependent shear-flow equilibria in function spaces up to Gevrey class 4. More precisely, we construct compactly supported smooth initial data, Gevrey class 4 in the tangential variable, for which the system admits no weak solution for any positive lifespan. In this regard, we improve previous results by showing that classical semigroup-type instabilities do not, by themselves, imply Hadamard ill-posedness in the non-autonomous case when the initial time is not a variable of the system. Our argument is based on a family of exact unstable modes whose $L^2$-norms grow, at tangential frequency $k$, like $\exp(c \sqrt k t)$ up to times of order $t \sim k^{-1/4}$. Their construction relies on an inner-outer gluing scheme, in the spirit of matched asymptotic expansions, which combines unstable inner solutions near a non-degenerate critical point with an exact outer solution and yields exponentially small matching errors for short times.

A generalised inverse first passage problem asks whether, given a probability measure $p$ on $[0,\infty]$, one can find a boundary $b:[0,\infty]\to \mathbb{R}$ such that the stopping time:\[\tau:=\inf\left\{t:\Lambda\int_0^t \omega(W_s-b(s))ds \geq U\right\}\] has distribution $p$, where $U\sim Exp(1)$, $\Lambda\in(0,\infty)$ and $\omega$ is a monotonic decreasing function. We construct a branching-selection particle system whose hydrodynamic limit is governed by a free boundary problem and connect this to the generalised inverse first passage problem. In the $N$-particle system, particles move as independent Brownian motions, branch at a prescribed rate, and are removed at a rate proportional to their location relative to a position $b^N(t)$ which is a function of the empirical distribution. We identify the limit of $b^N$ as the solution of the inverse first passage problem.

A steady Euler flow is localizable if the pressure function is constant along its stream lines. This property was used by Gavrilov to construct the first smooth compactly supported steady states of 3D Euler. We prove that any analytic localizable 3D Euler flow in a bounded domain $\Omega$ is axisymmetric and $\Omega$ is a rotationally symmetric domain whose transverse section is a disk or an annulus with convex boundary curves. To the best of our knowledge, this is the first symmetry theorem for 3D steady Euler flows. In the context of MHD equilibria, this result shows that Grad's conjecture holds true for magnetic fields satisfying the isodynamic condition, a property introduced by Palumbo in the 1960's to minimize the effect of particle drifts in plasma confinement devices.

Motivated by the polygonal Pólya–Szegő conjecture for torsional rigidity, we study two monotonicity problems for torsional rigidity. The first concerns a mixed torsion problem on fixed-area right triangles, with a Dirichlet condition on one leg and Neumann conditions on the other leg and on the hypotenuse. We prove that the mixed torsional rigidity strictly increases as the ratio of the Neumann leg to the Dirichlet leg increases. The proof uses a Hadamard shape derivative, a Pohozaev-type identity, and a monotonicity result for the mixed torsion function. We also prove a similar result for the mixed ground state of Laplacian. The second concerns regular polygons. If \(P_N\) denotes the regular \(N\)-gon of area \(\pi\), we prove, by a purely analytic Schwarz–Christoffel/Bergman analytic-content argument, that \[ T^D(P_{N+1})>T^D(P_N),\qquad N\ge3, \] where \(T^D\) is the Dirichlet torsional rigidity. We also obtain the asymptotic expansion \[ T^D(P_N)=\frac{\pi}{8}-\frac{\pi\zeta(3)}{N^3} +\frac{\pi^5}{45N^4}+O(N^{-5}). \]

We study the inverse scattering problem for the Kerr-nonlinear Helmholtz equation \[ \Delta u + k^2(1+q(x)|u|^2)u = 0 \quad \text{in }\mathbb{R}^n,\; n\geq 2, \] where the aim is to recover the unknown potential $q$ from the scattering amplitude. We obtain uniqueness for full data and partial data cases of backscattering, fixed angle scattering, and fixed energy scattering. For the linear Helmholtz equation, uniqueness in backscattering and fixed angle cases are classical and largely open problems. We are able to explicitly reconstruct individual Fourier modes of the potential, and if the measured directions and energies cover an open subset, we recover $q$. The simplicity of the approach leads to an efficient numerical method, and numerical experiments show accurate reconstructions, even in the presence of noise.

We prove a sharp quantitative stability result for Alexandrov's theorem in arbitrary dimension for bounded $C^1$ open sets in a small-excess regime. More precisely, if $E\subset \mathbb R^n$ is a bounded $C^1$ open set with the same volume as the unit ball $B$, small excess, and scalar distributional mean curvature $\mathcal H_{\partial E}\in L^2(\partial E)$, then, up to a translation, $$ \operatorname{Exc}(E)+|E\Delta B|^2+|\mu-(n-1)|^2 \le C(n)\|\mathcal H_{\partial E}-\mu\|_{L^2(\partial E)}^2 \qquad \forall\,\mu\in \mathbb R. $$ In other words, both the excess and the symmetric difference from the ball are controlled by the optimal $L^2$-oscillation of the mean curvature. This yields a sharp stability estimate in a genuinely non-parametric regime. The proof combines a $BV$ version of Fuglede's spectral-gap argument, a star-shaped rearrangement for sets of finite perimeter, quantitative estimates for the part of the boundary contained in the tentacles, and a polyhedral approximation argument for the non-graphical region. We note that the $C^1$ regularity assumption enters only as a qualitative technical ingredient of the proof, but all constants in the final estimate depend only on the dimension.

Let $M$ be a two-dimensional compact manifold without boundary and let $N$ be a compact manifold without boundary. We study the $L^2$-gradient flow of an energy functional that interpolates between the harmonic map energy and the intrinsic biharmonic map energy. The critical points of this functional are called sesqui-harmonic maps. We investigate regularity properties of this flow, generalizing Struwe's classical regularity result for harmonic maps.

We prove the global-in-time existence of weak solutions for the Maxwell–Stefan–Fourier system with physically motivated degenerate thermal conductivity $\kappa(\theta)=\theta^\alpha$, for $0<\alpha\leq2$. In contrast to existing nonisothermal compressible theories based on nondegenerate conductivities, the entropy inequality no longer gives control of the quantities $\nabla\theta$ and $\nabla \log\theta$. The main new ingredient is a renormalized energy estimate, which yields compactness of the temperature and allows the identification of the degenerate heat flux.

In this paper, we study the dynamical Prandtl equation, which plays an important role in the study of the vanishing viscosity limit of the Navier–Stokes equations. Our focus is on the (Sobolev) well-posedness regime, where the given data satisfy a crucial monotonicity condition. In this case, local classical solutions have been constructed in the pioneering works of Oleinik \cite{O68,OS99}. More recently, global weak solutions were obtained in \cite{XZ04} by Xin and Zhang, and in \cite{XZZ24} by Xin, Zhang, and Zhao, where the uniqueness and interior H"older estimates of the solutions were established (in Crocco coordinates). Using a precise description of the fundamental solution to the Kolmogorov equation in the half-space, we first obtain the H"older regularity of local weak solutions up-to-boundary. We also provide a detailed proof of higher-order regularity estimates together with $W^{2,p}$ Sobolev estimates and $H^{s}$ hypoelliptic estimates, which are nontrivial. Up-to-boundary smoothness of solutions (in Crocco coordinates) is important in order to conclude the smoothness of the Prandtl solutions in the physical variables, even in the interior. It is also physically significant for applications to the Boundary Layer Theory where the dynamical Prandtl equation is essential. Using these smoothing estimates, we then prove the global existence and regularity of classical solutions to the dynamical Prandtl equation under monotonicity assumptions, which was listed by Oleinik and Samokhin in \cite{OS99} as one of the open problems. We also develop a self-contained local existence theory using weighted energy estimates and further expand the theory of global weak solutions. The main point is to incorporate all physical types of asymptotic matching of the boundary layer with the outer flow, which is expected to be useful for applications to the Navier–Stokes equations.

We consider the uniqueness of the solution of the dissipative surface quasi-geostrophic equation, without assuming time-continuity and smallness of the solutions. We show that the uniqueness holds in the scale-critical Lebesgue spaces and non-homogeneous Besov spaces. The proof is based on the energy method, inspired by the approach introduced by Lions and Masmoudi (2001) in the study of uniqueness for the Navier-Stokes equations. A key ingredient of the argument is the justification of the energy inequality via the smoothing effect of the fractional heat semigroup together with an iteration scheme based on the structure of the integral equation.