Papers

We study phase-field and sharp-interface models for contact line dynamics of a liquid droplet on a solid substrate within a unified variational framework. The motion of the contact line, where liquid, gas, and solid phases meet, poses a fundamental difficulty in continuum modeling due to the classical stress singularity of no-slip hydrodynamics. Phase-field models regularize this singularity by introducing a thin transition layer of thickness and encoding interfacial effects through a Ginzburg-Landau free energy augmented by a wall energy on the substrate. Starting from the total free energy $E = E_b + E_w$, we analyze two phase-field models: the Allen-Cahn equation and the Cahn-Hilliard equation. Using matched asymptotic expansions as $\delta \to 0$, we recover their corresponding sharp interface limits. In the Allen-Cahn case, the limit yields motion by mean curvature with a contact line law driven by deviations of the dynamic contact angle from Young's angle. In the Cahn-Hilliard case, the limit leads to a Mullins-Sekerka problem with the same form of contact line dynamics. A central result of this work is the identification of consistent gradient-flow structures across both models. The Allen-Cahn dynamics correspond to an $L^2$-gradient flow, while the Cahn-Hilliard dynamics correspond to an $H^{-1}$-gradient flow, and both converge to sharp-interface evolutions that preserve the same energy-dissipation structure. This provides a unified interpretation of contact line motion as a consequence of a single variational principle. Finally, we develop energy-stable numerical schemes based on the minimizing movement principle and establish discrete energy dissipation and well-posedness of the fully discrete problem. Numerical examples confirm that both schemes relax toward the same stationary sharp interface solution while their dynamics reflect the different dissipation mechanisms.

We establish existence, uniqueness, and local \(L^1\)-stability for multidimensional scalar conservation laws with discontinuous heterogeneous flux, without imposing a non-degeneracy condition on the physical space–time flux. The discontinuity set may be a locally finite family of \(C^2\) hypersurfaces whose intersections and singular points are contained in a closed set of vanishing \((d-1)\)-dimensional Hausdorff measure. The main obstacle is the possible failure of multidimensional compactness when a nontrivial linear combination of the space–time flux components is constant on an interval of states. We show that, for interface problems, this obstruction can be reduced to the physical normal flux. Its flat intervals are collapsed by a normal-flux quotient that preserves the normal flux and hence the Rankine–Hugoniot relation. On the non-flat region, the time and tangential components are replaced by auxiliary polynomial functions of the normal flux, while the physical normal component is left unchanged. The resulting auxiliary space–time vector satisfies the interval non-degeneracy required by Panov's compactness theory. This yields strong compactness and strong one-sided traces of the quotient variables. The interface admissibility condition is obtained by projecting the vanishing-viscosity germ associated with the two one-sided physical normal fluxes. A localized viscous comparison shows that the quotient traces belong to this maximal \(L^1\)-dissipative germ, and a Young-measure contraction argument recovers strong convergence of the physical states. Curved interfaces are treated by flattening, localization, and finite-propagation patching. The theory is first constructed for \(BV\) initial data and then extended by continuity to \(L^1_{\mathrm{loc}}\cap L^\infty\) data taking values in the invariant interval.

We consider the uniform exponential stability analysis of infinite-dimensional impulsive systems defined on a Banach or Hilbert space, whose flow is governed by a fixed $C_0$-semigroup generator and whose jumps occur at a prescribed time sequence. While the flow and jump maps are themselves time-invariant, the time-triggered impulses render the propagator a genuinely time-varying evolution family, which is the source of the analysis difficulty addressed here. We combine ideas from hybrid systems theory and infinite-dimensional systems to produce operator-based stability conditions, which can be analytically or numerically checked via convex programming. Necessary and sufficient conditions for the uniform exponential stability of impulsive systems on Banach spaces are obtained in the context of a fixed impulse-times sequence but also of arbitrary, constant, minimum, and range dwell-times using both non-coercive and coercive Lyapunov functionals. Some of those results are then adapted to systems on a Hilbert space and quadratic Lyapunov functionals. As an application, linear switched systems are shown to be an exact special case: reformulated as impulsive systems with unit-norm selector jumps, they inherit non-coercive and clock-dependent dwell-time stability conditions on both Banach and Hilbert spaces. Theoretical and numerical examples are given for illustration, notably on the sampled-data control of time-delay systems.

We prove endpoint theorems for one-dimensional bilinear rough singular integrals. Our starting point is a sharp structural characterization of the associated angular multiplier. For every mean-zero $\Omega\in L^1(\mathbb{S}^1)$, the finite-part angular multiplier associated with $T_\Omega$ has bounded variation if and only if the antipodal even part of $\Omega$ belongs to $H^1(\mathbb{S}^1)$. This characterization identifies the precise rotational regularity required in the one-dimensional bilinear setting. It also yields a Stieltjes decomposition compatible with uniform estimates for the bilinear Hilbert transform. We then establish two boundedness criteria under critical kernel assumptions. First, if $\Omega\in L\log L(\mathbb{S}^1)$, then $T_\Omega$ is bounded from $ L^{p_1}(\mathbb{R})\times L^{p_2}(\mathbb{R})\text{to} L^p(\mathbb{R})$ whenever $1<p_1,p_2,p<\infty$ and $ \frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}.$ Moreover, the logarithmic exponent $1$ is optimal within the scale $L(\log L)^A$. Second, at the critical directional index, the same boundedness holds for $\Omega\in\mathcal{K}_{1/2,\beta}(\mathbb{S}^1)$, provided that $\beta>\frac{3}{2}\max\bigl\{p_1,p_1',p_2,p_2'\bigr\}-1.$The two critical kernel classes are incomparable. The $L\log L$ result is obtained by reducing the multiplier to a finite-part angular profile of bounded variation. The directional result follows from endpoint Fourier decay, product wavelet decompositions, and interpolation.

Let $(X,\omega_0)$ be a compact Kähler manifold of complex dimension $n$, and let $U\Subset X$ have $C^{3,1}$ uniformly strongly pseudoconvex boundary. Consider the signed parallel family $U_t=\{\rho<t\}$, where $\rho$ is a defining function agreeing near $\partial U$ with the signed distance, negative in $U$. Let $u_t$ be the equilibrium envelope with zero obstacle on $X\setminus U_t$, and write $\Sigma_t=\partial U_t$, with outward unit normal $\nu_t$. We prove that the normalized Aubin–Mabuchi energy $e(t)=E_{\omega_0}(u_t)$ belongs to $C^{1,1/2}([-\tau,\tau])$ for some $\tau>0$ and satisfies the Hadamard formula \[ e'(t)=\frac{1}{n+1}\int_{\Sigma_t}\left(-\partial_{\nu_t}u_t\right)^2\,d\sigma_{\rho,t}^{\mathrm{tot}},\qquad |t|<\tau. \] Here $d\sigma_{\rho,t}^{\mathrm{tot}}$ is the interior outward Anzellotti trace of \[ \frac{1}{V}\,d^c\rho\wedge \sum_{p=0}^{n-1}(p+1)\,\omega_{u_t}^{p}\wedge\omega_0^{n-1-p},\qquad V=\int_X\omega_0^n,\quad \omega_{u_t}=\omega_0+dd^c u_t. \] This measure is positive and uniformly comparable to the induced surface measure. The proof combines uniform $C^{1,1}$ estimates up to the boundary from the interior with $1/2$-Hölder stability of the normal derivatives under the normal-flow identifications. We identify the boundary part of the normalized Monge–Ampère measure with the negative outward trace of a divergence-measure flux current. The one-sided weak-* limits of the mixed Bedford–Taylor boundary measures yield the weighted total flux appearing in the variation formula.

The classical Kohn-Temam stress-strain pairing $({\bf A}:E{\bf u})$ for symmetric tensors ${\bf A}$ and ${\bf u}\in BD$ is typically formulated under summability assumptions on the divergence of ${\bf A}$. This excludes stress fields whose divergence has singular surface contributions, as occurs at cracks and material interfaces in continuum mechanics. We define and study stress-strain pairings for bounded symmetric divergence-measure tensor fields. For general ${\bf u}\in BD$, we introduce a slicing pairing $(({\bf A}:E{\bf u}))_\Xi$ for tensor fields satisfying a directional $BV$-type condition with respect to a finite frame $\Xi$. The definition is based on a one-dimensional disintegration strategy, and despite this construction, the new pairing enjoys analogous properties of the usual pairing $({\bf A}:E{\bf u})$, such as the absolutely continuity with respect to $|E{\bf u}|$ and the Gauss-Green formulas. We also identify several situations in which the pairing is independent of the choice of frame $\Xi$, including the relevant case in which the stress field ${\bf A}$ belongs to $BV$. While a distributional stress-strain pairing can be defined naturally for bounded $BD$ functions, it cannot be extended to the unbounded setting, since the truncation techniques available in $BV$ fail in $BD$. The slicing pairing is consistent with the distributional one whenever the latter is defined, while being more general even for bounded ${\bf u}$. Indeed, its existence does not require the compatibility condition $|{\rm Div}\,{\bf A}|(S_{{\bf u}}\setminus J_{\bf u})=0$ which is necessary for the distributional definition. This allows the treatment of stress fields interacting with diffuse micro-cracking.

We study Heisenberg uniqueness for the positive hyperbola branch and shifted lattice crosses in the supercritical regime $q=\alpha\gamma>1$. We resolve the infinite-dimensionality clause of the arbitrary-shift problem posed by Giri and Manna: for arbitrary shifts on both arms, the normalized pre-annihilator is infinite-dimensional. More precisely, every $v\in BV((1,q))$ has a global $BV$ pre-annihilating extension; the extension is unique unless both twisting phases are trivial, in which case its ambiguity is one-dimensional. The proof reduces the annihilation conditions to a graph equation for a twisted Perron–Frobenius operator and combines a phase-uniform Lasota–Yorke estimate with peripheral spectral rigidity. We also give an exact operator-theoretic normal form for the entire $L^1$ pre-annihilator in terms of the maximal convergence domain of the associated Green series. Writing $Q$ for the twisted product and $A$ for the forcing operator, we show that $Q$ has the closed unit disk as its spectrum on $L^1((0,1))$, that $\Ran(I-Q)$ is not closed, and that, outside a countable set of algebraic values of $q>1$, the operator $\sum_{j=0}^{N-1}Q^jA:L^1((1,q))\to L^1((0,1))$ has norm $2N$ for every $N\ge1$.

In this paper, we prove a uniqueness result for the partial data Calderón problem with quasilinear conductivity in two dimensions. The proof is based on higher-order linearization and the use of CGO solutions in dimension two that vanish on part of the boundary. Since derivatives of the solutions appear in the integral identity, we need improved remainder estimates, for which we introduce a modification in the choice of the phase, analogous to limiting Carleman weights. We also analyze how combinations of phases produce specific patterns in the products of solutions, which allows us to apply both stationary and nonstationary phase arguments to recover the conductivity.

In Einstein's seminal work [Ann. Physik, 17 (1905), 549-560], he pointed out that the temperature of a fluid influences the motion of suspended particles dramatically. To describe the effect of the temperature in this physical process more precisely, Boudin et al. [ESAIM Proc., 28 (2009), 195-210] introduced a new fluid-particle interaction model containing of the non-isentropic compressible Euler equations for the fluid and a nonlinear Vlasov-Fokker-Planck type equation for the particles. By adding some viscous and heat conductive terms to the fluid part of this model, Mu and Wang [Calc. Var. Partial Differential Equations, 59 (2020), Paper no. 110] established the global existence of classical solutions near an equilibrium state. In this paper, through establishing the uniform a priori estimates with respect to the viscosity and heat conductivity coefficients and taking the combined zero viscosity and heat conductivity limits, we show that the model introduced by Boudin et al. still admits a global classical solution and enjoys optimal decay rates thereby improving Mu and Wang's results and confirming Einstein's predications. Our work indicates that the presence of particles indeed emanates new dissipation effects on the non-isentropic compressible fluid-particle model via the differences between the macroscopic velocity of the particles and the fluid velocity, and the macroscopic temperature of the particles and the fluid temperature, which is significantly different from the case of pure non-isentropic compressible Euler equations. To achieve these goals, we have developed new ideas and techniques to surmount substantial obstacles caused by the absence of viscosity and heat conductivity, and the nonlinear interactions between the fluid and particles.

We consider the initial value problem for the vorticity equation in the endpoint critical Sobolev space $W^{d,1}(\mathbb{R}^{d})$ for $d = 2, 3$. In two dimensions, we prove global propagation of the $W^{2,1}(\mathbb{R}^{2})$ regularity of the vorticity. In three dimensions, for axisymmetric flows without swirl, we propagate $W^{3,1}(\mathbb{R}^{3})$ regularity of the vorticity for all times. These are in stark contrast to existing strong ill-posedness results in critical Sobolev spaces $W^{d/p,p}(\mathbb{R}^{d})$ for all $1 < p < \infty$, which were based on axisymmetric flows without swirl when $d = 3$.

We propose a reduced order modeling (ROM) framework for 1D conservative PDEs based on the cumulative distribution transform (CDT). The CDT maps nonnegative, equal-mass states into a Hilbert space in which 1D Wasserstein distances become weighted $L^2$ distances and translations become affine shifts. This makes the transform especially suited for transport-dominated dynamics, where Eulerian linear-subspace ROMs often suffer from slow decay of Kolmogorov widths. We study this phenomenon for scalar conservative dynamics by analyzing the solution manifold in CDT coordinates. For linear transport, the transformed solution manifold is contained in the 2-dimensional space spanned by the transformed initial datum and the constant function, and has zero Kolmogorov $2$-width. For nonlinear hyperbolic conservation laws, we prove two complementary types of estimates: robust $O(n^{-1})$ bounds that rely only on the conservative transport structure and remain meaningful after shock formation, and sharper $O(n^{-2})$ bounds in smooth pre-shock regimes. For conservative advection-diffusion, we show that the CDT trajectory remains within distance $O(\sqrt{DT})$ of the pure-transport plane, and we also obtain sharper $O(D^2T^2)$ estimates under additional regularity or away from initial layers. In both cases, the zero 2-width behavior of linear transport is recovered as the diffusion coefficient tends to zero. Motivated by these estimates, we develop a CDT-POD numerical scheme: snapshots are mapped to CDT space, Proper Orthogonal Decomposition (POD) is performed in transformed coordinates, and the inverse CDT is used to reconstruct physical states. Numerical experiments for several transport-dominated dynamics show that CDT-POD can capture solution manifolds with substantially fewer modes than Eulerian POD.

In this article, we establish Hölder regularity for viscosity solutions to a class of degenerate fully nonlinear elliptic equations of the form \[ F(D^2u,Du)=f(x)~~\text{in}~~B_1, \] where the operator is elliptic only in regions where the Hessian is sufficiently large. Such equations arise naturally in free boundary problems and models with partial ellipticity. The proof combines a modified cusp function with a decomposition of the contact set in a point-to-measure argument. As a consequence, interior Hölder continuity follows under natural structural assumptions.

We study viscosity solutions for a class of degenerate one-phase free boundary problems of the form $w\Delta w = h(\nabla w)$. We assume the existence of a star-shaped domain $D$ such that $h < 0$ in $D$, $h = 0$ on $\partial D$, and $h > 0$ in $\bar{D}^{c}$. This class of degenerate one-phase free boundary problems arises when a canonical transformation is performed to a semilinear equation $\Delta u = f(u)$, and $f$ behaves like $-\gamma u^{-(\gamma + 1)}$ for some $\gamma \in (0,2)$. In this case, known as the Alt-Phillips equation for negative power potentials, $h(\rho) = c(|\rho|^2 - 1)$. We show existence of a viscosity solution, Lipschitz regularity, and regularity of the free boundary at flat points. Additionally, we show that as $\gamma$ converges to $2$, the free boundary converges to a minimal surface.

We study Brenier's relaxed least-action problem in the unit disc $D:=\{x\in\mathbb{R}^2:\ |x|<1\}$ at the critical time $T=\pi$, with the endpoint pair $i_D$ (the identity on $D$) and $-i_D$. We prove that the critical energy shell \[ S^3=\{(x,v)\in\mathbb{R}^2\times\mathbb{R}^2: |x|^2+|v|^2=1\} \] supports stationary action-minimising generalised incompressible flows that are not invariant under physical rotations. This answers the question posed by Bernot, Figalli, and Santambrogio. Our construction relies on the normalised surface measure $\sigma$ on $S^3$ and the Hopf quotient \[ \Pi=(N,M,L):S^3\to\mathbb S^2_{1/2}, \qquad \mathbb S^2_{1/2}:=\{(n,m,\ell)\in\mathbb{R}^3: n^2+m^2+\ell^2=1/4\}. \] Here, \[ N=\frac12(x_1^2+v_1^2-x_2^2-v_2^2),\quad M=x_1x_2+v_1v_2,\quad L=x_1v_2-x_2v_1 . \] This quotient is a first integral of the harmonic-oscillator flow. Let $\tau:=\Pi_\#\sigma$, let $g$ be a bounded Borel function on $\mathbb S^2_{1/2}$ whose $L^\infty(\tau)$-class is odd under $(n,m,\ell)\mapsto(n,m,-\ell)$, and let $\delta\in\mathbb{R}$. If $1+\delta g\circ\Pi\ge0$ $\sigma$-a.e., then \[ d\mu_{\delta,g}=\pi(1+\delta g\circ\Pi)\,d\sigma \] has Lebesgue spatial marginal and is stationary. The induced path measure is then minimising. We also determine exactly which members of this tilted family are rotationally invariant. If $\delta\ne0$, this is equivalent to axisymmetry of the $L^\infty(\tau)$-class of $g$, modulo $\tau$-null sets. In particular, taking $g(n,m,\ell)=\ell m$ with $0<|\delta|<8$ gives strictly positive non-rotationally invariant minimisers.

We consider incompressible FENE dumbbells coupled to a reversible Markov operator on a compact continuum of internal states. At zero centre-of-mass diffusion we prove that state averaging is an exact factor map and that a Lagrangian state lift is its unique inverse on the natural energy-solution classes. Thus existence, multiplicity, and uniqueness of the state-resolved system are precisely those of its scalar FENE projection; the internal-state dynamics creates no additional large-data nonuniqueness. The lift is driven by a trace-free matrix fibre evolution and permits nonlinear local activities, including rates with linear dependence on the singular Kramers stress.We also prove a sequential form of this structure. For state-resolved regularizations with relative-entropy-prepared initial fibres and no viscous dissipation defect, the complete densities converge strongly without reconstructing them after the scalar limit. Consequently the full drag,state-dependent activity, singular stress, and non-atomic Jeffreys production all pass to the limit. The argument combines stability of regular Lagrangian flows with a relative-entropy estimate for simultaneously varying matrix drifts and jump rates. A stationary oscillation shows that the preparation cannot follow from the natural entropy bounds alone. With positive centre-of-mass diffusion we independently construct global large-data weak solutions and identify the same nonlinear terms. An explicit infinite-rank kernel proves that these results are not finite-species reductions.

There is significant interest in the study of calculus on graphs, especially regarding the use of gradient-based methods for applications in data driven problems such as classification, clustering and regularisation for inverse problems. Geometric graphs, whose vertices are take from from a Euclidean domain and whose edge structure is determined by the distance between the nodes in the domain, have been central in theoretical studies. Typical approaches for analysis, such as studying consistency and the existence of continuum limits, rely on $\Gamma$-convergence. This technique has some limitations, as it requires the typical length scale which determines the connectivity structure of the graph to be much larger than the scales frequently used for applications. Moreover, it may fail to provide quantitative results. This paper provides necessary and sufficient conditions on the asymptotic behaviour of this length scale for the existence of a uniform collection of Sobolev inequalities on a sequence of geometric graphs. Furthermore, these inequalities hold when the length scales are much smaller than what is typically assumed for $\Gamma$-convergence results and within the range of what is used for data-driven problems. The Sobolev inequalities provide a quantitative estimate on the $L^q$-regularisation effect of discrete gradients.

We investigate the asymptotic geometry of shifting superlevel sets $\Omega_t = \{x \in \Omega : u(x) > t\}$ generated by solutions to the elliptic Dirichlet problem $-\Delta u = f$ in $\Omega$, where the non-negative source $f \not\equiv 0$ is compactly supported within a strictly convex inner core $C \subset \Omega$. Under a quantitative radial monotonicity condition, each boundary $\partial\Omega_t$ is a smooth normal graph over $\partial C$ characterized by a thickness function $d_t \in C^{1,\alpha}(\partial C)$ tracking $d_0$ as $t \to 0$.A central contribution is a rigorous decomposition of the inward unit normal field along the level surfaces: $\mathbf{n}_{\Omega_t} = \nu - \nabla_{\partial C} d_t + \mathcal{G} + \mathcal{P}$, where $\nu$ is the static radial normal, $-\nabla_{\partial C} d_t$ is the kinematic driving vector, and $\mathcal{G}, \mathcal{P}$ are curvature and PDE Hessian remainder operators.In a thin-shell configuration $(\Vert{}d_0\Vert{}_{C^1} \ll 1)$, we formalize a discrete specular point-reflection mapping $F_n$ on $\partial C$. We prove the tangential displacement satisfies $F_n(p) - p = -2d_n(p)\nabla_{\partial C}d_n(p) + R_n(p)$, with quadratic control $\Vert{}R_n\Vert{}_{L^\infty} \le C\Vert{}d_n\Vert{}_{C^1}^2$. Using the energy $\mathcal{E}(t) = \int_{\partial C} d_t^2 \, d\mathcal{H}^{N-1}$, we show these orbits approximate a continuous gradient flow driven by $+\nabla_{\partial C} d_{\tilde{t}}(p)$ to first order. Finite element computations (FEniCS) validate these convergence rates.

Let $\Omega\subset\mathbb R^n$ be a bounded $C^2$ open set and let $P_t$ be the killed Dirichlet heat semigroup. We prove the sharp fixed-time power-scale modulus of $P_t$ for the Figalli–Gigli boundary-reservoir transport distances $W_{b,p}$. For every $t>0$, $P_t$ is globally Lipschitz with respect to $W_{b,1}$. For every $p>1$, and on every total-mass sublevel $\{\mu:\mu(\Omega)\le m\}$, it is $1/p$-Hölder: \[ W_{b,p}(P_t\mu,P_t\nu)^p \le C_{t,p,m,\Omega} W_{b,p}(\mu,\nu). \] For $p>1$, we show that the exponent $1/p$ is optimal in the scale of power moduli. On the full finite-measure space, $P_t$ is discontinuous at the zero measure. To establish the lower bound, we rely on the amplification of the boundary layer. More precisely, a unit mass initially placed at distance $\varepsilon$ from $\partial\Omega$ has input $W_{b,p}$-distance $O(\varepsilon)$ from zero, whereas after any fixed positive time, its $p$-th boundary moment is bounded below by $c\varepsilon$. As a result, in the quadratic case and in the original finite-measure $W_{b,2}$ metric, there does not exist a standard finite-$\lambda$ $\mathrm{EVI}_\lambda$ semigroup on a $W_{b,2}$-metric domain which would contain the affine constant-boundary data class and could restrict to the affine constant-boundary Dirichlet heat flow. Finally, we also describe the corresponding lower-bound obstruction for smooth uniformly elliptic perturbations in divergence form.

We introduce ideal Gårding polynomials, a convexity-enhanced subclass of Gårding polynomials whose Gårding components are recursively convex under partial differentiation. This class strictly contains real stable polynomials and, after translation and homogenization, lies in the Lorentzian class. Our main result is that ideal Gårding polynomials still admit a robust structure theory despite this additional convexity: they are preserved under polarization, satisfy natural closure properties, and support a linear preserver theory. A key contribution of this paper is a universal model for univariate Gårding polynomials, described by monotone root sequences and equivalently by volume polynomials of Pitman–Stanley polytopes. We establish quotient concavity, and Newton–Maclaurin type inequalities, which leads to the polarization theorem, and suggests further connections with convex geometry and Lorentzian polynomials.

We study continuous data assimilation for the two-dimensional incompressible Navier-Stokes equations on the periodic torus using a single signed scalar velocity projection in a prescribed spatially varying direction. We prove exponential synchronization in both L2 and H1 through two complementary mechanisms. In the regular resonant regime, nontrivial invisible currents are controlled through a moving-frame expansion, adapted coordinates, and Farhat-Lunasin-Titi logarithmic estimates, provided the resulting geometric shear defect is viscously absorbable. This recovers the periodic one-component mechanism for constant rational directions and applies to genuinely nonconstant projection fields. In the kernel-free regime, qualitative injectivity and compactness yield observability with arbitrarily small viscous leakage, giving synchronization for every sufficiently large gain without an upper gain restriction. For sufficiently regular projection fields, both mechanisms extend to L2-stable Type-I coarse scalar observations satisfying a first-order approximation property, under the usual gain-resolution condition. A common parabolic smoothing argument then upgrades all four L2-synchronization results to H1-synchronization without additional observation hypotheses.

We study the uniform-in-time stability and mean-field limit of an infinite Cucker–Smale-type (ICS-type) model in a fully noncompact setting. First, we clarify the main difficulties arising from the original ICS model and its corresponding kinetic Cucker–Smale (KCS) model, when the spatial support is noncompact. In particular, we show that the velocity diameter may remain constant in time, which invalidates the classical approach based on position and velocity diameters. Moreover, we construct a counterexample showing that the original KCS model does not satisfy uniform-in-time stability in the noncompact spatial setting. To overcome these obstacles, we introduce a new ICS-type model whose communication weight depends on a pairwise spatial moment. Under suitable assumptions, we also establish the uniform-in-time stability of the ICS-type model with respect to initial data in the noncompact setting. Finally, we derive the uniform-in-time mean-field limit for the ICS-type model, together with uniform-in-time stability for the corresponding KCS-type model.

In this paper, we study the low-regularity Cauchy problem for the Chern–Simons gauged $O(3)$ sigma model in $\mathbb{R}^{1+d}$ ($d=1,2$) under the Lorenz gauge. For $d=1$, we establish local well-posedness for initial data $(\boldsymbol{\phi}_0,\mathbf{A}_0)\in H^{s_1}(\mathbb{R})\times H^{s_1-1}(\mathbb{R})$ with $s_1>\frac12$. This improves the previous result of Jin and Huh \cite{HJ} by one quarter of a derivative and is almost optimal in view of the scaling-invariant regularities $\dot H^{1/2}(\mathbb{R})$ for the matter field and $\dot H^{-1/2}(\mathbb{R})$ for the gauge field. For $d=2$, we establish local well-posedness for initial data $(\boldsymbol{\phi}_0,\mathbf{A}_0)\in H^{s_2}(\mathbb{R}^2)\times H^{s_2-\frac34}(\mathbb{R}^2)$ with $s_2>1$. This improves the previous result of Jin and Zhang \cite{JZ} by one quarter of a derivative and brings the regularity threshold close to the scaling-invariant exponents $\dot H^{1}(\mathbb{R}^2)$ and $\dot H^{0}(\mathbb{R}^2)$ for the matter and gauge fields, respectively. The analysis relies on two main ingredients. In two space dimensions, we identify the complete null structure of the derivative nonlinearities, allowing the entire system to be treated within a unified null-form framework. In one space dimension, we establish a direct energy estimate in the function space introduced by Keel and Tao, avoiding the finite-propagation reduction to a small-data problem and enabling the low-regularity iteration for general initial data.

We consider the semilinear fully fractional heat equation \[ (\partial_t-\Delta)^\sigma u = |u|^{p-1}u \quad \text{in } \mathbb{R}^n \times \mathbb{R}_{-}, \qquad 0 < \sigma < 1. \] For $n\leq 2\sigma$ or $1<p\leq \frac{n+2\sigma}{n-2\sigma}$, we generalize the monotonicity formula and Liouville-type theorem when $\sigma=1$ proved by Giga and Kohn. In order to overcome the difficulty that this equation is nonlocal, we give a new interpretation of the classical Giga-Kohn's Pohozaev identity in terms of Hermite expansion. This insight is new and interesting even for $\sigma=1$. We further establish a space-time nonlocal monotonicity formula for the self-similar equation. As far as we are concerned, this is the first monotonicity formula for space-time nonlocal equations without using an extension by Stinga and Torrea.

We study the existence of solutions for systems of both elliptic and parabolic partial differential equations with potentially singular right-hand sides and free boundaries, posed in general smooth domains. Our results are established within a framework of "meta-theorems." This approach hinges on specific strong properties of the operators and their solutions (in approximate smooth settings) to guarantee the existence of a limit as the approximation parameter tends to zero. The primary challenge lies in applying these meta-theorems to prototype cases, which requires verifying that the necessary strong properties hold. For our analysis, we focus on fully nonlinear and $p$-Laplacian operators and a mixing of these operators, in both elliptic and parabolic contexts. While we focus on these specific cases, the meta-theorems remain valid for any other operators that satisfy the required properties. Beyond the complex proofs of our meta-theorems, and their applications to specific operators, a major challenge is the technical handling of the $p$-parabolic case, which requires proving the regularity of solutions of the $p$-parabolic equation with singular or degenerate right-hand side–addressed in the Appendix–along with several (new) properties, which is missing in the literature.

We investigate boundedness and regularity properties of weak solutions to a class of generalized logarithmic double phase equations with variable exponents. The considered operators arise from Musielak-Orlicz type energies of logarithmic double phase type and exhibit nonstandard growth features. Under general structural assumptions, we derive a priori boundedness estimates in the subcritical setting and establish boundedness of weak solutions also in the presence of critical growth terms. In addition, we prove global Hölder continuity up to the boundary by means of the De Giorgi iteration scheme, localization arguments, and the frozen functional technique. The obtained results extend several existing regularity results for double phase and related nonstandard growth problems.

We construct a new type II finite-time blow-up mechanism for the energy-supercritical heat equation \[ u_t=\Delta u+u^3, \qquad n\geq 5. \] The solution is positive and blows up only at the origin, but in a highly anisotropic fashion. As $t\nearrow T$, the solution concentrates in a thin tubular region around an $(n-4)$-dimensional sphere whose radius shrinks to zero at the self-similar scale \[ \xi_r(t)\sim \sqrt{2(n-4)(T-t)}. \] At the same time, concentration takes place transversely to the sphere at the much smaller scale \[ \lambda(t)\sim \kappa_* \frac{T-t}{|\log(T-t)|^{\frac n{n-2}}}, \] for some $\kappa_*>0$. More precisely, in cylindrical coordinates $r=|x'|$, $z\in\mathbb R^3$, the leading profile is \[ u(x,t) \sim \frac{1}{\lambda(t)} U\left( \frac{r-\xi_r(t)}{\lambda(t)}, \frac{z}{\lambda(t)} \right), \] where $U$ is the Aubin–Talenti bubble in $\mathbb R^4$. The construction reveals a two-scale singularity mechanism in which a critical transverse bubble concentrates around a geometric set that itself collapses. The concentration tube evolves at the parabolic scale $\sqrt{T-t}$, whereas its transverse thickness is governed by the much smaller type II scale $\lambda(t)$. The logarithmic blow-up law is determined by a nonlocal modulation equation arising from the interaction between the four-dimensional critical bubble and the axisymmetric heat kernel. To our knowledge, this seems to be the first Type II blowup that quantifies the effect of a self-similar collapsing tube. The exponent $p=3$ is energy-supercritical in dimensions $n\geq5$, but lies below the Joseph–Lundgren exponent for $5\leq n\leq 12$, in a regime where positive radial type II blow-up is ruled out. The present result provides the first example of a positive type II, single-point blow-up through a collapsing thin-tube geometry.

It is well-known that solutions to the incompressible Navier-Stokes system are limits in various contexts (weak or strong), of solutions to the Boltzmann equation, when the Mach and the Knudsen numbers go to zero. In particular the case of smooth solutions is by now rather well understood. Recent works have aimed at choosing initial data in function spaces as close as possible to those corresponding to well-posedness for the incompressible Navier-Stokes system. This paper tackles the case of measurevalued initial vorticity, in two space dimensions: in the special case when the initial vorticity is a Dirac mass, it is known that the unique solution to the Navier-Stokes system is the solution to the heat equation. We prove that the kinetic emanation of this initial vorticity (slightly smoothed out) leads to a solution of the Boltzmann equation which diverges in a strong way and in a very small time layer, from the expected hydrodynamic limit.

This paper studies the asymptotic behavior of positive solutions to the boundary Yamabe equation near an isolated singularity when the metric is not conformally flat. In dimensions 3 and 4, we establishes the sharp upper bound and, in the non-removable case, the matching lower bound, which also gives a necessary and sufficient condition for removability. Moreover, every solution with a non-removable singularity is shown to be asymptotically cylindrically symmetric, without relying on a global classification of Fowler-type singular solutions. These results aslo extend the flat half-space theory of Caffarelli-Jin-Sire-Xiong (2014) to non-flat boundary geometries and provide boundary analogues of the interior theories developed by Marques (2008) and Xiong-Zhang (2022).

We study the behaviour of heat semigroups on quantum graphs under Benjamini-Schramm convergence. For quantum graphs with uniformly bounded geometry, equipped with continuity and Kirchhoff vertex conditions, we show that the heat semigroup, transported to a common Hilbert space by a canonical breadth-first identification of the edges, depends continuously on the underlying rooted quantum graph with respect to a local Benjamini-Schramm-type metric. As a consequence, root-averaged pairings of the semigroup converge along Benjamini-Schramm convergent sequences of finite quantum graphs. Combining this with a Trotter-Kato-type approximation of the semigroup by semigroups on metric balls, we obtain a double-limit theorem interchanging the truncation radius and the graph limit.

We develop a new approach to the problem of the motion of a large number of rigid bodies immersed in a viscous fluid. The leading idea is the concept of cluster - a collection of individual rigid objects that may be grouped or even connected in such a way that their collective impact on the bulk motion of the system is similar to that of a single body. The applications of the new approach include: 1. Improving the critical value of the number of balls of small radius such that their cloud has no impact on the limit system represented by the incompressible Navier–Stokes equations. 2. The balls follow the fluid flow in the asymptotic limit of vanishing radius and increasing number even if a gravitational force is imposed.