Papers

This paper concerns the barotropic compressible Navier-Stokes equations in a two-dimensional half-space subject to Navier-slip boundary conditions with vacuum or non-vacuum far-field density. The global existence and large-time behavior of weak and strong solutions are established under the assumption that the bulk viscosity coefficient is sufficiently large. It should be remarked that this result is obtained without any restrictions on the size of the initial data. For strong solutions, we derive some a priori decay estimates for the spatial gradient of the velocity field that are uniform with respect to the bulk viscosity coefficient, which play a crucial role in establishing the time-uniform upper bound for the density. Furthermore, we prove that, as the bulk viscosity coefficient tends to infinity, the solutions of the compressible Navier-Stokes equations converge to those of the inhomogeneous incompressible Navier-Stokes equations. In particular, the incompressible limit for weak solutions holds without requiring the initial velocity to be divergence-free.

In this work, we establish the existence of mass-conserving weak solutions to a nonlinear collision-induced breakage equation in which binary collisions may trigger particle breakup. The result is proved for a class of product-type collision kernels whose small-size behavior is controlled by a power-law function of the form $\omega_0(x)\le A_1\,x^\ell$, while no growth restriction is imposed on the large-size factor $\omega_\infty$. The qualitative behavior of the solutions depends crucially on the exponent $\ell$ near the origin. Sublinear growth corresponding to $\ell<\tfrac12$ yields existence only on finite time intervals, whereas superlinear growth corresponding to $\ell>\tfrac12$ ensures global-in-time existence.

We derive the first-moment Shallow Water Exner Moment model with sediment entrainment and deposition (SWEMED1) and show that the full source term has a fully-settled water-at-rest equilibrium manifold. We prove that the model is only weakly hyperbolic at this equilibrium, which prevents the use of Yong's structural stability framework. However, a linear spectral analysis and numerical results do not indicate instability. Based on numerical results, we introduce a fast-slow scaling of the source term, and for the fast limit, we derive a new suspended water-at-rest equilibrium manifold, which has a different structure but is still only weakly hyperbolic. Our results show that the remaining obstruction is linked to the transport closure of the SWEMED1, and we give a constructive direction for the derivation of new closures leading to models with more desirable analytical properties.

It is well known that band inversion across a straight interface in a periodic medium gives rise to interface modes that are localized near the interface and propagate along it inside the bulk spectral gap. This phenomenon constitutes the key mechanism underlying the valley-Hall effect. In this paper, we address the long-standing problem of the robustness of such interface modes. We prove that, when the interface is bent through an angle of $\frac{2\pi}{3}$, the interface modes persist for every frequency in the bulk spectral gap where the group velocity is non-vanishing, except for a finite exceptional set. We also show that corner-localized modes, if they occur, can appear only at these exceptional frequencies and have finite multiplicity. To the best of our knowledge, this is the first rigorous mathematical theory of the bending immunity of valley-Hall interface modes.

This paper investigates fractional Riesz-Bessel equations with random initial conditions that exhibit either classical or cyclic long-range dependence. It studies zoom-in asymptotics for the corresponding solutions and establishes multiscaling limit theorems. It is known that for similar problems, non-degenerate multiscaling limits may not exist in general. The paper develops a kernel-smoothing approach for these equations and obtains non-degenerate limit fields under suitable normalisation and rescaling. It proves that the kernel-smoothed solutions converge weakly to Gaussian random fields, which are non-stationary in both time and space. Their spectral representations and covariance functions are derived. The paper also analyses the regularity and dependence structure of the limit fields. In particular, under appropriate general assumptions on the smoothing kernel, the limits exhibit long-range dependence in time and short-range dependence in space. Numerical examples for the case of Matérn-type kernels are provided to illustrate the theoretical results.

Let $L$ be the Dunkl Laplacian on the Euclidean space $\mathbb{R}^N$ associated with a normalized root system $R$ and a multiplicity function $k(\nu)\geq 0$, $\nu\in R$. We establish a Leibniz-type rule for the fractional powers of $L$ on Besov and Triebel–Lizorkin spaces in the Dunkl setting. Our approach exploits the interplay between spectral multipliers and the Dunkl transform, together with the support properties of the distributions associated with Dunkl translations. These results extend the corresponding Leibniz-type estimates previously established on $L^p$ spaces to the broader setting of Besov and Triebel–Lizorkin spaces.

We propose a high-order numerical methodology for computing the ground state and time evolution of the two-dimensional Gross-Pitaevskii equation with harmonic trapping potential. The ground state is obtained by combining normalized gradient flow with time-splitting schemes featuring strictly positive coefficients, which makes them suitable for both dissipative and Hamiltonian regimes and avoids the negative time steps required by classical symplectic methods. The computational cost of these methods grows quadratically with the order, in contrast to the exponential growth of symplectic alternatives. Numerical benchmarks assess ground state convergence under different initializations, long-time preservation of mass and Hamiltonian energy for varying mass constraints, and the cost-accuracy trade-off for orders q = 2, 4,..., 14.

We study the long-time behavior of the (critical) Kohn–Sham equation in two and three dimensions, i.e.,\[ \mathrm{i} \partial_t {\gamma} = \Big[-\frac{1}{2}\Delta + \lambda \, |\cdot|^{-1} \ast \rho_{{\gamma}} + \mu \, \rho_{{\gamma}}^{1/d}, {\gamma} \Big] \quad \text{for} \quad d=2,3. \] By introducing a suitable ”square root” of the density matrix and exploiting the pseudo-conformal transform, we establish global well-posedness for small initial data in an appropriate weighted Schatten norm. We also prove the optimal time decay of the particle density and establish modified scattering for small and localized solutions. In particular, our results provide a resolution to the open problems proposed by Pusateri and Sigal (2021) for the critical and subcritical regimes, rigorously proving their conjectures regarding modified scattering in the critical case and linear scattering in the subcritical cases. Our results place these scattering phenomena in the operator-valued setting of density matrices, thereby extending the classical scalar theory to a broader framework.

We study the anisotropic Finsler $p$-Laplacian equation \begin{equation*} \left\{ \begin{aligned} &-\Delta ^{H}_{p}u=f(u) \quad\,\,\, &{\rm{in}} \,\, \mathcal{C}, &{\bf{a}}(\nabla u)\cdot \nu =0 \quad\,\,\, &{\rm{on}} \,\, \partial\mathcal{C}, \end{aligned} \right. \end{equation*} where $N\geq3$, $1<p<N$, $\mathcal{C}\subseteq\mathbb{R}^{N}$ is an open convex cone and $\Delta ^{H}_{p}$ is the anisotropic Finsler $p$-Laplacian operator. If $f(u)$ is nonnegative and subcritical, we prove that every bounded nonnegative solution in $\mathcal{C}$ is identically zero. In particular, for $f(u)=u^{q}$ with $0<q<p^*-1$, we establish a pointwise decay estimate in $\mathcal{C}$ via the doubling argument and blowing-up method and prove that all nonnegative solutions must be zero without the boundedness assumption. Our results are the subcritical counterpart of the classification result for the critical case in \cite{CFR}, and extend the Liouville type theorems in $\mathbb{R}^{N}$ for the standard $p$-Laplacian in \cite{SZ} and for the anisotropic $p$-Laplacian in \cite{CFV, CHN} to general convex cones $\mathcal{C}$. In the critical case $f(u)=u^{p^*-1} $ and typical case $H(\xi)=|\xi|$, for $\frac{N+1}{3}<p<N$, we classified the positive solutions of the critical $p$-Laplacian equation in convex cones $\mathcal{C}$ without finite-energy assumption. This extends the classification result of \cite{Ou} in $\mathbb{R}^{N}$ to general convex cones $\mathcal{C}$, and removes the finite-energy assumption in \cite{CFR} in the typical case $H(\xi)=|\xi|$.

Motivated by the introduction of the Zakharov-Kuznetsov equation as a higher dimensional generalization of the Korteweg-de Vries equation, in this paper we introduce the modified Zakharov-Kuznetsov (mZK) equation as a 2-dimensional generalization of the complex-valued modified Korteweg-de Vries equation. We initiate the mathematical analysis of the mZK equation on $\mathbb{T}^2$ by proving a local well-posedness result in Sobolev spaces and by establishing a failure of uniform continuity. We also note that the real-valued version of the mZK equation has physical significance.

We prove existence of solutions to a nonlinear degenerate elliptic equation of the form \[ \begin{cases} -\Delta_{1} u+ \frac{|D u|}{(1-u)^{\gamma}}=g & \mbox{in $\Omega$,}\\ u=0 \hfill & \mbox{on $\partial\Omega$,} \end{cases} \] in a suitable sense, where $\Omega $ is a bounded open set of $\mathbb{R}^n$, $\gamma>0$ is a fixed parameter, $g\geq 0 $ is a function in some Lebesgue space.

In this paper, we analyze the solvability of the discrete nonlinear Schrödinger equation \begin{equation*} i\beta(\Delta_t+\nabla_t)\phi(t,k) +\gamma |\phi(t,k)|^2\phi(t,k) +\varepsilon \Delta_k^2\phi(t,k-1) = g(t,\phi(t,k)), \end{equation*} where $\Delta_t$ and $\Delta_k$ denote the standard forward difference operators in the variables $t$ and $k$, respectively, $\nabla_t$ denotes the standard backward difference operator in $t$, and \begin{equation*} \Delta_k^2\phi(t,k-1) = \phi(t,k+1)-2\phi(t,k)+\phi(t,k-1) \end{equation*} is the discrete Laplacian operator in the spatial variable $k$. Throughout, we will assume the parameters $\beta$ and $\varepsilon$ are positive real numbers, the parameter $\gamma$ is a nonzero real number, and the potential function $g:\mathbb{Z}\times\mathbb{C}\to \mathbb{C}$ is continuous.

Spherical caps play a crucial role in establishing a criterion for the existence of solutions to the Yamabe problem on a compact Riemannian manifold with boundary, similar to the role played by the standard sphere in the problem on a closed Riemannian manifold. This problem is expressed in terms of a nonlinear boundary-value problem, where both the nonlinearity and the boundary condition are critical in the Sobolev sense. This work focuses on the existence of multiple solutions to the Yamabe problem on spherical caps. We show that if the spherical cap is contained in a hemisphere of the standard $n$-sphere and $n = 5$ or $n \geq 7$, the Yamabe problem has infinitely many sign-changing solutions. Our approach takes advantage of symmetries and is based on a careful analysis of the loss of compactness of the variational problem.

Let $\mathbf{\eta}$ be a positive Radon measure on the space of continuous paths from R into a locally compact Polish space $Y$, and assume that $\mathbf{\eta}$ admits an invariant measure. We explicit conditions on $\mathbf{\eta}$ such that successive Markovianisation of $\mathbf{\eta}$ over any dense and countable set of times have all limit points (in the weak-star topology on Radon measures) satisfying the strong Markov property. We show that if Y is a locally compact Polish group, and $\mathbf{\eta}$ is left or right translation invariant, then $\mathbf{\eta}$ satisfies such conditions. Our proof uses the Zermalo-Fraenkel and Axiom of Dependant Choice axiomatisation of set theory, in which countable products of compacts are compact.

We settle the problem of finding the sharp constant in the log Sobolev inequality on the $n$-cycle for all $n\ge 4$, by showing that it is equal to half of the spectral gap. We deduce this result from an optimal cubic Sobolev inequality.

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 $p=2$.

This work addresses the mean-field limit of inertial particle systems with singular interactions in a perturbative regime around Gibbs equilibrium. We prove that small fluctuations around equilibrium are asymptotically governed by the linearized Vlasov equation. The result applies to a broad class of singular interaction kernels, including the Coulomb case in dimensions $d\le3$. In particular, this provides a rigorous derivation of the linearized mean-field dynamics near equilibrium in settings where the corresponding nonlinear mean-field limit remains out of reach.

We study finite-time blow-up for the focusing Calogero–Sutherland derivative NLS on the torus with smooth initial data in the Hardy space $L^2_+(\mathbb{T})$. For finite-gap potentials as initial data, the explicit solution formula reduces the dynamics to a finite-dimensional analytic family of contractions. This yields a complete description of every finite-time blow-up solution in this class: near blow-up time $T >0$, the solution decomposes into finitely many concentrating universal blow-up profiles and a smooth remainder, each bubble carries one unit of $L^2$-mass, the concentration points are distinct, and the blow-up rates are quantized according to $$ \|u(t) \|_{H^s(\mathbb{T})} \sim_{s,u_0} \frac{1}{(T-t)^{2s\nu}} \quad \mbox{as} \quad t \nearrow T \quad \mbox{for any $s > 0$} $$ with some integer $\nu\geq1$. Within an explicit subclass of finite-gap potentials, we identify an exact resonance condition that yields the existence of finite-time blow-up solutions. By a non-perturbative method, we construct single- and multi-bubble blow-up solutions with rate $\nu=1$ for every prescribed $L^2$-mass in $(1,\infty)\setminus \{ 2 \}$. In the complementary non-resonant regime, we prove global existence with uniform Sobolev bounds. The construction also shows instability of the blow-up solutions and non-chiral finite-time blow-up examples. To the best of our knowledge, these results provide the first explicit, non-perturbative construction of finite-time blow-up for an integrable NLS-type equation on the torus, together with a complete classification of the singular dynamics within a natural finite-gap class.

We study the large-scale behaviour of a family of stochastic reaction-diffusion equations driven by long-range correlated noise in a weakly nonlinear regime. Depending on the decay of correlations of the noise and the strength of the nonlinearity, the universal behaviour is governed by a version of the dynamical $\Phi^p$ model with long-range correlated noise, and with a coupling constant determined by the reaction term of the microscopic model. Our main result establishes the stochastic estimates and convergence of models required in the theory of regularity structures. We adapt the multiindex-based approach to regularity structures using a suitable expansion of the reaction term tailored to the law of the noise. This yields a systematic weak universality result, allowing for a particularly simple identification of the macroscopic limit throughout the full subcritical regime and beyond Gaussian noise. The method appears robust and applicable to a broader class of singular stochastic PDEs.

Let $n\geq 3$ and let $\Omega \subset \mathbb{R}^n$ be a $\mathcal{C}^1$ bounded domain which is diffeomorphic to a ball. We investigate here the problem of finding critical points of the $n$-energy in the space $\mathcal{I}=\{v\in W^{1,n}(\Omega,\mathbb{R}^n) ; \ |\mathrm{tr}_{|\partial \Omega}v|=1\}$. Maps in $\mathcal{I}$ have a well-defined topological degree on $\partial \Omega$ but this degree is not continuous for the weak convergence in $W^{1,n}$. Hence finding critical points with prescribed degrees results in a problem of lack of compactness. We first prove that minimizers of the $n$-energy exist only when $\Omega$ is a round ball and when the prescribed degree is $-1,0$ or $1$. We then develop a mountain pass approach for the $(n+\alpha)$-energies and study the convergence, when $\alpha$ goes to zero, of the resulting critical points via a bubbling analysis. We exclude the existence of bubbles in the case where $\Omega$ is close to a ball by proving an energy gap result for free boundary $n$-harmonic maps from $\mathbb{B}^n$ to $\mathbb{B}^n$. We thus obtain the existence of critical points of the $n$-energy with prescribed degree $1$ when $\Omega$ is close to a ball.

The existence of positive, pointwise decaying at infinity, weak solutions to a fractional $p$-Laplacian problem in the whole space and with singular reaction is established. Truncation arguments, variational methods, as well as suitable a priori estimates are exploited.

We give examples of proper minimal immersions in Euclidean space with very rapid area growth. The first is a proper embedding into $\bf{R}^4$ that yields a stable minimal surface, while the second is a proper immersion into $\bf{R}^3$. These results are motivated by [CM1] that shows that proper minimal submanifolds confined in space satisfy strong structural constraints.

We prove a weak fundamental principle for $\lambda$-homogeneous solutions of homogeneous constant-coefficient systems on open pointed convex cones. Starting with the solution family $S_{\mathcal B}$ arising in the Ehrenpreis–Palamodov theory, we construct a corresponding family $S_{\mathcal B,\lambda}$ by replacing the exponential kernels $e^{\langle x,z\rangle}$ with homogeneous kernels $(-\langle x,z\rangle)^\lambda$. The key tool is a Mellin-type operator on Paley–Wiener spaces, which links the classical theory to the Euler-constrained setting. For $\lambda\in \mathbb{C}\setminus \mathbb{N}_0$ and under a visibility assumption, we show that the span of $S_{\mathcal B,\lambda}$ is dense in the space of $\lambda$-homogeneous solutions.

This article presents an overview of quasineutral limits in plasma models. Starting from the Vlasov-Poisson system, it explains the role of the Debye length, the emergence of a kinetic incompressibility constraint, and the stability issues caused by fast oscillations and singular electric fields. A central theme is that the geometry of the kinetic flow should be reflected in the way perturbations are measured. This leads to kinetic Wasserstein distances adapted to phase-space dynamics, which provide refined stability estimates for quasineutral limits. The article also discusses related models with thermalized electrons and the additional challenges of the electromagnetic Vlasov-Maxwell setting.

Several numerical reconstruction algorithms for the inverse boundary value problem of the 1-dimensional wave equation exist. In this paper we revisit two of them, the Sondhi-Gopinath (SG) method from 1971 and the Korpela-Lassas-Oksanen (KLO) method from 2016. Our main contribution is a systematic and statistical comparison of the performance of these two well established methods. The stability of the SG algorithm, even in practical applications, is well known. KLO has a regularisation scheme with a theoretical proof, and is an evolution of the boundary control method. Both are based on the idea of constructing solutions that are characteristic functions of a set at a given time. This similarity has been pointed out before, but no systematic comparison has been published. We compare the performance of the two algorithms with noisy simulated data. The application in our mind is the reconstruction of the internal cross-sectional area of a pressurised fluid pipe. The area corresponds to the first order $\partial_x$-term of the wave equation. Instead of just observing the performance on a few test cases, we generate $n=1000$ random area profiles of various smoothness levels and measurement noise up to $10\%$ of the signal energy and perform statistical tests. SG and KLO have a difference of one time-derivative in their standard boundary data, which complicates the analysis. Our results show that SG performs better in the low noise regime, and KLO with high noise. SG is easier to implement and runs faster.

This paper studies the core problems in the $L_p$ dual Brunn-Minkowski theory, encompassing the $L_p$ Minkowski problem and $L_p$ Brunn-Minkowski inequality for dual quermassintegrals. For the case $0<p<q\leq n$, we establish $C^0$ estimates for the $L_p$ dual Minkowski problem without symmetric assumptions, thereby resolving a related problem proposed by Böröczky-Chen-Liu-Saroglou in the smooth sense. We further prove the uniqueness of smooth solutions under appropriate conditions, provided the density function is sufficiently close to a constant in the Hölder norm. Finally, exploiting the fact that the uniqueness of the Minkowski type problem is equivalent to the validity of the Brunn-Minkowski inequality in a certain sense, we study the $L_p$ Brunn-Minkowski inequality for dual quermassintegrals for origin-symmetric convex bodies with $p<q$.

We establish a Liouville type theorem for fully nonlinear uniformly elliptic equations in exterior domains in half spaces under quadratic boundary data and a quadratic growth condition, that is, any viscosity solution tends to a quadratic polynomial plus lower order terms at infinity with the rate at least $|x|^{-\alpha-n}$ for any $\alpha\in(0,1)$. As applications, this result can lead to Liouville type theorems for Monge-Amp`ere equations, k-Hessian equations and special Lagrangian equations with critical and supercritical phases.

We establish scaling limit results for fluid dynamics equations driven by pseudo-transport noise. The behaviour of noise at small scales is governed by a parameter a. This extends previous results by Flandoli and Luo (2020) and Galeati (2020), which correspond to a=0 in our setting. Depending on the value of a, we prove that the noise delays the potential blow-up of both the 3D Euler and Navier-Stokes (NS) equations with high probability.

We design observer-based controllers to stabilise abstract linear boundary control systems on Hilbert spaces. Our main results introduce conditions for exponential, strong, and polynomial stability, and establish external well-posedness of the closed-loop system. We design controllers for a one-dimensional wave equation, a two-dimensional wave equation with distributed control and observation, and a non-uniform SCOLE model.

We study the asymptotic dynamics of multi-bubble solutions to the focusing energy-critical wave equation in five dimensions. Assuming that the solution asymptotically decomposes into a finite superposition of spatially separated bubbles with comparable scales, we prove a rigidity result that describes the precise long-time behavior of these scales. More precisely, we show that all scaling parameters are necessarily of order $t^{-2}$, and that the corresponding renormalized modulation vector converges to a connected component of a finite-dimensional algebraic set determined by the limiting spatial configuration of the bubbles. This algebraic system encodes the strong interactions between the polynomial tails of the bubbles and governs the effective asymptotic dynamics of the multi-bubble regime.