Papers

In this paper we study a nonlinear Dirac equation with intrinsic Yukawa potential on compact manifolds. In dimension 2, and in dimension 2 after projection onto the positive-energy sector, we prove global well-posedness together with exponential upper bounds on the growth of higher-order Sobolev norms. The main ingredient that allows to extend the local solutions, which are constructed by standard contraction argument, is a commutator-based higher-order energy method, combined with Grönwall's lemma, which provides the necessary a priori Sobolev bounds on the solutions.

This work focuses on the two-dimensional, nine-velocity (D2Q9) lattice Boltzmann model. First, we show that the D2Q9 scheme cannot achieve secondorder accuracy unless the cubic velocity terms are neglected, and we explain how some of these parasitic terms can be eliminated. Second, we demonstrate that the standard choice of the equilibrium distribution has no effect on the equivalent PDE at second order. Finally, we numerically investigate the effect of these cubic terms and study different choices of equilibrium distributions using a new benchmark called the Oblique Dipole Benchmark, which describes obliquely propagating 2D vortex dipoles with periodic boundary conditions.

For each integer $K\geq2$ when $n\geq4$, and for $K=2,3,4$ when $n=3$, we construct an almost-calibrated Lagrangian mean curvature flow $L_K(t)$ in $\mathbb{C}^{n}$, starting from initial data arbitrarily close to being special Lagrangian, which develops a finite-time Type II singularity at time $T$ with the explicit curvature blow up rate \[ \sup_{L_{K}(t)} |\mathbf{A}_{L_{K}(t)}| \sim (T-t)^{-K/2} \qquad \text{as } t\nearrow T . \] The tangent flow at the singularity is a transverse pair of cohomogeneity-one special Lagrangian cones, while the Type II blow-up limit is a smooth cohomogeneity-one special Lagrangian desingularization. This gives a quantitative construction of Type II blow-up for a fully nonlinear parabolic PDE arising from cohomogeneity-one Lagrangian mean curvature flow. Our construction is based on a modulation analysis around a shrinking family of cohomogeneity-one special Lagrangian desingularizations, using the perturbative spectral theory developed in the companion paper.

We deal with a Kirchhoff problem on a smooth bounded domain of $\mathbb{R}^4$ with competing critical and concave terms. By using new approximation techniques and the Nehari manifold analysis, we derive several existence results, complementing earlier ones obtained in the critical-convex case. Compared to similar results in the literature, we provide explicit bounds of the range of the parameters leading to the existence of solutions.

We study the homogenization, in the critical scaling regime, of a boundary value problem with a nonlinear dynamic Signorini-type condition posed on a rapidly oscillating portion of the boundary. The source term is time-periodic and we look for time-periodic solutions. Using the method of oscillating test functions (Tartar), compactness, and monotonicity arguments, we identify the homogenized problem and the effective nonlinear boundary operator. In contrast with the evolutionary (initial value) setting, the periodic framework eliminates memory effects and yields an instantaneous time-periodic operator defined through a periodic-in-time cell problem.

We establish the equivalence between weak and viscosity solutions for a broad class of nonlocal $p$-Laplace type equations on the Heisenberg group whose kernels satisfy standard symmetry, ellipticity, and left-translation invariance assumptions. As a particular case, our results apply to the fractional Heisenberg $p$-Laplacian. The proof combines intrinsic approximation by modified infimal convolutions, Heisenberg mollification, and a nonlocal integration-by-parts argument adapted to the sub-Riemannian geometry. The main analytical difficulty stems from the noncommutative structure of the Heisenberg group, which prevents a direct extension of the Euclidean theory and requires new localization and approximation techniques.

We prove that if a forward or backward complete Berwaldian or reversible Finslerian manifold $(M,F)$ admits a non-trivial (non-homothety) conformal concircular transformation (circle-preserving transformation or {\cpt} for short), $e^{\sigma} F$, where $\sigma$ has at least one critical point, then, $(M,F)$ is Riemannian. Consequently, $(M,g)$ is conformally diffeomorphic to either 1) the standard sphere, 2) the Euclidean space, or 3) the hyperbolic space. In particular, a compact Berwaldian or reversible Finslerian manifold does not admit any non-trivial conformal {\cpt}s, unless it is conformally diffeomorphic to the standard sphere.

We study the spectral asymptotics of elliptic operators with quasiperiodic coefficients by exploiting projections from higher-dimensional periodic functions. Using the framework of two-scale convergence adapted to cut-and-project quasiperiodic structures we establish that, in the low-frequency (homogenization) regime, the spectrum converges to that of a homogenized operator with effective coefficients determined by a cell problem on the higher-dimensional torus. In the high-frequency regime, we introduce a rescaling approach that transforms the problem to an expanding domain with asymptotically frozen coefficients. In the critical scaling, the rescaled spectrum converges to the union of the Bloch spectra arising from the quasiperiodic bulk and a boundary layer spectrum consisting of eigenfunctions concentrated near the boundary of the macroscopic domain. This boundary spectrum can be characterised as a subset of the spectrum of a family of half-space operators with frozen macroscopic coefficients. For any non-critical scaling, the rescaled spectrum fills the positive real line.

We consider the aggregation-diffusion equation in the whole space with a mildly singular interaction kernel K = K(x) which behaves like |x|^k near the origin for some k $\in$ (0, 2). This equation, supplemented with nonnegative, bounded, and integrable initial data, possesses a global-in-time solution. We prove that the family of nonnegative, radially symmetric solutions of this equation, all sharing the same initial datum, focuses around the origin over a common finite time interval as $\epsilon$ ___ 0.

We investigate quantitative rapid stabilization for the one-dimensional Allen–Cahn equation and develop a quantitative modal decomposition approach that makes explicit the dependence of the feedback laws and stabilization costs on the prescribed decay rate. We construct an explicit feedback law on the finite-dimensional unstable modes via Ackermann's formula. The explicit structure of the feedback allows us to derive quantitative low-frequency estimates, which, combined with the frequency Lyapunov method, yield quantitative stabilization estimates. Together with the stabilization framework of [37], the resulting estimates can be adapted to a broader class of one-dimensional parabolic models. We further construct piecewise feedback laws that yield the null controllability with control costs and finite-time stabilization.

We study the long-time asymptotics of the three-dimensional Hartree equation with Yukawa potential \[ V_\mu(x)=\frac{e^{-\mu|x|}}{|x|}, \qquad 0\leq\mu\leq1. \] The Coulomb case corresponds to $\mu=0$, while $\mu>0$ introduces the screening length $\mu^{-1}$. In the limit $\mu\to0$ and $t\to\infty$, the asymptotic behavior depends on the comparison between the observation scale $t$ and the screening length $\mu^{-1}$, equivalently on the parameter $\mu t$. This leads to three distinct asymptotic regimes, according as $\mu t\to0$, $\mu t\to L\in(0,\infty)$, or $\mu t\to\infty$, with different modified scattering phases in each case.

We develop a two-dimensional structural local-defect theory for scalar non-divergence advection–diffusion operators \(Lu=-a:D^2u+b\cdot\nabla u\), where \(a=a^{\mathrm{per}}+a^e\) and \(b=b^{\mathrm{per}}+b^e\). The periodic coefficients and the bounded local defects are uniformly Hölder continuous, the interpolating matrices \(a_t=a^{\mathrm{per}}+t a^e\) are symmetric and uniformly elliptic, and \(a^e\in L^r(\mathbb{R}^2)\), \(b^e\in L^s(\mathbb{R}^2)\), where \(1<r,s<2\). Under the periodic centering condition \(\langle m^{\mathrm{per}}b^{\mathrm{per}}\rangle=0\), global harmonic coordinates remove the periodic drift. After blow-down, the transformed defect drift is small in the critical local \(L^2\) space. Critical-drift compactness then yields a finite-energy Liouville theorem and closes the continuation argument, giving a whole-space estimate for \(1<q<2\) and \(1/q^*=1/q-1/2\). This estimate provides defect correctors and, by duality, a positive invariant density \(m=m^{\mathrm{per}}+m^e\). A planar Hodge construction in harmonic coordinates, followed by a Piola pull-back, produces a skew-symmetric field \(B=B^{\mathrm{per}}+B^e\) such that \(mLu=-\operatorname{div}((ma-B)\nabla u)\). If \(M=\max\{r,s\}\) and \(M^*=2M/(2-M)\), then, for some \(\beta>0\), the defects \(B^e\) and \(A^e:=ma-B-(m^{\mathrm{per}}a^{\mathrm{per}}-B^{\mathrm{per}})\) belong to \(L^{M^*}\cap L^\infty\cap C_{\mathrm{unif}}^{0,\beta}\) and vanish uniformly at infinity. This supplies the missing two-dimensional structural reduction in the scalar regular non-endpoint regime.

We characterize solutions of the regularized $p$-Laplace equation \[ \operatorname{div}\!\left((1+|Dv|^2)^{p/2-1}Dv\right)=0, \qquad 1<p<\infty, \] in a bounded domain $\Omega\subset\mathbb{R}^n$ by a pointwise asymptotic mean value identity. For $v\in C^2(\Omega)$, solving the equation is equivalent to \[ v(x) = \frac{\widetilde{\alpha}}{2} \left( \mathcal{S}_{\varepsilon}^{+}[v](x) + \mathcal{S}_{\varepsilon}^{-}[v](x) \right) + \widetilde{\beta} \int_{B_\varepsilon(0)} v(x+h)\rho_\varepsilon(h)\,dh + o(\varepsilon^2), \] where \[ \widetilde{\alpha} = \frac{p-2}{p+n+1}, \qquad \widetilde{\beta} = \frac{n+3}{p+n+1}. \] The kernel $\rho_\varepsilon$ is the semicircular marginal of normalized Lebesgue measure on the $(n+1)$-dimensional ball, and $\mathcal{S}_{\varepsilon}^{+}$ and $\mathcal{S}_{\varepsilon}^{-}$ are the tilted strategic functionals arising from the affine lift \[ w(x,s)=v(x)+s. \] The lifted gradient $(Dv,1)$ never vanishes, so the extremal second-order expansion is valid at every gradient regime. The characterization holds for the full range $1<p<\infty$. By standard interior regularity for nondegenerate regularized $p$-growth equations, weak solutions are smooth in the interior; the weak and viscosity viewpoints for related quasilinear $p$-Laplace equations are connected in \cite{JLM01}. The convergence of the associated projected dynamic programming scheme is established in the companion paper \cite{Moosavi26}.

We study a nonlinear size-structured transport equation where the endogenous scalar output $E(t)=\int_{l_0}^{l_m}\chi(l)x(t,l)\,dl$ feeds back into velocity and mortality. This principal-coefficient feedback precludes a semilinear perturbation framework. Freezing the feedback path yields a non-autonomous linear evolution, reducing the closed-loop problem to a scalar Volterra fixed point $E=\mathcal K(E)$. Mass balance provides an intrinsic feedback interval, while a Bielecki-norm contraction ensures unique nonnegative weak solutions. Stationary equilibria satisfy a scalar closure equation $E=\Phi(E)$. We prove uniqueness below the sharp margin $1-\Phi'(E)>0$ and identify $\Phi'(E)=1$ as a nondegenerate fold threshold. Linearization yields a finite-memory renewal equation with characteristic equation $\mathcal E(\lambda)=1$, whose root set determines the feedback spectrum and stability. Finally, the stationary harvesting adjoint reduces to a rank-one perturbation formula. At zero discount, we establish the identity $\mathcal E(0)=\Phi'(E^*)=B(0)$, linking closure resonance, spectral crossing, and adjoint loop gain.

We study the Bhatnagar-Gross-Krook (BGK) equation in a bounded domain with in-flow boundary condition. The BGK model is a simple relaxation of the Boltzmann equation through replacing the quadratic nonlinearity by the so-called local Maxwellian, which consists of the density, the velocity, and the temperature. The aim of this paper is twofold. First, we establish the Dirichlet boundary value problem for the BGK equation by showing a local existence result in the weighted $L^\infty$ norm if the initial data and boundary condition are close to the global Maxwellian. For the purpose of studying the inverse problem, we further derive an expansion of the BGK solution with respect to a small parameter. This allows us to decompose the nonlinear BGK into purely linear transport equations based on various order of the expansion. Second, we investigate an inverse problem of determining a general collision frequency, depending on both density and temperature, in the BGK equation from the boundary measurement operator, which maps from the in-flow boundary data to the out-flow data. We apply highly concentrated in-flow test data to the solution of the resulting linearized BGK equations to extract hidden information of the collision frequency.

We propose a kinetic model of opinion dynamics under selective media influence on both graphon-based and real-world networks. The media action, inspired by Hallin's theory of spheres, is incorporated through a model predictive control strategy designed to steer agents' opinions toward a desired target opinion. For the resulting Boltzmann-type description, we analyse the evolution of the moments and, in the quasi-invariant interaction limit, derive a Fokker–Planck-type equation together with a characterisation of its stationary states. We also prove exponential convergence to equilibrium in the Fourier metric. Numerical experiments are performed on networks generated by a Gaussian graphon and on real-world, single-issue Twitter networks, allowing us to investigate the role of control and interaction parameters, as well as the impact of the subset of agents subject to media influence. Using real-world social networks data to initialise opinions and infer the interaction structure, we then compare the dynamics obtained on the original networks with those produced by Gaussian graphons fitted to their adjacency matrices, thereby assessing the descriptive power of the graphon approach for real-world opinion dynamics.

The Fisher-KPP equation is the canonical reaction-diffusion equation used in the study of invasive phenomena in mathematical biology. While moving boundaries have already been included to introduce biologically realistic sharp interfaces, these flux-based conditions assume loss of mass as a byproduct of boundary movement. We consider a mass-conserving boundary condition at a free boundary of the Fisher-KPP equation to allow interface movement without population consumption. We use a combination of phase plane analysis, perturbation analysis, and numerical simulation to show that the model supports both invading and receding travelling wave solutions with distinct wave speeds and front densities. Prescribing the free boundary velocity as a linear function of the front density, we show the existence of multiple stable and unstable travelling wave solutions, as well as receding population blow-up. Our results indicate that in regions where multiple steady states exist, those with larger wave speeds are stable. This is corroborated by considering the parameter regime associated with the uniform steady state solution and performing linear stability analysis. The framework we develop is readily extendable to more general boundary velocity functions, offering a means of studying increasingly complex and biologically relevant boundary dynamics.

In this paper we consider solutions $\boldsymbol{u}$ of the three-dimensional Navier-Stokes system and investigate sharpness of the a priori bound \begin{align*} \frac{d}{dt}\|\boldsymbol{u}\|_q^q \leq C\|\boldsymbol{u}\|_q^{q\frac{q-1}{q-3}}, \qquad q > 3. \end{align*} This bound is closely related to the Ladyzhenskaya-Prodi-Serrin conditions characterizing classical solutions of the Navier-Stokes system. Velocity fields maximizing the rate of growth $(d/dt)\|\boldsymbol{u}\|_q^q$ under certain constraints are found as solutions of a suitable optimization problem which is solved numerically using a Riemannian conjugate gradient approach. The results obtained for different $q$ and increasing values of $\|\boldsymbol{u}\|_q$ indicate that the bound is indeed sharp, up to a numerical prefactor, and therefore cannot be fundamentally improved. Additionally, the results also suggest that the rate of growth $(d/dt)\|\boldsymbol{u}\|_q^q$ diverges as $q\to 3$.

We establish the boundedness of exponential Schrödinger-Calderón-Zygmund operators on weighted $BMO_\rho^\alpha(w)$ spaces via a $T1$ criterion, where the weights belong to classes that capture the exponential decay of the operators, and $\rho$ is a critical radius function. Specifically, we prove that the boundedness of such an operator $T$ on $BMO_\rho^\alpha(w)$ is equivalent to a natural oscillation condition on $T1$ over sub-critical balls. The weight classes considered, introduced in connection with $\rho$, include and extend the classical $A_p^\rho$ weights, and are well-adapted to the exponential decay of the kernels. As applications, we derive weighted endpoint estimates for several operators associated to the generalized Schrödinger operator $\mathcal{L}_\mu=-\Delta+\mu$, including Riesz transforms, Laplace transform-type multipliers, maximal operators for the heat and Poisson semigroups, Littlewood-Paley functions and fractional integral operators. When $d\mu(x)=V(x)dx$, the results above extend the known endpoint estimates to larger classes of weights.

We study the long-time stability of global self-similar solutions to two energy supercritical nonlinear wave equations, namely, the cubic nonlinear wave equation in $6$ dimensions and the corotational wave maps equation in $4$ dimensions. We prove the stability of self-similar solutions under perturbations that are small in the critical Sobolev spaces. The proof is based on Strichartz estimates for wave equations with potentials in similarity variables.

We study the spectral Barron regularity of Coulombic electronic eigenfunctions after extraction of a cut-off Jastrow factor. Let \(H=-\Delta+V\) be an \(N\)-electron Coulomb Hamiltonian with clamped nuclei, and let \(\psi\) be an eigenfunction associated with a discrete eigenvalue below the bottom of the essential spectrum. For the cut-off Jastrow factor \(F_{\rm cut}\) of Fournais–Hoffmann-Ostenhof–Hoffmann-Ostenhof–Sørensen, we set \[ \phi=e^{-F_{\rm cut}}\psi . \] Whereas the original wave function satisfies the sharp global threshold \(\psi\in \mathcal B_{\rm sp}^s(\mathbb R^{3N})\) for every \(0\leq s<1\), we prove that the Jastrow quotient gains one full order: \[ \phi\in \mathcal B_{\rm sp}^s(\mathbb R^{3N}) \qquad \text{for every } 0\le s<2 . \] The endpoint \(s=2\) is shown to be natural through an explicit hydrogen-like eigenfunction. The many-body proof is a global Fourier-side resolvent argument. After conjugation by the cut-off Jastrow factor, the Coulomb singularities are converted into localized angular coefficient blocks with admissible Fourier-control measures. Low frequencies are controlled by the a priori \(H^1\)-bound, while high frequencies are recovered by a Neumann fixed-point argument using the resolvent multiplier and annular estimates for the coefficient measures.

We prove regularity of a fully nonlinear equation that arises from the study of geodesics in the space of mixed volume forms on Hermitian manifolds admitting a balanced metric. Under conditions for ellipticity, we prove that this degenerate equation has a $C^{1,1}$ solution on Hermitian manifolds. We derive uniform Laplacian estimates for the perturbed equation, and also construct explicit subsolutions. In particular, this shows the existence of a unique $C^{1,1}$ solution to the Donaldson equation on Hermitian manifolds.

The Clifford torus in a Berger sphere with parameter $\tau$ is a critical point of the Willmore functional for every $\tau>0$, yielding a smooth path of Willmore surfaces. By estimating the Morse index along this path, we apply bifurcation theory to produce new symmetric Willmore tori emerging from the Clifford torus.

We consider a quasilinear Allen–Cahn system which arises when the gradient energy term in the Ginzburg–Landau energy also contains zero order terms. Such systems offer significant advantages in applications, since surface tensions and mobilities can be easily calibrated. The analysis for these systems is highly challenging, partly due to the fact that the gradient term in the energy is non-convex and since gradient terms appear quadratically in the weak formulation. This explains why an existence theory has been lacking for nearly thirty years. In this paper, we give the first existence and uniqueness results for such systems. Firstly, we prove existence and uniqueness of local-in-time strong solutions using the theory of maximal regularity. Here, non-standard techniques have to be applied due to the fact that linear constraints on the solution are involved and due to nonlinear boundary conditions. Secondly, using a minimizing movement approach we show the existence of global-in-time weak solutions. Here, the main difficulty arises from the fact that the underlying energy is not $\lambda$-convex. We overcome this issue by proving higher integrability of the gradient of the solution, first showing that solutions are bounded and then using an approach by Giaquinta and Modica. This finally allows us to pass to the limit in the time-discrete approximation. Using the de Giorgi interpolation technique, we are also able to show a sharp energy decay property despite the lack of convexity of the energy.

Let $\{\eta^i\}_{i=1}^2$ be a an orthonormal coframe on a domain $U$ on a smooth surface $(\Sigma,g)$. When $\eta^i$ is smooth, it is well-known that there is a unique connection 1-form $\omega$ verifying Cartan's first structural equations $d\eta^i = (*\eta^i) \wedge \omega$, and Cartan's second structural equation $d\omega = K_g dvol_g$. We prove that this statement remains valid when the frame is $C^0 \cap H^{\frac12}$, where the structural equations are understood in the sense of distributions. From this, we deduce that the Gauss equation $\mathrm{Det}\, D^2 f = K_g (1+|Df|^2)^2$ holds for every graphical representation $f$ of an isometric embedding of regularity $C^1 \cap W^{1+\frac23,3}$ or $c^{1,\frac12} \cap BV^2$. As an application, we prove regularity and convexity results for isometric embeddings of closed surfaces and convex caps with $K_g \geq 0$.

We study a Brown–York type mass for closed hypersurfaces in four-dimensional asymptotically flat manifolds. The reference mean curvature is defined intrinsically as the trace of the positive solution of the contracted Gauss equation. For large uniformly convex hypersurfaces with controlled scale, we derive an expansion consisting of a boundary term converging to the ADM mass and a shape-dependent correction. For the four-dimensional analogue of the nearly round surfaces of Shi–Wang–Wu, this correction vanishes under a natural decay compatibility condition.

The solution space of the homogeneous Dirichlet problem for the fractional Laplacian $(-\Delta )^{a}$ ($0<a<1$) or a pseudodifferential generalization $P$, on a bounded open set $\Omega \subset R^n$ with $C^{1+\tau }$-boundary, $$ Pu=f \text{ on }\Omega ,\quad u=0 \text{ on }R^n\setminus \Omega , $$ is analysed in detail. It is shown, both for solutions in Sobolev spaces of Bessel-potential type $H_q^t$ and in Hölder-Zygmund spaces $C_*^t$, that the solution space for $f$ of regularity $s\in [0,\tau -2a)$ is the direct sum of a component $\dot H_q^{2a+s}(\bar\Omega)$ resp. $\dot C_*^{2a+s}(\bar\Omega)$ with full regularity $2a+s$ and a component of the form $d^a$ times a lifting of boundary values by Poisson operators. Here $d(x)=dist(x,\partial\Omega )$. This extends to non-smooth problems results known in the $C^\infty $ setting. The knowledge is used to establish gradient estimates for $a>1/2$, e.g. estimating $d^{1-a+s}\nabla (u/d^a)$ in terms of norms of $f$ and $u$, both in $H_q^t$-spaces and $C_*^t$-spaces. This is entirely new in the case of Bessel-potential spaces; it extends previous results by Fall and Jarohs in Hölder spaces. A new tool is introduced: $\dot H^{s+t}_q(\bar\Omega)\subset d^s\dot H^{t}_q(\bar\Omega)$ holds for $s,t\ge 0$ with $s+t<\tau $.

We study the critical coagulation–fragmentation equation with multiplicative coagulation kernel $a(s,\hat s)=s\hat s$ and constant fragmentation kernel $b(s,\hat s)=1$. Under the Bernstein transform, mass-conserving solutions correspond to solutions of a singular Hamilton–Jacobi equation studied by Tran and Van (Comm. Pure Appl. Math. 75 (2022), no. 6, 1292–1331). Through this correspondence they proved that mass-conserving solutions are unique on the full critical range $0<m\le1$, but could establish their existence only for $0<m<\tfrac12$. We identify a one-sided, convexity-type invariant that holds for Bernstein-transform data and is propagated by their viscous scheme as a genuine maximum-principle bound. We call it the half-slope invariant. It sharpens the curvature barrier and thereby extends mass-conserving existence to the entire critical range $0<m\le1$. Hence $m=1$ is the critical mass, confirming the threshold predicted by Vigil and Ziff (J. Colloid Interface Sci. 133 (1989), no. 1, 257–264). The same invariant appears in the radial partial-mass formulation of the two-dimensional Keller–Segel equation, whose critical mass is $8\pi$.

We prove the existence of globally stable quasistatic evolutions for a cohesive fracture model with unprescribed crack path and without any topological restriction, in arbitrary dimension. The surface energy density is assumed to be concave and to exhibit an activation threshold, modeling depinning effects and fracture process zones in quasi-brittle materials. We devise a new notion of convergence for memory variables supported on evolving crack sets, inspired by $\sigma$-convergence in brittle fracture, guaranteeing compactness and lower semicontinuity properties. In contrast to the brittle case, global stability is not preserved under passage to the limit because of oscillation and branching phenomena in the approximating cracks. To overcome this difficulty, we deviate from the classical scheme for proving energetic solutions by first proving the energy balance and convergence of the surface energies, and only afterwards recovering the global stability condition.

We prove a global well–posedness result for the quintic NLS on $\mathbb{T}$ for initial data in $H^{s}(\mathbb{T})$, $s > 1/3$. This improves the previous best bound of $s > 2/5$.