Papers

We introduce graph-compatible powers determined by the coupling graph for multicomponent weakly coupled elliptic systems and establish concavity of the associated transformed components through a concave-envelope method and a weighted viscosity comparison principle. We further establish a componentwise constant-rank theorem for the transformed Hessians, which yields strict power concavity under additional assumptions.

This paper discusses about an inverse scattering problem to the nonlocal nonlinear Schrodinger equation. We extend the previous result treated by Y. Zhao and E. Fan of inverse scattering result at the point with slowly decaying data. More precisely, we show the global existence of solution in $H^{1,s}$ for $s>1/2$ by assuming small data in $L^1$. The proof relies on the inverse scattering method based on the ZS-AKNS (Zakharov-Shabat, Ablowitz-Kaup-Newell-Segur) eigenvalue problem on the spectral theory of ordinary differential equations upon the framework of slowly decaying data.

We consider nonlinear diffusion equations on an interval where the diffusion coefficient in a small region near the center is scaled such that it approximates a transmission condition at a membrane. While the limiting behavior of the solutions is well-understood, we study the convergence in the sense of the energy-dissipation principle (EDP) of associated gradient structures given in terms of a free energy and a dissipation potential. EDP-convergence provides a uniquely specified limiting gradient structure that reformulates the transmission condition in terms of an effective kinetic relation for the membrane, which relates the jump of the chemical potential and the flux through the membrane. We show how \AAA properties of the chosen free energy and the mobility of small-scale diffusion migrate to the effective kinetic relation. A surprising result is that starting from the linear Onsager relation of Otto's gradient structure for the linear diffusion equation, one obtains an exponentially growing kinetic relation, the so-called Marcelin-De Donder kinetic.

We introduce a notion of pairing between essentially bounded tensor fields with divergence measure and vector-valued functions of bounded variation, extending the classical theory to the tensorial setting. This naturally leads to an adaptation of the definition of normal trace for tensor fields with measure divergence even on a rectifiable set. As a consequence, we establish tensorial Gauss-Green formulas that remain valid on sets with low regularity, including sets of finite perimeter. These results yield a unified and robust framework for integration by parts in the presence of irregular tensor fields and domains.

We prove the Harnack inequality (with tails) and local Hölder regularity for the fractional logarithmic $p$-Laplace operator, which is derived by differentiating the fractional $p$-Laplace operator with respect to its order. To be more precise, for a suitable function $u,$ the operator reads as the first order derivative \begin{align*} (-\Delta_p)^{s+\log} u:= \frac{d}{dt}(-\Delta_p)^t u \Big|_{t=s} \end{align*} at any arbitrary order $s\in (0, 1).$ The kernel of this operator involves a logarithmic factor. As a consequence, it changes sign at large scales and, near the diagonal, is more singular than the kernel of the fractional $p$-Laplacian. To achieve our regularity estimates, we adopt the classical De Giorgi-Nash-Moser techniques in this setting. We also construct an example showing that the Harnack inequality fails without tail terms. Our results are new even in the linear setup $p=2$.

We prove the minimisers of the Lieb-Oxford bound (if any) for a fixed (and finite) number of particles $N \geq 1$ are necessarily compactly supported, extending a result proved by E. H. Lieb and S. Oxford in the one-particle case.

We study Nakao's problem in two and three space dimensions, a weakly coupled system consisting of a semilinear damped wave equation and a semilinear undamped wave equation. We establish the global in-time existence and uniqueness of small data mild Sobolev solutions in new admissible ranges, together with time-dependent estimates matching those for the corresponding linearized problems. The proof combines diffusion-type $L^m-L^r$ estimates for the damped component with the Poisson and Kirchhoff formulas for the undamped component. Dimension-dependent solution spaces are introduced to incorporate the weaker wave decay and logarithmic $L^2$ growth in two space dimensions. In three space dimensions, a two-level fixed point argument avoids an artificial restriction on $p$ by establishing the self-map property in a strong space and the contraction in a weaker metric. Furthermore, the undamped component scatters to a free wave in $H^2\times H^1$.

This study presents a three-phase suspension lava flow model with a Herschel-Bulkley rheology. The suspension contains crystals and gas bubbles, and two closures for the evolution of the crystal volume fraction are considered. The first closure minimizes the complexity of the system by treating crystal fraction as a transported quantity subject to relaxation towards an equilibrium state. This closure avoids a parametrization of the many heat transfer mechanisms a lava flow is subjected to. The other closure is on the lava temperature considering four heat transfer mechanisms and a prescribed temperature–crystallinity relationship. We deduce from this system and solve numerically a one-dimensional depth-averaged model. A comparison with a pre-existing model based on real lava flow data suggests that the prediction of flow parameters done with the multi-parametric evolution of temperature yields more accurate results than those obtained with the other closure. The transient nature of our model correctly predicts that confined lava traveling down an irregular steep slope yields a series of cascading fill-then-breakout lumps that causes the overall flow to be pulsatory. These pulses dominate the local dynamics and preclude a strict steady state to be reached. Theoretically, taking gas bubbles into account is best done with a general rheological relationship valid at any capillary number. In the conditions explored herein, bubbles modulate viscosity within a factor 2 with a shear thinning behavior, decelerating slow flows and accelerating fast flows. When a simplified rheology treating bubbles as hard spheres was used, the only dynamic parameter affected was bulk viscosity.

Let $H=-\Delta+V(|x|)$ be a nonnegative radial Schrödinger operator on $\mathbb{R}^d$, $d\ge 2$, whose positive harmonic function satisfies $U(r)\simeq r^{-\sigma_0}$ for $0<r\le 1$ and $U(r)\simeq r^{-\sigma_\infty}$ for $r\ge 1$, with $-d/2<\sigma_0,\sigma_\infty<d/2$. Assuming the two-sided ground-state heat-kernel estimate of Ishige, Kabeya, and Ouhabaz, we determine the maximal open range in which the kernel of $H^{-s/2}$ admits the clean two-sided estimate $K_s^H(x,y)\simeq |x-y|^{s-d}U(|x|)U(|y|)/[U(|x|+|x-y|)U(|y|+|x-y|)]$, namely $0<s<\min\{d,d-2\sigma_0,d-2\sigma_\infty\}$. In this range we give a complete necessary-and-sufficient classification of the broken-power estimate $\|w_{-\beta_0,-\beta_\infty}H^{-s/2}f\|_{L^{q,v}}\lesssim \|w_{\alpha_0,\alpha_\infty}f\|_{L^{p,u}}$ for $1<p,q<\infty$ and $1\le u,v\le\infty$. The result covers signed ground-state exponents, the full range $q<p$, all one-sided weight equalities, both scale equalities, and simultaneous endpoint corners. Scale equality is governed by $u\le v$, including when $q<p$; an input or output power endpoint requires respectively $u=1$ or $v=\infty$; and at a same-side power/scale corner the only admissible pair is $(u,v)=(1,\infty)$. The proof combines clean-kernel analysis, local Lorentz-Hardy-Littlewood-Sobolev estimates, rank-one endpoint arguments, geometric annular sequence spaces, a triangular matrix theorem, and a nine-block decomposition.

In this paper, we investigate the existence, multiplicity, and semiclassical concentration of normalized solutions to a critical Schrödinger–Poisson system with a singular Hardy potential in \(\mathbb{R}^3\). More precisely, we consider \[ \begin{cases} -\varepsilon^2\Delta u+ \left(V(x)-\dfrac{\kappa\varepsilon^2}{|x|^2}\right)u -\phi |u|^3u =\lambda u+\mu |u|^{q-2}u+|u|^4u, & \text{in } \mathbb{R}^3, \\[1mm] -\varepsilon^2\Delta\phi=|u|^5, & \text{in } \mathbb{R}^3, \end{cases} \] under the prescribed mass constraint \[ \int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3, \] where \(a,\mu>0\), \(q\in(2,10/3)\), \(\varepsilon>0\) is a small semiclassical parameter, and \(0<\kappa<1/4\). The parameter \(\lambda\in\mathbb{R}\) appears as a Lagrange multiplier associated with the mass constraint, while \(V:\mathbb{R}^3\to(0,+\infty)\) is a continuous electric potential whose minimum set is assumed to be nonempty and compact. The main difficulty stems from the simultaneous presence of the inverse-square Hardy singularity, the mass constraint, the critical local nonlinearity, and the nonlocal Poisson interaction. By combining the Hardy inequality, constrained variational methods, suitable truncation arguments, and concentration-compactness techniques, we first establish the existence of a normalized ground state for sufficiently small mass and sufficiently small \(\varepsilon\). We then employ Ljusternik–Schnirelmann category theory to obtain multiple normalized solutions whose number is related to the topology of the minimum set of \(V\). Finally, we show that the corresponding semiclassical states concentrate near the global minimum set of the electric potential as \(\varepsilon\to0\).

In this paper, we introduce and analyse a model in which a shared, age-structured resource is exploited by two competing populations through age-dependent interaction kernels. It provides a general framework for studying resource-mediated competition in epidemiology, immunology and ecology. We use integrated semigroups theory to establish the well-posedness of the model and its fundamental properties, including positivity and the existence of a global attractor. Then, by means of Lyapunov functionals, persistence results and generalised LaSalle principle, we fully characterize the asymptotic dynamics in terms of the competitors's basic reproduction numbers and invasive fitnesses. Our results demonstrate that in addition to the classical competitive exclusion, the system can exhibit globally asymptotically stable coexistence, an outcome allowed by a mechanism of age-mediated niche partitioning.

We make use of the free data formalism developed in \cite{CK25,CK26} to provide a direct construction of short-pulse type Cauchy data. The construction of the spacelike short-pulse follows from the local existence result established in \cite{CK26}. The forward integration construction in \cite{CK26} allows us to show that such data can be extended to a set of asymptotically flat Cauchy data. This greatly extends the result of Li–Yu \cite{LiYu}.

In this paper, we investigate the weak observability of second-order infinite-dimensional evolution systems generated by skew-adjoint operators of the form $iA_0$, where $A_0$ is a self-adjoint elliptic operator. We first establish a spectral characterization of weak observability by introducing the notion of spectral coercivity for the observation operator and proving its equivalence to a suitable resolvent estimate. Our main result reveals a direct link between resolvent estimates for the elliptic operator $A_0$ and the weak observability of the associated evolution generator $A$. More precisely, we prove that a resolvent inequality for $A_0$ implies a Hautus-type spectral observability estimate for $A$, which guarantees the weak observability of the system. This provides a unified spectral framework for weak observability based on the coercivity properties of the observation operator. As an application, we establish explicit weak observability estimates for the wave equation on a rectangular domain under several geometric configurations of the observation region. The analysis combines frequency-domain methods, resolvent estimates, and Fourier analysis, yielding new insights into the interplay between resolvent inequalities, spectral coercivity, and weak observability in infinite-dimensional systems.

We prove existence and uniqueness in BV for a Vlasov-Poisson boundary value problem which takes into account a nonlinear boundary condition at the boundary on the electric potential. It models the dynamical interaction between a plasma and an electrically isolated metallic wall. The nonlinear boundary condition is the so-called floating potential equation of plasma physics.

We consider the inhomogeneous nonlinear heat equation \[ \partial_t u-\Delta u=|u|^{p-1}u+f(x),\qquad x\in\mathbb{R}^3,\quad p>5, \] where \(f\in L^\infty\cap C^{0,1}(\mathbb{R}^3)\). For every sufficiently large integer \(n\), we construct a codimension-\(n\) Lipschitz manifold of non-radial initial data whose corresponding solutions blow up in finite time and whose rescaled profiles converge to the prescribed self-similar profile \(\Phi_n\) of the homogeneous equation. The main novelty is to show that the finite-codimensional stability mechanism for self-similar blow-up, developed in the work of Collot, Raphaël and Szeftel [Mem. Amer. Math. Soc. (2019)] for the homogeneous equation, is robust under the addition of a bounded, Lipschitz spatially inhomogeneous source term. In contrast with the homogeneous problem, the equation considered here has no exact scaling invariance, which is a key ingredient in many previous constructions. We expect that the framework developed in this work may also be useful for related problems in which exact scaling invariance is broken.

In this paper, we consider the famous Brezis-Nirenberg equation \begin{eqnarray*} \left\{ \aligned &-\Delta u=\lambda u+|u|^{\frac{4}{N-2}}u,\quad&\mbox{in}\,\, \Omega,\\ &u=0,\quad&\mbox{on}\,\, \partial\Omega, \endaligned \right. \end{eqnarray*} where $N\geq3$ is the dimension, $\Omega\subset\mathbb{R}^N$ is a bounded domain with smooth boundary $\partial\Omega$ and $\lambda>0$ is a parameter. By developing a refined blow-up analysis based on the inverse reduction argument developed in \cite{WW2019,WW2019-2}, we classify, for the fist time, the Struwe decomposition of the Brezis-Nirenberg equation in the one-bubble case as the parameter $\lambda$ varies for $N\geq4$. As applications, we prove that the $4d$ Brezis-Nirenberg equation has a nontrivial solution (least energy solution) for $\lambda\in\sigma(-\Delta)$ in general bounded domains, where $\sigma(-\Delta)$ is the spectrum of $-\Delta$ in $H^1_0(\Omega)$. Our result completes the existence theory of the Brezis-Nirenberg equation for $N\geq4$ in \cite{AP2025,CFP1985,CFS,CSS1986,CW2005,CSZ2012,SWW2009,TYZ2022} since 1984.

This paper establishes the direct scattering transform for the Calogero-Moser derivative nonlinear Schrödinger (CMDNLS) equation on the real line. For potentials in a weighted Hardy-Sobolev space, we prove the existence and uniqueness of the Jost functions associated with the Lax operator, and construct the corresponding scattering coefficients including the transmission coefficient and the reflection coefficient. Under the CMDNLS flow, we determine the time evolution of these Jost functions and scattering data, revealing a simple dynamics: the transmission coefficient is invariant, and the reflection coefficient acquires a purely rotational phase. Furthermore, we establish the invariance of a suitable weighted Sobolev space under the flow, which guarantees the persistence of the required regularity for the solutions.

For every integer $n\ge27$, we construct a smooth metric on $\mathbb S^n$ that is invariant under the antipodal map and is not locally conformally flat. For this fixed background metric, the normalized $\sigma_2$-Yamabe equation admits a noncompact family of positive $\Gamma_2^+$-admissible solutions. The main difficulty is the possible loss of ellipticity of the linearized operator. This difficulty does not occur for the scalar Yamabe equation, whose linearization has a fixed Laplace-type principal part. In the $\sigma_2$ problem, the positive Newton tensor of a standard bubble decays in the far field, while the terms produced by the background metric need not decay at the same rate. Our construction provides the relative decay needed to keep the conformal metrics inside the ellipticity cone. A quartic profile in the finite-dimensional reduction yields the endpoint dimension $n=27$.

We determine the rate of convergence of the value functions of the $N$-particle stochastic optimal control problem to the value function of the corresponding mean field control problem, for mean field costs that are merely Lipschitz continuous in the 1-Wasserstein distance, a class that covers problems whose mean field optimizers are neither unique nor stable. For $d\geq2$, the optimal rate is the empirical-measure rate ($N^{-1/d}$ for $d\geq3$, $N^{-1/2}\sqrt{\log N}$ for $d=2$): this proves the rate conjectured by Daudin, Delarue and Jackson, and removes the semiconcavity hypothesis made there. In dimension one, we discover that the empirical-measure benchmark is not optimal: cooperating particles beat it, and the optimal polynomial exponent is $4/7$, strictly between the accuracy of independent samples and that of quantization by freely placed points. The proofs are control-theoretic: from each realization of an $N$-particle control we build a pathwise Fokker-Planck flow (a "shadow flow") which, repeatedly recoupled to the particles by optimal transport, shadows the empirical measure at the optimal rate. The one-dimensional rate requires additional constructions: we correct the shadow flow with a filter built on the future of the discarded noise, draw the cooperating particles from a Gibbs law, and prove the optimality of the exponent by a Schrodinger ground-state estimate. The empirical-measure rate also holds under additive common noise, uniformly in its intensity.

We investigate the existence of normalized solutions to the gradient-type Schrödinger system \begin{equation*} \begin{cases} -\Delta u+ V_1(x)u+\lambda u= uv^2 & \text{ in } \Omega,\\ -\Delta v+ V_2(x)v+\lambda v= u^2v & \text{ in } \Omega %\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=0 \, & \text{ on } \partial \Omega \end{cases} \end{equation*} subject to the mass constraint $\int_{\Omega}\left(|u|^2+|v|^2 \right)dx=a>0$ and Neumann boundary conditions, where $\Omega\subset \mathbb{R}^3$ is a smooth bounded domain, each $V_i$ is continuous, and $\lambda$ is a Lagrange multiplier. Applying a minimax principle that incorporates Morse index information, we establish the existence of nontrivial normalized solutions of mountain pass type. The proof is based on a refined blow-up analysis adapted to such gradient-type systems, together with new Liouville-type theorems for finite Morse index solutions of the associated limit systems in $\mathbb{R}^3$ and $\mathbb{R}^3_+$.

This paper provides the study of the homogenization of the 3D non-homogeneous incompressible heat-conducting fluid in perforated domains with holes of subcritical size, where the viscosity and the heat conductivity coefficient are assumed to depend on the temperature. The diameter of the holes is of order $\varepsilon^{\alpha} \ (\alpha>3)$, where $\varepsilon > 0$ is a small parameter that measures the mutual distance between the holes. We prove that as $\varepsilon\to 0$, the limit behavior of velocity, density and temperature is governed by the original system in homogeneous domain without holes.

In this paper, we study the initial value problem of the sixth-order Boussinesq equation with quadratic and cubic nonlinearities in arbitrary spatial dimensions. First, by using the Fourier restriction norm method and a high-low frequency decomposition, we establish the nonlinear smoothing for this equation, namely, the integral form of the solution to the Duhamel formulation enjoys higher regularity than its linear counterpart. Finally, by using the nonlinear smoothing, we establish the uniform convergence of the integral term and its spatial decay for each fixed $t$.

We prove the existence of global weak entropy solutions to the $p$-system with the piecewise affine flux function. The construction is based on a wave-front tracking scheme for which we identify a necessary and sufficient condition for the occurrence of infinitely many fronts. For smooth fluxes, the theory is well established. However, the lower Lipschitz regularity of the flux requires a new method to obtain the interaction estimates. We also prove the stability of the solution in the sense of \cite{bia-col-02}, showing that solutions are Lipschitz continuous with respect to the $L^{\infty}$ norm of the difference of the flux derivatives. We obtain the convergence rate of the solutions from the piecewise affine flux to smooth flux by using the stability estimate.

We prove that heat flow generically produces Morse-minimal functions on round lens spaces. The result gives a rigorous explanation of a phenomenon first suggested by computational experiments: after high-frequency modes have been suppressed by diffusion, generic heat evolutions on lens spaces tend to settle into Morse functions with the smallest possible number of critical points. Precisely, for every lens space \(L(p,q)\) with \(p\geq2\), \(1\leq q\leq p/2\), and \((p,q)=1\), there is an open dense set of initial data \(f\in L^2(L(p,q),\mathbb R)\) such that the heat evolution \(e^{t\Delta}f\), where \(\Delta\) is taken in the non-positive sign convention, is, for all sufficiently large \(t\), a Morse function with exactly four critical points, of indices \(0,1,2,3\). The proof analyzes the asymptotic spectral expansion of the heat flow. In the most delicate case \(1<q<p/2\), the leading term is Morse–Bott with two critical circles, and the higher heat modes break these circles through a resonant Fourier mechanism. An arithmetic estimate shows that the fundamental reduced frequency dominates all higher multiples, and generic nonvanishing of the corresponding resonant coefficients gives the minimal critical-point count.

For nonlinear Fokker-Planck equations with mobility, the Wasserstein gradient flow structure is described by the generalized relative entropy as the energy functional and the modified Wasserstein metric $W_h$ as the associated metric structure. This work investigates the nonlinear effects induced by mobility and establishes the corresponding inequalities. For nonlinear diffusion, we establish a logarithmic Sobolev inequality, which yields the convergence rate of the free energy functional and the Talagrand inequality. By further exploiting the relationship between the weighted homogeneous Sobolev norm and the $W_h$ metric, we derive an HWI inequality relating the relative entropy, the $W_h$ metric, and the Fisher information. In the case of mobility dependent drift and linear diffusion, the convergence rate in the $W_h$ metric is also obtained by applying the Girsanov theorem.

We propose a four-field auxiliary reformulation of a time-discrete Cahn-Hilliard step. The construction is motivated by the scalar trace structure underlying two-dimensional Rafetseder-Zulehner decompositions and represents the scalar quantity $-\Delta c$ in the form $2p+div\,u$ through an auxiliary scalar field $p$ and an auxiliary vector field $u$. The resulting auxiliary problem is a mixed second-order Stokes/elasticity-type system, while the evolution equation for the phase field retains its standard mass-conserving gradient-flow structure. We derive a continuous four-field formulation that is equivalent to the classical convex-splitting mixed time step. We also state a conforming finite element discretization and prove one-step spatial estimates for the phase-field variables together with a stable auxiliary block estimate containing an explicit weak-Laplacian recovery defect. The numerical experiments verify the expected phase-field convergence in a manufactured setting, mass conservation, energy decay, and projected consistency of the auxiliary block.

It is well known that the McKean-Vlasov stochastic differential equation with a symmetric double-well potential exhibits a continuous phase transition. In contrast, for an asymmetric double-well potential, the system undergoes a discontinuous phase transition, in which an invariant measure abruptly appears in the shallower well and subsequently splits into two as the temperature decreases. In this work, we systematically investigate the types of phase transitions driven by non-convex confining potentials and cooperative interactions in $\mathbb{R}^n$. Our main results consist of a pitchfork bifurcation theorem characterizing continuous phase transitions and a saddle-node bifurcation theorem governing discontinuous phase transitions. In addition, despite the dimension-dependent nature of phase transitions, we are able to extend Dawson's criterion for phase transition points – originally formulated for quadratic interactions in $1$-dimensional space – to $n$-dimensional space. This extension directly relates the critical parameter and bifurcation direction to the eigenvalue and eigenvector of the critical covariance matrix, respectively. Finally, we apply our theoretical results to the aforementioned double-well models, to a class of four-well models in two-dimensional Euclidean space that exhibit multiple phase transitions, and to an attractive Gaussian interaction model.

We consider a general class of first order ODE systems for the evolution of $N$ interacting particles (in Euclidean space $\mathbb{R}^d$) in a mean-field regime. The class of interactions treated includes singular interactions of inverse power type up to power $d+1$, attractive or repulsive, and not necessarily deriving from a potential – unlike, for instance, the modulated energy method. We introduce a new method to prove quantitative convergence of the discrete system to solutions of the mean-field equation. It relies on studying the evolution of a metric encoding a multiscale control of the difference between the empirical measure and its limit, via mollification by heat kernels. We prove that the desired convergence holds (i) up to the maximal time of existence of the smooth solution to the limiting equation if the singularity is sub-coulombic in any dimension, or coulombic in dimensions 1 and 2 (where, to do so, we introduce a notion of weak solution to the ODE system), or (ii) for short time in the case of Coulomb singularity in dimension 3 and above and (iii) up to a short $N$-dependent timescale for super-coulombic interactions in all dimensions. The latter two results are demonstrated to be optimal as we prove that collisions occur within the same timescale for a class of attractive interactions.

We study the density patch problem for the two-dimensional inhomogeneous incompressible Navier–Stokes system with vacuum, for initial data consisting of a Lipschitz density patch and a divergence-free velocity field in $L^2(\mathbb{R}^2)$. We establish uniqueness of solutions at the natural energy level, thereby concluding the global well-posedness for $L^2$ data. Furthermore, we prove a log-Lipschitz estimate for the velocity field, extending the classical result of Chemin-Lerner to the inhomogeneous setting. As a consequence, the associated flow belongs to $L^\infty_t C_x^{1-\varepsilon}$. for any $\varepsilon \in (0,1)$, ensuring that the patch boundary remains a continuous curve of Hausdorff dimension $1$, thus preserving its initial dimension for all time.

We prove the existence of spectral submanifolds in nonlinear partial differential equations (PDEs) describing forced-damped continuum vibrations. Our results cover structural vibrations subject to generalized structural damping. Based on these results, backbone curves and forced response curves can be rigorously extracted from PDEs, without a priori discretization. On two examples involving an elastic beam and a thin plate, we show that backbone and forced response curves can even be calculated by hand directly from the PDEs.