Papers

We develop a Banach-space framework for first-order time-dependent mean-field games with local couplings, using monotone operator theory and low-order $p$-Laplacian regularization to avoid high-order elliptic smoothing. Under monotonicity and power-growth assumptions, together with either a Lagrangian lower bound or strict positivity of the initial density, we prove existence of weak variational-inequality solutions by Minty's method. The constructed solutions satisfy uniform $L^\beta$ estimates on the density, $L^\alpha$ estimates on the spatial gradient of the value function, and space-time shift estimates sufficient to identify the limiting PDE system. We prove that any variational-inequality solution satisfying these bounds, regardless of how it is obtained, is a MFG solution satisfying the Hamilton–Jacobi and transport equations in the $BV$ sense. This separates the construction of VI-solutions from the verification of the PDE system, a feature not directly available in the existing stationary Banach-space framework. Finally, for each fixed density $m$, we establish a maximal value function among Hamilton–Jacobi subsolutions; every MFG value function coincides with this maximal representative on $\{m>0\}$ and initially on $\{m_0>0\}$. Under semi-strict monotonicity, the density $m$ itself is unique. Our results apply to non-separable Hamiltonians with power growth and impose no dimension restrictions.

We investigate the dynamics of classical solutions to the dust fluid model under spherical symmetry in Schwarzschild spacetime. According to whether the solution will persist globally or develop a finite-time singularity, a precise classification of initial data is provided. Moreover, a detailed analysis on the exact blowup profile near the singularity is given.

We are concerned with the geometric structures of pseudo-sonic curves in two-dimensional self-similar solutions for the Euler equations for potential flow, allowing for non-uniform supersonic states. Mathematically, the governing second-order potential flow equation is of mixed hyperbolic-elliptic type, with degeneracy occurring along the pseudo-sonic curve. In this paper, we develop rigorous analytical approaches to analyze the geometric structures of pseudo-sonic curves in such self-similar solutions. We first show that the pseudo-sonic curve is necessarily a circle if the pseudo-velocity at each point is a normal to the curve. We then analyze the general case in which the pseudo-velocity on the pseudo-sonic point is not a normal to the curve, and study the geometric properties of streamlines in a neighborhood of the pseudo-sonic curve. Next, we establish two theorems that provide sufficient conditions ensuring that the pseudo-velocity at a pseudo-sonic point is normal to the curve, under natural assumptions on the local behavior of the solution. These results yield a precise characterization of the geometry of pseudo-sonic curves. Finally, we apply the developed theory to the shock reflection-diffraction problem with non-uniform incoming flow. We prove that the pseudo-sonic curve must be an arc if the solution is a $C^2$-small perturbation, either in the pseudo-supersonic or pseudo-subsonic region, of a solution with uniform incoming flow. In particular, the density and velocity must be constant, corresponding to the radius and the center of the pseudo-sonic arc, respectively. Moreover, we prove that the solution is $C^{2,\alpha}$-regular in the pseudo-subsonic region up to the sonic arc (except at point $P_1$). The techniques and ideas developed in this paper are expected to be applicable to other nonlinear problems involving similar mixed-type degeneracies.

This paper proves a finite-chain counting theorem for Caffarelli–Kohn–Nirenberg bad scales of suitable weak solutions to the three-dimensional incompressible Navier–Stokes equations. The main standard-PDE result bounds the weighted size of a finite set of CKN-bad scales by nonnegative channel costs consisting of vertical one-component concentration, annular leakage, pressure-tail terms, and pressure–flux–energy residuals. The proved closing mechanism is qualitative one-component compactness under a full local critical bound: small vertical component forces CKN smallness at a smaller radius. The paper then gives a canonical detector realization of the same finite-window counting philosophy. The original abstract detector is not identified with standard PDE channels; instead, a new amended canonical detector is defined using energy, flux, pressure-tail, retained low-pressure-mode, and finite-dimensional residual coordinates. For this amended detector, we prove upper realization, lower audit, CKN extraction, and finite-chain bad-scale counting.

This paper pertains to the general theory of boundary value problems for general linear differential equations with smooth coefficients in a bounded domain with a smooth boundary and contains new advances in the general theory related to the boundary properties of solutions. Specifically, conditions on the traces of a solution to a general differential equation on the boundary of the domain are found and studied, allowing the solution to be reconstructed from its traces and the right-hand side of the equation. For the case of a general equation with constant coefficients, the resulting conditions on the traces of the solution take the form of a generalized moment problem.

We study the Cauchy problem associated with the higher-order Benjamin-Ono-Schrödinger system \begin{equation*} \begin{cases} \partial_{t}r-a\partial_{x}^{3}r-b\mathcal{H}\partial_{x}^{2}r =cr\partial_{x}r -d\partial_{x}(r\mathcal{H}\partial_{x}r+\mathcal{H}(r\partial_{x}r)) +\beta \partial_{x}(|q|^{2}), \quad x,t\in \mathbb{R},\\ i\partial_{t}q-\alpha \partial_{x}^{2}q=-\beta qr , \end{cases} \end{equation*} where $b,c,d,\alpha,\beta$ are positive constants and $a\neq 0$ is a real constant. This system was introduced by Kairzhan, Kennedy, and Sulem in \cite{Higher-order-Benjamin-Ono-NLS-System-Sulem}. We prove that this system is locally well-posed in the energy space $H^{1}(\mathbb{R})\times H^{1}(\mathbb{R})$. Furthermore, in the case $a<0$, this result extends globally for initial data $(r_{0},q_0)$ with sufficiently small $H^{1}\times H^1$-norm. The proof combines compactness arguments with energy methods. To provide smooth solutions, we have to deal with the lost of the derivatives phenomenon introduced by higher-order derivatives and the Hilbert transform in the nonlinear terms when performing energy estimates. This is overcome by introducing a modified energy functional that cancels the problematic terms arising in the standard energy estimates. Once this is done, we extend the method put forward by Molinet and Pilod in \cite{HOBOinH1-Didier-Molinet} to study a single higher-order Benjamin-Ono equation. Their procedure includes the use of a gauge transformation of Tao's type \cite{GWP-BO-Tao}, and delicate bilinear estimates in Bourgain type spaces.

Systems in which the network structure and particle states co-evolve in mutual influence are increasingly recognized as essential for modeling complex adaptive systems. However, traditional models of non-exchangeable interacting particle systems frequently assume a fixed network topology, a simplification that fails to capture the dynamical nature of many real-world phenomena. In this paper, we rigorously establish the mean-field limit for systems of non-exchangeable interacting diffusions on co-evolutionary networks. The primary analytical challenge arises from the coupling between the network dynamics and the agents' states, which induces non-Markovian dynamics where the system's evolution depends on its entire history. Consequently, the macroscopic limit is not governed by a classical partial differential equation, but rather by a coupled system of path-dependent McKean-Vlasov SDEs for the particles' states together with a transport equation for the distribution of the weights. A further difficulty stems from the non-linear weight dynamics, which requires an adequate choice for the limiting network structure. To overcome the structural limitations of classical graphon theory, we employ the framework of \(\mathcal{K}\)-graphons (Lovász and Szegedy, 2010), also termed probability-graphons (Abraham, Delmas, and Weibel, 2025). To the best of our knowledge, this is the first example in the literature in which the natural topology of probability-graphons is used in the context of mean-field limits, providing a natural and rigorous framework that is fully compatible with non-linear network adaptivity.

We develop a quantitative partial regularity theory for stable solutions of \[ -\Delta u=f(u), \] where $f:\mathbb R \to [0,+\infty]$ is increasing and convex. The theory is uniform in the nonlinearity and allows for a finite or infinite blow-up level $T_f\in(-\infty,+\infty].$ Our first result is a universal $\varepsilon$-regularity criterion that answers a celebrated question of Brezis: smallness of the scale-invariant mass of the stability potential $f'(u)$ forces Hölder regularity. Moreover, if $T_f<+\infty$, the same smallness condition forces almost quadratic contact between the solution and the blow-up level $T_f$. This result is optimal and, in particular, covers the case of MEMS-type nonlinearities. Our second result identifies a critical exponent $q_f\ge1$, given explicitly in terms of the asymptotic behavior of $f$, $f'$, and $f''$, such that \[ f'(u)\in L^q_{\text{loc}}\text{ for every }q<q_f . \] Combined with our $\varepsilon$-regularity theorem, this yields quantitative bounds for the singular set, in particular \[ \dim_{\mathcal H}\Sigma(u)\le n-2q_f. \] Remarkably, our exponent $q_f$ recovers the sharp thresholds for all standard model nonlinearities, including $f(t)=(1+t)^p$, $e^t$, and $(1-t)^{-p}$. Also, this result provides the first general quantitative singular-set estimates for stable semilinear equations beyond the model nonlinearities. Finally, in the two-dimensional case, we provide a complete picture by proving the universal Hessian estimate \[ \|D^2u\|_{L^\infty(B_{1/2})}\le C\|u\|_{L^1(B_1)}, \] where $C$ depends neither on $u$ nor on $f$. This $C^{1,1}$ regularity is essentially optimal: one cannot expect $C^{2,\alpha}$ estimates for any $\alpha>0$, and in general even $C^2$ regularity should fail.

Let $\mathcal C$ be a $G$-invariant special Lagrangian cone admitting a scaled family of $G$-invariant special Lagrangian desingularizations $a \overline L$ which converge to $\mathcal C$ as $a\searrow 0$. We study the linearized self-shrinker operator on $a\overline L$ in a Gaussian weighted $L^2$ space of $G$-equivariant functions. For $0<a\ll1$, we construct any prescribed finite number of eigenfunctions whose eigenvalues converge to those of the limiting conical operator, and we prove a spectral gap estimate on the orthogonal complement of these modes. We also identify the lowest eigenfunction with the scaling mode of the special Lagrangian desingularization. This spectral basis provides the analytic foundation for the construction of Type II blow-up solutions of Lagrangian mean curvature flow in the companion paper.

The search for extremal geometries is a central theme in several areas of mathematics. Here, we address the following question: among all n-dimensional orthogonal tori of unit volume, which one maximizes the zeta regularized determinant of the Laplacian? We prove that the equilateral torus is the unique maximizer in each dimension n, for all n greater than or equal to 2, validating Sarnak's conjecture in this context. We also investigate the analogous question for the Laplacian on Euclidean boxes with the Neumann and Dirichlet boundary conditions. For orthogonal flat tori of unit volume and dimension n, we show further that the determinant is strictly decreasing with the dimension and tends to zero as the dimension n tends to infinity.

We study a finite-horizon optimal consumption and portfolio problem with labor supply flexibility and an irreversible early retirement option under a borrowing constraint. The agent chooses consumption, risky investment, and leisure before retirement, while after retirement labor income disappears and leisure is fixed at its maximal level. Preferences are described by a Cobb–Douglas utility, and wealth must remain nonnegative. {Using a dual martingale method, we transform the primal problem into a zero-sum stopper–singular-controller game. The associated dual value is characterized by a min–max parabolic variational inequality with obstacle and gradient constraints. We show that the maximal strong solution of the resulting variational inequality is the unique admissible strong solution whose gradient-constrained free boundary, namely the binding boundary, is monotone increasing in calendar time. A verification argument then identifies this strong solution with the value of the stopper–singular-controller game, and duality recovers the optimal retirement, consumption, leisure, and portfolio policies.} The numerical analysis recovers the value function and optimal policies, and illustrates how labor supply flexibility affects consumption, portfolio choice, and retirement timing under borrowing constraints.

An inverse source problem for a class of fractional elliptic equations with singular coefficients is investigated in this paper. For the corresponding direct problem, a formal solution is derived and the well-posedness of the solution is established. For the inverse problem, a Hölder-type conditional stability estimate is obtained in a Hilbert scale associated with exponential operators. Based on this stability framework, two regularization methods are proposed for reconstructing the unknown source term: the exponential-type Tikhonov regularization method and the exponential quasi-boundary value regularization method. Convergence estimates for the regularized solutions are derived under both a priori and a posteriori choices of the regularization parameter. In addition, finite-dimensional spectral approximation results show that the proposed methods are also applicable to general square-integrable source terms, without requiring the exact source to satisfy an exponential-type source condition. Numerical experiments demonstrate that the proposed methods provide stable and accurate reconstructions for both smooth and piecewise smooth sources even under low signal-to-noise ratio conditions.

In this note, we are interested in the linear Rayleigh-Taylor instability problem for the incompressible fluid with an upper free surface. Using the spectral theory of self-adjoint and compact operator, we rigorously prove the existence of infinitely many normal mode solutions to the linearized equations. Some numerical computations are presented to support our theoretical study.

The aim of this paper is to account for viscosity, surface tension, and interactions between micro-bubbles in approximating their resonant behavior in the ultrasonic regime. Original asymptotic formulas for the resonance frequencies are derived in terms of the difference in the acoustic impedance at the interface between the gas and the fluid. Both low-frequency resonances (Minnaert resonances) and higher-frequency resonances, i.e., beyond the subwavelength regime, are considered. We also provide a resonant characterization for a system of several micro-bubbles.

In this paper we study the logarithm of order $m$ of the Schrödinger operator $\mathcal L_V$ in $\mathbb R^d$, for certain nonnegative potentials $V$. First, the operator $\log^m\mathcal L_V$, $m\in \mathbb N$, is defined by using the spectral measure associated with the self-adjoint operator $\mathcal L_V$ on a suitable subspace of $L^2(\mathbb R^d)$. Then, the semigroup of operators $\{T_t^V\}_{t>0}$ generated by $\mathcal L_V$ allows us to extend the definition of $\log^m\mathcal L_V$ to a wider class of Lipschitz functions. By using logarithmic operators $\log^m\mathcal L_V$, $m\in \mathbb N$, we prove Taylor expansions for the fractional powers $\mathcal L_V^s$ and $\mathcal L_V^{-s}$ with respect to the order $s\in (0,1)$, where the convergence is understood in $L^p(\mathbb R^d)$, $1<p<\infty$.

We consider the compressible 3D magnetohydrodynamic (MHD) equations under general pressure laws. For all hypo-viscosities $(-\Delta)^{\alpha_1}$ and hypo-resistivity $(-\Delta)^{\alpha_2}$ with $\alpha_1,\alpha_2\in(0,1)$, we prove the non-uniqueness of weak solutions to 3D MHD equations which reveals that there exist infinitely many weak solutions with the same initial data. Also, for the weak solutions in $C^{\widetilde{\beta}}_{t,x}$ to the compressible ideal MHD, where $\widetilde{\beta}>0$, we prove that they are the strong vanishing viscosity and resistivity limit of the weak solutions to the hypo-dissipative compressible MHD.

In this work, the derivation and multiscale analysis of a mathematical model for plant tissue biomechanics is considered. The microscopic model consists of a coupled systems of equations of linear elasticity, describing mechanical deformation, and reaction-diffusion and ordinary differential equations modelling the dynamics of load-bearing cross-links in cell walls and middle lamella, connecting individual cells in a plant tissue. It takes into account a two-way coupling between mechanical deformation and chemical processes in cell walls and middle lamella, where elastic properties depend on the density of cross-links and reactions terms, defining the stretching and breakage of the cross-links, depend on the deformation gradient. The nonlinear dependence of solutions of the ordinary differential equation on the displacement gradient and of the boundary conditions in reaction-diffusion problem on the displacement induces novel approaches in the derivation of a priori estimates. Using homogenization techniques of locally periodic (l-p) two-scale convergence, a macroscopic model for plant tissue biomechanics is derived, representing the first result in the multiscale analysis of a biomechanical model for tissues with non-periodic fibrous microstructure. For multiscale analysis and derivation of the macroscopic model, the non-periodic microstructure of plant cell walls, characterised by rotated planes of parallel aligned microfibrils, is approximated by the corresponding locally periodic microstructure. The two-way coupling between equations of linear elasticity and reaction-diffusion and ordinary differential equations requires the proof of the strong l-p two-scale convergence for the deformation gradient and for the density of stretched cross-links, in order to pass to the limit in the nonlinear functions.

We develop a finite-window recursive audit framework for Navier–Stokes-generated packages. On a fixed window, the underlying anti-phantom certificate asserts that a baseline-visible package is either detected locally or charged to an explicit residual ledger. The first component of the framework is a broad one-step admissibility criterion for pressure-admissible finite-energy Navier–Stokes-generated audit packages under an explicit ledger of synchronization, localization, projection, harmonic-tail, chart, clean-gap, gate/slack, and detector mismatches. The second component is a finite-chain propagation theorem: once one-step admissibility supplies a finite renormalized chain of packages and a static finite-window audit certificate is available at each scale, variable-coefficient error recursion and weighted summation yield a recursive finite-window anti-phantom lower bound. We also give compact/effective pressure projection criteria, reduced quotient chart visibility, clean detector gaps, a pressure–flux–energy matrix kernel condition, a smooth reduced NS-generated verification class, and a conditional Caffarelli–Kohn–Nirenberg compatible defect-extraction criterion.

We consider the Poisson Problem for the stability operator of the helicoid on a vertical strip, under the assumption that the source term is supported on a strip of fixed height. We prove that solutions decay at a definite rate away from the support assuming natural orthogonality conditions on the source term.

The paper establishes integrated local energy decay (ILED) estimates for the damped wave equation on certain non-stationary spacetimes. The main technical result is a high frequency estimate that holds in great generality, provided that null geodesics trapped in a compact region are sufficiently damped.

This note gives a complete classification of the asymptotic behavior of radial solutions to the two-dimensional parabolic-elliptic Keller-Segel system on the whole space, for general initial data in the large. We review previous separate results, and unify them within a single classification framework. Depending on the mass, the flow exhibits three distinct asymptotic regimes. For a subcritical mass, solutions converge toward the unique self-similar expander of same mass. At the critical mass $8\pi$, solutions concentrate in infinite time around the stationary state with a universal logarithmic rate. The determination of this behaviour was the last missing step for achieving a complete radial classification, and we prove it in a companion paper (in fact without radial assumption). For a supercritical mass, solutions undergo type II finite-time blow-up with an explicit universal asymptotic rate and the stationary state as profile. This trichotomy holds for all radial initial data with finite second momentum. For non-radial data or infinite second momentum, it is known that other dynamics can be possible; for each of these three universal regimes we review the known results showing how they persist.

We consider the parabolic-elliptic Keller-Segel equation in two dimensions on the whole space. We prove that for arbitrary initial data with critical mass $8\pi$ and finite second momentum, all solutions have the same universal behaviour. They are globally defined and, asymptotically for large times, they converge to a renormalized stationary state of the equation which concentrates around the center of mass of the solution at a universal logarithmic-in-time scale. Our result holds for general solutions without symmetry assumptions, and we furthermore provide explicit convergence rates. In the radial case, by combining our result with previous ones (by Blanchet-Dolbeault-Perthame, Mizoguchi, and related works), this achieves a complete classification of the possible dynamics: for subcritical masses solutions converge to a self-similar expander, at the critical mass they concentrate a stationary state in infinite time at the aforementioned universal scale, and for supercritical masses they blow up in finite time by type II concentration of a stationary state at another universal scale. Our proof starts with soliton resolution, then controls the motion of the stationary state and the remainder in new spaces, and devises new techniques for the multi-scale linearized analysis, which eventually enables the stability and modulation analysis around an approximate solution.

For negative values of the spreading coefficient (that is, in the so-called “partial wetting” regime), we prove that the thin-film equation with zero slip and repulsive potentials $P$ of the form $P(h)\approx h^{1-m}$ as $h\to 0$, $m>1$, admits for any positive speed a one-parameter family of travelling-wave solutions with a contact line and (as in standard slippage models) a logarithmically-corrected linear behaviour as $h\to +\infty$. These waves have locally finite rate of dissipation for any $m>1$ and locally finite energy for any $m\in (1,3)$. The result thus confirms that mildly repulsive potentials effectively resolve the no-slip paradox. The family is parametrized by a thermodynamically consistent contact-line condition which reduces to the classical fixed microscopic contact-angle one if $P\equiv 0$.

We study a nonlinear Helmholtz system with cubic nonlinearity on high-contrast inclusions in three dimensions, and the solitons that emerge as the contrast $\delta$ tends to zero. Using the Dirichlet-to-Neumann operator and a capacitance formalism, we develop a perturbative cascade that expands the resonant frequency and field in powers of $\sqrt{\delta}$. Our main result is a rigorous two-way correspondence with a finite discrete nonlinear capacitance system: every discrete solution lifts to a continuous soliton (a convergent expansion, analytic in $\sqrt{\delta}$), and every continuous family with the natural subwavelength scaling reduces to a discrete one. The construction is algorithmic, giving higher-order corrections in both the subwavelength and non-subwavelength regimes, the latter via a frequency-dependent capacitance matrix. We illustrate the theory numerically and characterise a symmetry-breaking bifurcation in a symmetric dimer.

We study self-adjoint realizations of conic Laplacians on compact Riemann surfaces with radial conic metrics whose cone angles are integral multiples of \(2\pi\). Their critical asymptotic coefficients span a finite-dimensional space with a nondegenerate skew-Hermitian Green form, whose Lagrangian subspaces parametrize the self-adjoint realizations. We construct the associated Weyl functions and derive Kreĭn's resolvent formula, a resolvent trace identity, and, under explicit zeta-regularity hypotheses, a comparison formula for positive-spectrum zeta determinants. The boundary data of formal solutions define a holomorphic Weyl curve in the Grassmannian, with the Weyl functions as its local graph coordinates. Its real restriction is a positive curve in the Lagrangian Grassmannian, and its tangent form is identified with the \(L^2\)-inner product through the Poisson operator. We further identify the boundary determinants with the transition functions of the determinant line bundle induced by the Weyl curve.

We investigate the long time behaviour of the solutions of the cubic Szeg}o equation on the line in the Sobolev space H^{1/2}(R). We prove that, for every datum of which the Lax operator has simple positive spectrum, the solution asymptotically decouples as an infinite sum of traveling quasi-periodic breather solutions. Under an additional generic condition on the data, we prove that these traveling breather solutions are in fact soliton solutions, leading to a soliton resolution theorem.

We consider a family of optimization problems, based on a mean-field description of particles interacting through Coulomb forces in a quadratic trap. In addition, the particles are constrained to lie in a halfspace and we are interested in the way the particle distribution changes as the halfspace varies. In particular, we can prove the existence of a phase transition, thereby settling a recent conjecture by Byun, Forrester, Majumdar and Schehr.

We investigate the spectral stability of small-amplitude, periodic, traveling-wave solutions of the modified Camassa-Holm equation with cubic nonlinearities. More precisely, we analyze the $L^2(\mr)$-spectrum of the associated linearized operator in a neighborhood of the origin in the spectral plane. Inspired by a recently novel method based on Kato's perturbation theory [Berti et al, Full description of Benjamin-Feir instability of Stokes waves in deep water, Invent. Math., 230 (2022), 651-711.], we provide a complete description of the spectrum near the origin of the linearized operator–an integro-differential operator with periodic coefficients–and thus prove that such waves are not subject to modulational instability. Moreover, a spectral analysis reveals a remarkable threshold phenomenon: such waves with wave number $k^2\leq 3$ exhibit spectral stability, while instability emerges when $k^2>3$.

This work concerns regularity properties of weak solutions to elliptic equations in divergence form -div(a$\nabla$u) = div F , under low regularity assumptions on both the coefficient a and the source term F . We introduce generalized Morrey and Campanato spaces extending the classical definitions by replacing uniform boundedness requirements with suitable integrability conditions. Within this framework, we establish regularity estimates for the gradient of weak solutions in these generalized spaces. As applications, we recover classical Hölder and Lebesgue estimates and derive fractional Sobolev regularity results. In particular, the proposed approach yields fractional Sobolev estimates in situations where the coefficient may be discontinuous and the gradient of the solution is not expected to be locally bounded.

In this paper, we study a class of quasilinear elliptic problems involving the $(1,q)-$Laplacian operator and a discontinuous superlinear nonlinearity governed by the Heaviside function. The main difficulty of the problem arises from the presence of the $1$-Laplacian operator, whose natural setting is the Space of Functions of Bounded Variation. Our approach is based on an approximation method involving $(p,q)-$Laplacian problems as $p\to1^+$. As a consequence, we prove the existence of a nontrivial and nonnegative solution belonging to $W^{1,p}_0(\Omega)$, in an appropriate weak sense. Moreover, we investigate the asymptotic behavior of the solutions as $\beta\to0^+$, showing that the family of solutions converges to a solution of the limit problem without discontinuity.