arXiv:2606.30927v2 Announce Type: replace-cross Abstract: We introduce a discrete-energy Sobolev space $\mathcal{W}^{1,p}_{\mathscr V}(T)$ on Ahlfors regular snowtrees, a class of metric trees where every arc is a snowflake of the same type. Our main result shows that, for every partition $\mathscr V$ and every $1

<\infty$, this discrete space coincides quantitatively with the Korevaar--Schoen space on $T$. This fact and the independence of the space on the particular partition used to define $\mathcal{W}^{1,p}_{\mathscr V}(T)$ are both novel even for the class of geodesic trees. We also determine the critical Korevaar-Schoen exponent for Ahlfors regular snowtrees and prove capacity attainment and upper estimates, which reveal the appropriate walk dimension needed for the corresponding probabilistic profile on these trees.

arXiv:2607.07171v2 Announce Type: replace Abstract: For radial initial data, we construct explicit higher-order \(L^p(\mathbb R^N)\)-asymptotic profiles for the heat equation with Hardy potential. These profiles, denoted $A_n$ are obtained from the small-argument expansion, up to an arbitrary order \(n\), of the modified Bessel function appearing in the radial Hardy heat kernel. If $u$ is the mild solution generated by this kernel, we prove that the corresponding remainder $u(x,t)-A_n(x,t)$ admits a polynomial decay depending on $n$ in \(L^p(\mathbb R^N)\) as \(t\to\infty\). We also treat the non-radial case through spherical harmonics: each angular mode evolves according to a radial Hardy heat equation with a modified parameter, leading to finite and infinite angular expansion versions of the asymptotic profile under suitable summability assumptions.

arXiv:2607.03141v2 Announce Type: replace Abstract: 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.

arXiv:2607.00617v2 Announce Type: replace Abstract: We consider weak solutions to $p$-Laplace equations in cylindrical domains under mixed homogeneous Dirichlet-Neumann boundary conditions. We assume that the right-hand side is positive and locally Lipschitz continuous and we prove that any positive solution is monotone increasing in the $x_N$ direction for any $p>1$. As an application we prove that solutions to Allen-Cahn type equations are one-dimensional as well as a Liouville type result for Lane-Emden type equations.

arXiv:2606.10425v2 Announce Type: replace Abstract: We study the Cauchy problem for inhomogeneous evolution equations with time-dependent, potentially degenerate, and unbounded coefficients. A key feature of our work is allowing the principal coefficients to undergo arbitrary blow-up at both the initial and terminal times.

arXiv:2606.08222v2 Announce Type: replace Abstract: We consider a family of pure Neumann $p$-Laplacian problems, including eigenvalue problems, Lane-Emden type equations, and extremal cases such as sign nonlinearities and the $1$-Laplacian. Using variational methods, we develop a unified framework that establishes existence of solutions and characterizes their asymptotic behavior as the parameters vary. This approach reveals a natural asymptotic connection between pure Neumann $p$-Laplacian equations and a relative isoperimetric problem known as the Neumann-Cheeger problem. We describe the shape of minimizers in domains with different geometries and obtain results on regularity, uniqueness, multiplicity, symmetry, and symmetry breaking phenomena.

arXiv:2604.13736v2 Announce Type: replace Abstract: In this paper we consider the volume-constrained minimization of a variant of the Ohta-Kawasaki functional with an anisotropic surface energy replacing the standard perimeter. Following and suitably adapting the second variation approach devised in arXiv:1211.0164, we prove local minimality results for the horizontal lamellar configuration, in analogy with the isotropic case, under the assumption that the anisotropy is uniformly elliptic. If instead the Wulff shape of the anisotropy has upper and lower horizontal facets, we prove that the lamella exhibits a rigid behavior and is an isolated local minimizer for all parameter values. We conclude by showing some global minimality results, mostly focusing on the planar case.

arXiv:2603.29989v5 Announce Type: replace Abstract: In this paper we prove a Brunn-Minkowski inequality for the first Dirichlet eigenvalue of a Schr\"odinger type operator $\mathcal{H}_V:=-\operatorname{div}(A\nabla)+V$, where $V$ is convex and Kato decomposable, using the trace class property of the generated semigroup. As a consequence, we obtain the log-concavity of the ground state using the ultracontractivity of the semigroup, and also the strong log-concavity under additional assumptions on $\Omega$ and $V$.

arXiv:2602.05647v2 Announce Type: replace Abstract: We introduce and study a new class of higher order differential operators defined on $\mathbb{R}^{n}$, which are built with H\"{o}rmander vector fields, homogeneous w.r.t. a family of dilations (but not left invariant w.r.t. any structure of Lie group) and have a structure such that a suitably lifted version of the operator is hypoelliptic. We call these operators ''generalized Rockland operators''. We prove that these operators are themselves hypoelliptic and, under a natural condition on the homogeneity degree, possess a global fundamental solution $\Gamma\left( x,y\right) $ which is jointly homogeneous in $\left( x,y\right) $ and satisfies sharp pointwise estimates. Our theory can be applied also to some higher order heat-type operators and their fundamental solutions.

arXiv:2601.19456v2 Announce Type: replace Abstract: There has been significant recent interest in understanding the dependence on the wavenumber, $k$, of boundary integral operators (BIOs), supported on some set $\Gamma\subset \mathbb{R}^n$, that arise in the solution of the Helmholtz equation, $\Delta u + k^2 u=0$. Recently, for the Dirichlet boundary value problem with data $g$, Caetano et al (Proc. R. Soc. A, 481:20230650, 2025) have proposed a novel integral equation $A_k\phi=g$ that applies for arbitrary compact $\Gamma$. In this paper we study the dependence of $A_k$ on $k$, showing that, for $k\geq k_0>0$, $\|A_k\|\leq ck$ while $\|A_k^{-1}\| \leq c'k$ if $\Gamma$ is star-shaped, where $c, c'>0$ depend only on $k_0$ and $\Gamma$. Amongst other bounds we show that: (i) on the one hand, given any mildly increasing unbounded positive sequence $(k_m)$ and any unbounded sequence $(a_m)$, there exists $\Gamma$, with connected complement, such that $\|A_{k_m}^{-1}\|\geq a_m$ for every $m$; (ii) on the other hand, for every $\Gamma\subset \mathbb{R}^n$ and $k_0,\varepsilon, \delta>0$, there exists $c>0$ and $E\subset [k_0,\infty)$, with Lebesgue measure $m(E)\leq \varepsilon$, such that $\|A_{k}^{-1}\|\leq c k^{2n+2+\delta}$ on $[k_0,\infty)\setminus E$, i.e., the growth of $\|A_{k}^{-1}\|$ is at worst polynomial in $k$ if one avoids a set $E$ of arbitrarily small measure. As a corollary we obtain the first $k$-explicit bounds on the condition number of $S_k$, where $S_k$ is the standard single-layer BIO on $\Gamma$ when $\Gamma$ is the boundary of a Lipschitz domain, and analogous estimates when $\Gamma$ is a $d$-set (and so of Hausdorff dimension $d$), for non-integer values of $d$.

arXiv:2508.14388v2 Announce Type: replace Abstract: We prove a Carleman-type estimate for Dirichlet-stationary multivalued functions and apply it to give a different proof of the optimal dimension of the singular set of Dir-minimizing multivalued functions, originally due to Almgren and to De Lellis-Spadaro.

arXiv:2310.04868v3 Announce Type: replace Abstract: We establish a new class of weighted $L^2$ Poincar\'e and elliptic functional inequalities on smooth two-manifolds with explicit constants, for a family of weights satisfying a differential equation. This family includes, in particular, weights comparable to products of positive powers of the geodesic distance to finitely many points. Our primary motivation is the derivation of estimates associated with a weighted Hodge decomposition for one-forms.

arXiv:1406.7675v3 Announce Type: replace Abstract: (Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) We study dispersive equations with a time non-homogeneous modulation acting on the linear dispersion term. As primary models, we consider the Korteweg-de Vries equation (KdV) and related equations such as the Benjamin-Ono equation (BO) and the intermediate long wave equation (ILW), imposing certain irregularity conditions on the time non-homogeneous modulation. In this work, we establish phenomena called regularization by noise in three-folds: (i) When the modulation is sufficiently irregular, we show that the modulated KdV on both the circle and the real line is locally well-posed in the regime where the (unmodulated) KdV equation is known to be ill-posed. In particular, given any $s \in \mathbb R$, we show that the modulated KdV on the circle with a sufficiently irregular modulation is locally well-posed in $H^s(\mathbb T)$. Moreover, by adapting the $I$-method to the current modulated setting, we prove global well-posedness of the modulated KdV in negative Sobolev spaces. (ii) It is known that certain (semilinear) dispersive equations such as BO and ILW exhibit quasilinear nature. We show that sufficiently irregular modulations make the modulated versions of these equations semilinear by establishing their local well-posedness by a contraction argument, providing local Lipschitz continuity of the solution map. (iii) We also prove nonlinear smoothing for these modulated equations, where we show that a gain of regularity of the nonlinear part becomes (arbitrarily) larger for more irregular modulations. As applications of our approach, we also include further examples.

arXiv:2607.08631v1 Announce Type: cross Abstract: We prove that $S^3$ endowed with an arbitrary Riemannian metric $g$ admits at least two embedded minimal spheres. The proof is based on an iterative scheme of relative min-max constructions.

arXiv:2607.08509v1 Announce Type: cross Abstract: This paper investigates the behaviour of a fluid characterized by a viscosity simultaneously depending on pressure and shear rate within a Hele-Shaw cell featuring a sharp corner geometry. The study extends previous analyses conducted on purely pressure-dependent (piezo-viscous) and yield-stress fluids, providing a new perspective on confined complex flows. Motivated by practical applications related to designing biomedical devices and flows of relevance to biomedicine area, thin film technologies, injection molding -- to name only a few -- the flow configuration considered here can highlight essential features of complex fluid behavior in narrow-gap geometries around a sharp edge. Starting from the governing equations for an incompressible generalized Newtonian fluid and employing an appropriate rheological model, we derive the modified flow equations adapted to the Hele-Shaw flow. A particular solution is obtained near the corner region. Numerical simulations complement the theoretical results, illustrating the influence of the rheological parameters on the flow behavior.

arXiv:2607.08325v1 Announce Type: cross Abstract: We study the geometric regularization of a positive closed current by the (twisted) K\"ahler-Ricci flow on a compact K\"ahler manifold. We conjecture that the local Arnold multiplicities linearly decrease to zero, while the flow produces complete K\"ahler metrics in the Zariski open subset of points that have small Lelong numbers. We prove this conjecture in complex dimension 1 and provide several partial results in higher dimension.

arXiv:2607.08277v1 Announce Type: cross Abstract: Let $n\in\mathbb N\cap[2,\infty)$ and $\Omega\in L^1(\mathbb S^{n-1})$ with $\Omega\not\equiv 0$. In this article, we introduce a new family of lifted rough maximal operators $\{\mathcal{M}_\theta^\Omega\}_{\theta\in(0,\infty)}$ in the upper-half plane and establish their optimal weak-type estimates. Specifically, we prove that, for any $p \in (1, \infty)$, the estimate, with the positive equivalence constants independent of $f$, \[ \sup_{\theta,\lambda\in(0,\infty)}\lambda^p \underset{{\mathcal M}^\Omega_\theta(f)(x,t) > \lambda t^\frac{\gamma}{p}} {\int_{\mathbb R^n}\int_0^\infty} t^{\gamma-1}\,dt\,dx \sim \|f\|_{L^p(\mathbb{R}^n)}^p \] holds for all $f\in L^p(\mathbb R^n)$ if and only if $\gamma\in\mathbb R\setminus\{0\}$. For the endpoint case $p=1$ and $\Omega \in L(\log L)(\mathbb{S}^{n-1})$, we prove that the above estimate holds if and only if $\gamma \in (-\infty, -n) \cup (0, \infty)$. As applications, we obtain weak-type estimates for generalized Poisson integrals without any logarithmic integrability assumptions, which gives an affirmative answer to the question posed by Sj\"ogren and Soria in page 228 of [Israel J. Math. 95 (1996)]. Moreover, although the operator $M^\ast_\Omega$, arising from the method of rotation of Calder\'on and Zygmund, is not of weak type $(1,1)$, we find that its lifted variant is weak type $(1,1)$. In addition, we establish a new characterization of Hardy spaces in terms of truncated rough singular integrals.

arXiv:2607.08209v1 Announce Type: cross Abstract: The proximal operator is a fundamental tool in variational analysis and optimization. In the setting of a Hilbert space, given a proper, lower semicontinuous convex functional, its proximal operator is non-expansive, that is, 1-Lipschitz continuous. In the Wasserstein setting, the contraction properties of this operator have been investigated from different perspectives by Carlen and Craig and Adve and M\'esz\'aros, among others, and are not completely understood. In this paper, we study the stability properties of proximal maps, with a particular focus on non-expansivity, under various notions of convexity of the functional that can be considered in the Wasserstein space.

arXiv:2607.08067v1 Announce Type: cross Abstract: This work is concerned with an inverse medium problem for elastic waves, in which unknown inhomogeneities are reconstructed from time-resolved boundary measurements. We propose a novel time-domain direct sampling method for locating scatterers from a single incident source, without imposing specific assumptions on the temporal profile of the excitation. In particular, the imaging functional introduces a time-shifted correlation strategy that replaces the traditional $P$-$S$ wave decomposition with a travel-time alignment mechanism, thereby enabling direct imaging from the coupled elastic wave field. To analyze the proposed time-domain imaging functional, we employ Parseval's identity for the Fourier--Laplace transform and reformulate the functional in the frequency domain. By exploiting properties of modified Bessel functions, we characterize the asymptotic behavior of the imaging functional and show that it attains its maximum at the target location, which enables reliable identification of the scatterer. Rigorous theoretical justifications are provided to substantiate the effectiveness of the proposed method. Numerical experiments are also presented to demonstrate its performance and applicability.

arXiv:2607.07910v1 Announce Type: cross Abstract: The framed Beltrami--Vekua equation $\Phi(w_{\bar z} - \mu w_z) + \Psi(\overline{w_z} - \mu\overline{w_{\bar z}}) + \mathfrak{a}w + \mathfrak{b}\bar w = \mathfrak{f}$, with $|\mu|<1$ and $|\Phi|>|\Psi|$, carries a numerator field $N = \Phi\mathfrak{b} - \Psi\mathfrak{a} - W_L(\Phi,\Psi)$ whose weighted modulus integrates to the pseudo-analytic mass. This paper extracts the integer carried by the same field. When the zero set of $N$ is compactly contained in a bounded simply connected domain, the winding number of $N$ along any enclosing curve -- the pseudo-analytic charge $n \in \mathbb{Z}$ -- is invariant under every recombination $w = \varphi w' + \psi\bar w'$ of the unknown, every scaling of the equation, and every orientation-preserving $C^1$ change of variables: recombinations multiply $N$ by the positive factor $|\varphi|^2 - |\psi|^2$, so their invariance is exact, while on multiply connected domains the other two actions fix the component charges only in $\mathbb{Z}/2\mathbb{Z}$ and the total charge exactly. The charge is a Brouwer degree: it localizes at the zeros of $N$, vortices which no action of the class creates or destroys; an isolated vortex persists under perturbation of the data precisely when its local charge is non-zero. It involves the Beltrami coefficient only through the $L$-Wronskian of the frame, and is $\mu$-independent wherever $W_\partial(\Phi,\Psi) \equiv 0$ -- in particular at the trivial frame, where $N = \mathcal{B}$ and the charge is the gauge-invariant winding of the coefficient of the Beltrami--Vekua equation. Mass and charge are independent: every pair in $(0,\infty)\times\mathbb{Z}$ is realized.

arXiv:2607.08685v1 Announce Type: new Abstract: We give a self-contained short proof of optimal regularity for minimizers of the Alt-Phillips Free Boundary Problem for $\gamma \in (0, 1)$. We adopt a dichotomy argument that originates from [DS20].

arXiv:2607.08598v1 Announce Type: new Abstract: This work investigates time-harmonic electromagnetic scattering governed by the Maxwell system in the presence of bounded anisotropic electromagnetic scatterers embedded in an intermediate anisotropic electromagnetic layer. We focus on the localized enhancement of the gradients of the total electric and magnetic fields in small boundary-attached neighborhoods of finitely many prescribed points on the outer interface of the surrounding layer. We show that, through a suitable construction of incident electromagnetic waves, the gradients of both the total electric field and the total magnetic field can be made arbitrarily large in these neighborhoods. Moreover, the localization radius may be chosen according to the prescribed gradient magnitude, thereby describing a localized high-gradient concentration mechanism for electromagnetic fields near anisotropic scatterers. The main strategy is based on the introduction of auxiliary boundary-attached electromagnetic neighborhoods and the associated electric and magnetic fields, which exhibit strong gradient variation near the prescribed points. Using the approximation property of Maxwell Herglotz wave functions, these auxiliary fields are then approximated by physically admissible incident waves in the neighborhood of the scatterers. Together with the well-posedness and continuous dependence of the anisotropic scattering problem, this implies that the corresponding scattered field can be controlled to be sufficiently weak in the relevant layer region. Consequently, the total field is dominated by the incident field near the prescribed points and inherits its large-gradient behavior.

arXiv:2607.08498v1 Announce Type: new Abstract: The dynamics of a mathematical model for a chemostat-type reactor is investigated. The model describes the temporal evolution of suspended and wall-attached bacterial populations, with the latter represented as a one-dimensional biofilm, subject to a non-reproducing growth-limiting substrate and a reaction product formed through bacterial utilization of the substrate. In particular, it is shown that, in the regime where the trivial (washout) equilibrium is unstable, there exists a unique nontrivial equilibrium that is locally asymptotically stable. Under slightly stronger assumptions, uniform persistence and global asymptotic stability of the nontrivial equilibrium are established.

arXiv:2607.08433v1 Announce Type: new Abstract: The SIGEST paper Nielsen and Strako\v{s} (2024) characterized the spectrum of the preconditioned operator $\Delta^{-1}[\nabla \cdot (K\nabla u)]$ in a bounded open two-dimensional domain $\Omega$, where $\Delta$ denotes the Laplacian and $K(x,y)$ is a continuous symmetric matrix-valued function. An important part of the analysis states that for a diagonal tensor $K$ constant in an open subdomain $S \subset \Omega$, the closed interval defined by its diagonal elements belongs to the spectrum of the preconditioned operator. This result is correct, but the proof in Nielsen and Strako\v{s} (2024) must be refined. This paper presents a refined proof and extends the previous work. As shown in the cited papers, for any point $\lambda$ in the open interval defined by the elements of the diagonal tensor constant in $S$ and any point $(x_0,y_0)\in S$, a rectangular subdomain $\Sigma_l\subset S$ can be constructed such that the generalized eigenvalue problem associated with the preconditioned operator restricted to $\Sigma_l$, of arbitrarily small size, has the eigenvalue $\lambda$ and infinitely many eigenfunctions. These are given by solutions of a locally defined wave equation. However, such solutions of the locally restricted generalized eigenvalue problem cannot be extended to the whole domain $\Omega$. Using instead rectangular subdomains whose size shrinks to zero, the present paper constructs a Weyl singular sequence of \emph{approximate} eigenfunctions associated with $\lambda$, proving that $\lambda$ belongs to the spectrum of the preconditioned operator. Since self-adjoint operators in a separable Hilbert space can have at most a countable set of eigenvalues, this shows that the eigenvalues of the locally defined operator converge to points of the continuous spectrum of the preconditioned operator on the entire domain.

arXiv:2607.08405v1 Announce Type: new Abstract: We initiate the study of Lyapunov-type inequalities for Dirichlet problems driven by the discrete p-Laplacian on weighted graphs. The approach is capacitary and is based on point p-capacities and the associated capacitary radii. First, we prove general Lyapunov-type inequalities on arbitrary connected locally finite weighted graphs. These inequalities provide intrinsic lower bounds, expressed in terms of the capacitary radii, for the positive part of the potential whenever the corresponding Dirichlet problem admits a nontrivial solution. Next, we estimate these capacitary radii in several geometric settings and prove the sharpness of the resulting Lyapunov-type inequalities. As an application, we derive lower bounds for the first weighted Dirichlet eigenvalue of the discrete p-Laplacian.

arXiv:2607.08401v1 Announce Type: new Abstract: We investigate the stability and asymptotic behavior of spatially periodic cnoidal waves in the Korteweg-de Vries equation subject to localized perturbations. Standard stability arguments in Hamiltonian systems break down in this setting, since localized perturbations preclude a characterization of stable periodic waves as strict minimizers of a suitable energy functional subject to finitely many constraints. As a result, the nonlinear stability of periodic waves under localized perturbations has remained a long-standing open problem in Hamiltonian systems, with previous results only addressing plane waves that can be reduced to constant states by passing to polar coordinates. In this paper, we develop a novel method that resolves this obstruction by combining variational arguments, Floquet-Bloch theory, and Duhamel-based estimates with spatiotemporal modulation. Our framework applies to general periodic waves in Hamiltonian systems with symmetry and reduces the nonlinear stability problem to verifying diffusive spectral stability conditions for the second variation of a suitable conserved energy. Applying our approach to cnoidal waves in the Korteweg-de Vries equation, we obtain the first nonlinear stability result for periodic waves in Hamiltonian systems under localized perturbations that cannot be reduced to constant states.

arXiv:2607.08385v1 Announce Type: new Abstract: We investigate pathwise well-posedness of the stochastic modulated Korteweg-de Vries equation (KdV) on the circle with a multiplicative noise, where a time non-homogeneous modulation acts on the linear dispersion term. (i) In the Young case (= fractional-in-time case with Hurst parameter greater than $\frac 12$), we establish a new regularization-by-noise phenomenon on the stochastic convolution in a pathwise manner, where a gain of spatial regularity becomes (arbitrarily) larger for more irregular modulations. We then prove that, given any $s \in \mathbb R$ and any multiplicative Young noise, however rough it is in space, the stochastic modulated KdV is pathwise locally well-posed in $H^s(\mathbb T)$, provided that the modulation is sufficiently irregular. (ii) In the rough case (= white-in-time case), irregularity of the modulation does not induce any smoothing on the stochastic convolution, and in fact, there is a slight loss in the spatial regularity. In this case, by slightly regularizing the multiplicative noise term, we prove pathwise local well-posedness in $H^s(\mathbb T)$ for any given $s \in \mathbb R$, provided that the noise is sufficiently smooth in space. We achieve these goals by combining (i) the sewing lemma approach to the nonlinear Young integration theory, introduced by Chouk and the second author (2014), and (ii) the pathwise construction of stochastic convolutions as Young or rough integrals via the random tensor estimate and the sewing lemma, introduced by the first, fourth, and fifth authors (2026). In the appendix, we also present an example of regularization by noise for a stochastic modulated Schr\"odinger equation with a multiplicative Young noise.

arXiv:2607.08381v1 Announce Type: new Abstract: We study the existence, multiplicity, and concentration of normalized semiclassical states for a magnetic Schr\"odinger--Poisson system in $\mathbb{R}^3$ featuring both the Sobolev-critical local nonlinearity $|u|^4u$ and a critical nonlocal Poisson interaction. The problem is considered under the prescribed mass constraint $\int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3,$ where $a>0$ denotes the prescribed mass and $\varepsilon>0$ is the semiclassical parameter. By combining constrained variational methods, a suitable penalization scheme, concentration--compactness arguments, and Ljusternik--Schnirelmann theory, we first prove the existence of a normalized semiclassical solution for sufficiently small $a$ and $\varepsilon$. We then establish a multiplicity result showing that, for every sufficiently small $\varepsilon>0$, the number of distinct normalized solutions is bounded from below by the Ljusternik--Schnirelmann category of the minimum set \[ \mathcal M = \{x\in\mathbb{R}^3:V(x)=\min_{\mathbb{R}^3}V\}. \] Finally, we describe the semiclassical concentration phenomenon by showing that the maximum points of the resulting solutions approach $\mathcal M$ as $\varepsilon\to0$.

arXiv:2607.08190v1 Announce Type: new Abstract: The effect of a moving impurity in a dilute Bose-Einstein condensate is investigated by means of the one-dimensional Gross-Pitaevskii model (GP) with non-zero boundary conditions at infinity. The impurity is modeled as a localized external potential, that travels at constant speed $v \in \mathbf{R}$. In a co-moving reference frame, we study the existence and stability of time-independent solutions. The latter are of physical relevance, being associated with the superfluid behavior of the condensate. For every non-zero velocity $v$ in the subsonic regime, we show the existence of a family of time-independent solutions which bifurcates from a (displaced) gray soliton $\phi_{0,v}(x-s_0)$, with $s_0 \in \mathbf{R}$, of the GP equation. The position $s_0$ is determined as an extremal point of an effective potential explicitly defined. Moreover, we study the spectral stability of these states. For small values of the potential strength, we show that the families originating from the maxima of the effective potential are spectrally unstable. For this last result, we employ an Evans function approach. Finally, we formally apply the instability result to the case of a repulsive delta potential.

arXiv:2607.08140v1 Announce Type: new Abstract: We consider a nonlinear PDE describing a nonlinear electrostatic medium with nonlocal dielectricity. The existence proof for the corresponding equation is based on Schauder's theorem and a new compactness theorem for moving coefficients (``Helga's Theorem''). This technique uses insights from (operator-theoretic/topological) homogenisation theory. Surprisingly, even though monotonicity assumptions are neither used nor valid, the underlying domain is only required to be weak Lipschitz and no assumption on the derivatives of the nonlinearity is needed.

arXiv:2607.08061v1 Announce Type: new Abstract: We consider a family of semilinear parabolic equations with homogeneous Neumann boundary conditions on a family of varying non-smooth domains $\{\Omega_\mu\}_{\mu \in \Lambda} \subset \mathbb{R}^n$. Assuming only that the domains have uniformly bounded volumes, satisfy a uniform Jones condition, and possess uniform ellipticity bounds, we establish the well-posedness of the problem in an appropriate scale of fractional Banach spaces and prove the existence of global attractors. Using a Moser-Alikakos bootstrap iteration in tandem with the uniform Gronwall lemma and the uniform properties of the Jones extension operator, we show that the family of attractors is uniformly bounded in $L^\infty(\Omega_\mu)$. Finally, assuming the volume convergence of the domains, $|\Omega_\mu \triangle \Omega_0| \to 0$, we construct a framework of connecting maps to prove that the family of attractors is upper semicontinuous at $\mu = 0$ in the strong $H^1$ topology.

arXiv:2607.08060v1 Announce Type: new Abstract: We study finite-time breakdown of classical solutions to the Euler-alignment system through the degeneration of the associated Lagrangian flow. This approach allows us to characterize singularity formation in terms of the loss of local invertibility of the flow and the resulting concentration of density along characteristics. For the case of constant communication kernels, we derive an explicit formula for the flow and obtain an exact pointwise breakdown criterion in arbitrary dimension. In two dimensions, this criterion admits a closed-form reformulation in terms of the symmetric part of the initial velocity gradient and the initial vorticity. For general non-constant kernels, we derive sufficient conditions for finite-time degeneracy by combining a leading compressive mechanism with perturbative control of the nonlocal remainder. These conditions provide quantitative supercritical breakdown criteria in arbitrary dimension, complementing the existing subcritical global-regularity theory for multidimensional Euler-alignment systems.

arXiv:2607.08021v1 Announce Type: new Abstract: In this paper, we study the following elliptic system \begin{equation}\label{main_1} \begin{cases} -\Delta u = |v|^{p-1} v + \epsilon (\alpha u + \beta_1 v), & \text{in } \Omega, \\ -\Delta v = |u|^{q-1} u + \epsilon (\beta_2 u + \alpha v), & \text{in } \Omega, \\ u = v = 0, & \text{on } \partial \Omega, \end{cases} \tag{*} \end{equation} where \(\Omega\) is the unit ball in $\mathbb{R}^N$, \(\epsilon\) is a small parameter, \(\alpha\), \(\beta_1\) and \(\beta_2\) are real numbers, \((p, q)\) is a pair of positive numbers lying on the critical hyperbola \begin{equation} \frac{1}{p+1} + \frac{1}{q+1} = \frac{N-2}{N}.\nonumber \end{equation} Under suitable assumptions and suitable restrictions on $(p,q)$ and $N$, we construct infinitely many sign-changing solutions to \eqref{main_1} which look like a positive radial solution to \eqref{main_1} crowned by $k$ negative bubbles arranged on a regular polygon of a suitable radius, whose energy can be arbitrarily large.

arXiv:2607.07966v1 Announce Type: new Abstract: We extend some results of weak KAM theory to Lagrangians that are defined only on the horizontal distribution of a subriemannian manifold and depend on the unknown function

arXiv:2606.03373v3 Announce Type: replace-cross Abstract: We construct a natural Legendrian cycle $\mathcal{N}_\mathcal{S}$ associated with any $F_nW^{2,n}$-set $\mathcal{S}$, that is, a closed set locally described as a finite union of graphs of $(C^0\cap W^{2,n})$-regular functions with integer multiplicity. The construction relies on the fact that $\mathcal{S}$ is countably $\mathcal{H}^n$-rectifiable of class $C^2$ and, at $\mathcal{H}^n$-almost every point $p\in\mathcal{S}$, the proximal unit normal bundle at $p$, denoted by $\operatorname{nor}(\mathcal{S},p)$, consists of exactly two antipodal vectors $\{u,-u\}$, even in the presence of overlapping $W^{2,n}$-graphs. As a consequence, we prove Reilly-type variational formulae for the higher-order mean curvature integrals of $\mathcal{S}$, extending the classical results of Reilly to this non-smooth setting.

arXiv:2602.15810v2 Announce Type: replace-cross Abstract: We use Gaussian measure on $\mathbb{R}^N$ to define the coefficients of an elliptic diffusion and show that it lives in an open cone of $\mathbb{R}^2$. One component represents enstrophy and the other energy. We establish the existence and uniqueness of a stationary distribution for this diffusion. Owing to the special properties of the coefficients of this diffusion, we derive a condensation bound, which controls the distance to $1$ of the ratio of the expected energy to the expected enstrophy (this ratio is at most $1$ with our normalization). In a companion article, as a ``proof of concept'', we show that the diffusion constructed in this work is the inviscid limit of the laws of the ``enstrophy-energy'' process of a stationary $N$-dimensional Galerkin-Navier-Stokes type evolution with Brownian forcing and random stirring (the strength of which can be made to go to zero in the inviscid limit, and which plays the role of a regularization).

arXiv:2602.15805v2 Announce Type: replace-cross Abstract: In this article we consider a stationary $N$-dimensional Galerkin-Navier-Stokes type evolution with Brownian forcing and random stirring (of arbitrarily small strength that plays the role of a regularization). We show, as a ``proof of concept'', that the stationary diffusion in an open two-dimensional cone constructed in a companion article, stands as the inviscid limit of the laws of the ``enstrophy-energy'' process of the $N$-dimensional diffusion process considered here, this regardless of the strength of the stirring. With the help of the quantitative condensation bounds of the companion article, we infer quantitative inviscid condensation bounds, which for suitable forcings show an attrition of all but the lowest modes in the inviscid limit.

arXiv:2512.01566v3 Announce Type: replace-cross Abstract: We study reparametrization-invariant Sobolev-type Riemannian metrics on the space of immersed surfaces and establish conditions ensuring metric and geodesic completeness as well as the existence of minimizing geodesics. This provides the first extension of completeness results for immersed curves, originating from works of Bruveris, Michor, and Mumford, and validates an earlier conjecture of Mumford on completeness properties of general spaces of immersions in this important case. The result is obtained by recasting earlier approaches to completeness on manifolds of mappings as a general completeness criterion for infinite-dimensional Riemannian manifolds that are open subsets of a complete Riemannian manifold and by combining it with geometric estimates based on the Michael--Simon--Sobolev inequality to establish the completeness for specific Sobolev metrics on immersed surfaces. We expect that this approach will be useful for obtaining completeness results for other manifolds of mappings.

arXiv:2606.30284v5 Announce Type: replace Abstract: One proves that the the vorticity flow of 2D Navier Stokes equation can be identified with an absolutely continuous curve in Wasserstein space W_{p} where p\in [1,2}.

arXiv:2605.28982v2 Announce Type: replace Abstract: The goal of this short note is to prove qualitative stability for a family of trace Sobolev inequalities first proven by Carlen \& Loss for $p=2$ and by Maggi and the author for $p\in (1,n)$. This answers an open problem raised in a recent paper of Fan, Li \& Zhang, and in conjunction with their local analysis, yields sharp quantitative stability for this family of inequalities when $p=2$.

arXiv:2604.19535v2 Announce Type: replace Abstract: We consider the cubic nonlinear Schr\"{o}dinger system with Rashba type Spin-Orbit Coupling (SOC) on $\mathbb{R}^2$. The system describes SO-coupled spinor BEC in physics. In the literature of physics, the small semi-vortex solutions, small ground state, and the so-called mixed mode, which are mixture of semi-vortex solutions, are investigated. The semi-vortex solutions cause from the resonance on the essential spectrum of the linear operator. In the present paper, we give mathematical proofs of the existence of the semi-vortex and the ground state by finding minimizers of the energy under small mass constraint based on concentration compactness argument. Moreover, we also discuss the mixed modes in the case where all the coefficients of the nonlinear terms are equal.

arXiv:2604.11260v2 Announce Type: replace Abstract: This paper concerns a coupled semilinear SPDE-ODE system modelling the electropermeabilization phenomenon, which designates a transient increase in cell membrane permeability induced by short, high-voltage electric pulses. We present a stochastically perturbed electroporation model that couples electrostatic equations for the electric potential in the extra- and intracellular domains and a nonlinear evolution law for the transmembrane potential jump with an ordinary differential equation describing the porosity degree of the membrane. We prove the existence and uniqueness of a variational solution of the resulting coupled stochastic PDE-ODE system. Its long-time behavior is governed by the corresponding invariant measure for which we establish the regularity of its support. The ergodicity of this invariant measure is further established for a truncated nonlinear reaction term, corresponding to the case of a bounded electric potential. The main technical challenge arises from the nonlinear reaction term, which is neither Lipschitz continuous nor locally monotone. We also present a numerical example, computing the solution and its time averages for both additive and multiplicative noise, that provides an indication for the existence of an invariant measure.

arXiv:2603.11700v2 Announce Type: replace Abstract: This article is concerned with the inverse problem on determining the temporal component of the source term in a coupled time-fractional diffusion system by a single point observation. Under a non-degeneracy condition on the known spatial component, we establish the Lipschitz stability by observing all solution components by a series representation of the mild solution. To reduce the observation data, we prove the strict positivity of some fractional integral of the solution to the homogeneous problem by a modified Picard iteration. This, together with a coupled Duhamel's principle, lead us to the uniqueness of the inverse problem by observing any single solution component under a specific structural constraint on the unknown. Numerically, we propose an iterative regularizing ensemble Kalman method (IREKM) for the simultaneous recovery of the temporal sources. Through extensive numerical tests, we demonstrate its accuracy, robustness against noise, and scalability with respect to the number of components. Our findings highlight the essential roles of the non-degeneracy condition, measurement configuration, and fractional structural constraints in ensuring reliable reconstructions. The proposed framework provides both rigorous theoretical guarantees and a practical algorithmic approach for multi-component source identification in fractional diffusion systems.

arXiv:2602.16293v3 Announce Type: replace Abstract: In this paper, our first objective is to investigate the decay rates and the global (in time) existence of solutions to the semilinear damped wave equation with the Riesz potential-type power nonlinearity $\mathcal{I}_\gamma\left(|u|^p\right)$, where $\gamma\in[0,n)$, in terms of the decay character of the initial data. This approach enables us to establish global existence results for several classes of initial data. Our second objective is to show, via a blow-up argument, that the conditions imposed on the nonlinearity in the global existence theorem are sharp for initial data belonging to the pseudo-measure space $\mathcal{Y}^q$. As a consequence, we derive the new critical exponent $$ p_{\mathrm{crit}}(n,q,\gamma):=1+\frac{2+\gamma}{n-q} $$ for $1\leq n\leq 4$ and $0\leq \gamma

arXiv:2601.04117v2 Announce Type: replace Abstract: As a first step towards resolving a vanishing cosmological constant black hole stability conjecture, we prove energy, Morawetz and rp-weighted estimates for solutions to the Teukolsky equations on a slowly-rotating Kerr-de Sitter background, which we derive using an extension of the non-integrable formalism of [GKS24]. The main feature of our estimates is their uniformity with respect to the cosmological constant $\Lambda>0$ (thus allowed to tend to 0), while they hold on the whole domain of outer communications, extending up to $\Lambda^{-\frac{1}{2}}$. As an application of our result, we recover well-known corresponding estimates for solutions to Teukolsky on a slowly-rotating Kerr background in the limit $\Lambda\to 0$.

arXiv:2512.16332v2 Announce Type: replace Abstract: This paper combines the decay of high modes with the smallness introduced by high orders, leading to a normal form lemma for infinite-dimensional Hamiltonian systems under ultra-differentiable regularity. We prove the sub-exponential stability time of a wide class of Hamiltonian PDEs, including the Schr\"odinger equation with convolution potentials, fractional-order Schr\"odinger equations, and beam equations with metrics. When the conditions are equivalent to previous ones, the stability time we obtain reaches Bourgain's predicted optimal bound. Furthermore, we approach earlier results under lower conditions. These results are discussed within a general framework we propose, which applies to the ultra-differential class.

arXiv:2512.16299v3 Announce Type: replace Abstract: This paper investigates Nekhoroshev-type stability for solutions of ultra-differentiable regularity in Schr\"odinger equations with non-local nonlinear terms, employing the method of rational normal forms. We establish the first rigorous results for logarithmic ultra-differentiable regularity in infinite-dimensional Hamiltonian systems without external parameters. Under Gevrey class regularity assumptions, we achieve the stability times matching Bourgain's conjectured optimal stability time in \cite{B04}. Furthermore, we introduce a novel global vector field norm adapted to the rational normal form framework. This norm eliminate the need for degree tracking during the iteration process, thereby enabling a unified treatment of nonlinear terms.

arXiv:2501.08405v2 Announce Type: replace Abstract: Solutions to nonlinear integro-differential systems are regular outside a negligible closed subset whose Hausdorff dimension can be explicitly bounded from above. This subset can be characterized using quantitative, universal energy thresholds for nonlocal excess functionals. The analysis is carried out via the use of nonlinear potentials and allows to derive fine properties of solutions under sharp assumptions on data and kernel coefficients.

arXiv:2401.05328v2 Announce Type: replace Abstract: We prove the existence of weak solutions to steady, compressible non-Newtonian Navier-Stokes system on a bounded, two- or three-dimensional domain. Assuming the viscous stress tensor is monotone satisfying a power-law growth with power $r$ and the pressure is given by $\varrho^\gamma$, we construct a solution provided that $r>\frac{3d}{d+2}$ and $\gamma$ is sufficiently large, depending on the values of $r$. Additionally, we also show the existence for time-discretized model for Herschel-Bulkley fluids, where the viscosity has a singular part.

arXiv:2112.03995v2 Announce Type: replace Abstract: We study the inflow-outflow boundary value problem on an interval, the analog of the 1D shock tube problem for gas dynamics, for general systems of hyperbolic-parabolic conservation laws. In a first set of investigations, we study existence, uniqueness, and stability, showing in particular local existence, uniqueness, and stability of small amplitude solutions for general symmetrizable systems. In a second set of investigations, we investigate structure and behavior in the small- and large-viscosity limits. A phenomenon of particular interest is the generic appearance of characteristic boundary layers in the inviscid limit, arising from noncharacteristic data for the viscous problem, even of arbitrarily small amplitude. This induces an interesting new type of ``transcharacteristic'' hyperbolic boundary condition governing the formal inviscid limit.