Papers

Although there are many important contributions on compressible and incompressible fluid-particle interaction models respectively, how to connect the two-type fluid-particle models via the low Mach number limit remains a challenging open problem. In this paper, we resolve it for the compressible isentropic fluid-particle model (Euler–Vlasov–Fokker–Planck (Euler–VFP) system) in the whole space $\mathbb{R}^3$. First, we establish the global-in-time a priori estimates of strong solutions that are uniform with respect to the Mach number $\varepsilon$ near the global Maxwellian. The proof relies on a refined energy method that combines the relaxation structure $b^\varepsilon-u^\varepsilon$ induced by the fluid-particle interaction and the symmetrized acoustic structure of the compressible Euler part in the model. Under the assumption of well-prepared initial data, we derive a global-in-time uniform error estimate in the $H^2$ framework between the solution of the compressible Euler–VFP system and that of the limiting incompressible Euler–VFP system. A key point is to introduce the corrected acoustic variable $ q^\varepsilon-\varepsilon [P'(1)]^{-1}\pi$, which captures the pressure corrector in the low Mach number limit. This also allows us to exploit the exact cancellation of the singular acoustic terms and to close the global-in-time error estimate. The damping term $b^\varepsilon-u^\varepsilon$, which is absent in the pure Euler equations, plays an essential role in recovering the relative velocity dissipation and in controlling the coupled fluid-particle dynamics. As a consequence, we prove the low Mach number limit of the compressible Euler–VFP system with the convergence rate $\mathcal O(\varepsilon)$ in the time-continuous $H^2$ topology.

In this paper, we continue our recent study on anisotropic minimal surface equation with free boundary condition in Wang-Wei-Xia-Zhang (Arch Ration Mech Anal 250:42, 2026). The first main result solves the corresponding mixed boundary value problem. As an application, we prove the following Bernstein-type theorem: any anisotropic minimal graph over $\mathbb{R}^3_+$ with free boundary must be flat. These extend the classical results of Simon (Indiana Univ Math J 25:821–855, 1976; Math Z 154:265–273, 1977) to an appropriate free boundary setting.

In this paper, we study the well-posedness and large-time behavior of a pseudo-parabolic problem associated with a generalized total variation energy in the $BV$-framework. A main mathematical issue arises from the mismatch between the Sobolev regularity of solutions at finite times and the $BV$-structure of the corresponding steady-state problem. We prove the existence and uniqueness of solutions and investigate the relationship between their large-time behavior and solutions to the steady-state problem. Moreover, our results include the convergence of the solution trajectory to the unique steady-state solution under a typical setting arising in image processing.

In this paper, we investigate the asymptotic behavior of minimizers $u_\Lambda$ of the Bernoulli functional $\int_D |\nabla v|^2 \, dx +\Lambda|\Omega_v|$ for large $\Lambda$ with a boundary datum $g \ge 0$. In particular, in the case $g>0$, we show graphicality of the free boundary over $\partial D$ and obtain a second order asymptotic expansion in $\Lambda$ of the graph.

We obtain the existence results of least energy sign-changing solutions and ground state solutions for a class of asymptotically cubic Kirchhoff equations with Dirichlet boundary value on a locally finite graph $G=(V,E)$, and obtain the least energy of sign-changing solutions is strictly larger than twice of the ground state energy.

We study long-time behavior of the solutions to the final state problem for the defocusing nonlinear Schrödinger equation (NLS) in one space dimension with the power nonlinearity $|u|^{2\sigma}u$ in the subcritical long-range regime $\frac{2}{\sqrt{7}}<\sigma<1$. Given a prescribed asymptotic profile in a weighted $L^2$-space, without size restriction, obtained by modifying the free solution with a nonlinear polynomial phase correction, we construct a unique global solution of the NLS that scatters to this profile, thereby proving the existence of modified wave operators. The proof relies on two new ingredients. Extending our previous work for the cubic case, we incorporate the leading part of the nonlinear term into the linear part as a linear potential by linearizing the NLS around the asymptotic profile and prove a global modified energy estimate for the linearized equation. We also exploit a specific structure of the nonlinearity arising from the linearization, which gives rise to a crucial cancellation when estimating the nonlinear terms in the modified energy space and enables us to control the polynomial growth of the nonlinear phase correction in the subcritical case.

In this paper we investigate the exact controllability of $n \times n$ first order one-dimensional quasilinear hyperbolic systems by $m<n$ internal controls that are localized in space in some part of the domain. We distinguish two situations. The first one is when the equations of the system have the same speed. In this case, we can use the method of characteristics and obtain a simple and complete characterization for linear systems. Thanks to a linear test this also provides some sufficient conditions for the local exact controllability around the trajectories of semilinear systems. However, when the speed of the equations are not anymore the same, we see that we encounter the problem of loss of derivatives if we try to control quasilinear systems with a reduced number of controls. To solve this problem, as in a prior article by J.-M. Coron and P. Lissy on a Navier-Stokes control system, we first use the notion of algebraic solvability due to M. Gromov. However, in contrast with this prior article where a standard fixed point argument could be used to treat the nonlinearities, we use here a fixed point theorem of Nash-Moser type due to M. Gromov in order to handle the problem of loss of derivatives.

We study the compressible Euler-alignment system with pressure under a singular pressure scaling, where the hypersingular communication weight induces a fractional alignment operator of order $2\alpha$, $0<\alpha<1$. The scaling corresponds to a large-time and small-velocity regime and leads to a low Mach number problem in which the density is forced to remain close to a constant state. Our main result is a uniform regularity theory for this scaled pressure system. We establish uniform estimates with respect to the scaling parameter and construct global strong solutions near the constant state. A key feature of the analysis is that, in the low-order fractional regime $0<\alpha\le\frac12$, the estimates close under the lower Sobolev condition $s>\frac d2+1-2\alpha$, gaining $\alpha$ derivatives over the threshold $s>\frac d2 + 1 - \alpha$ arising from a direct use of the fractional alignment dissipation. This is achieved by combining refined commutator estimates for the singular alignment operator with the density dissipation induced by the pressure scaling. As an application of the uniform estimates, we justify the low Mach number limit toward the incompressible Navier–Stokes system with fractional dissipation. For general small, possibly ill-prepared initial data, a Helmholtz decomposition combined with dispersive estimates for the acoustic component yields subsequential strong convergence locally in space-time to a distributional solution of the limiting system. For well-prepared initial data, a relative-energy argument further identifies the limit with a prescribed sufficiently regular solution and yields global strong convergence in the fractional dissipation norm.

We study small solutions to the one-dimensional nonlinear Schrödinger family \[ \mathrm{i}\partial_t u_\delta=-\partial_x^2u_\delta+\kappa|u_\delta|^{2+\delta}u_\delta, \qquad 0\leq\delta\leq\delta_0, \] uniformly as $\delta\to0^+$ and $t\to\infty$, across the threshold between cubic modified scattering and nearby-power ordinary scattering. For initial data small in $H^{0,1}(\mathbb{R})$, with one spatial weight and no physical derivative assumed, we prove a joint-limit asymptotic formula governed by an explicit nonlinear clock. The variable $ \delta\log t$ gives three regimes according as it tends to zero, a positive finite limit, or infinity. Within the regime $\delta\log t\to0$, successive Taylor terms of the clock become visible at an infinite hierarchy of time scales, each requiring finer absolute phase accuracy. These scales accumulate at the transition scale, where the exact clock describes them together. In the transition and saturated regimes, the clock is of order $\delta^{-1}$, so the first-order variation of its coefficient contributes a finite phase correction.

In the spirit of the recent work by Demmel and Wiedemann \cite{[DW]}, we investigate the validity of the energy balance law for weak solutions to the three-dimensional Navier-Stokes equations with only horizontal dissipation. We prove that famous Lions-Shinbrot type energy conservation criteria, originally established for the classical isotropic Navier-Stokes equations, continue to hold in this anisotropic setting. Furthermore, we present two distinct approaches that handle the torus case and the whole space case, respectively.

The minimal control time of one-dimensional first-order linear hyperbolic systems is by now well-known when the boundary coupling matrix is full row rank. The goal of this article is to characterize the minimal control time for a class of systems with row rank deficient boundary coupling matrices.

At the occasion of the 70th birthday anniversary of Jean-Michel Coron, we review the literature on the control of 1D first-order linear hyperbolic systems, a topic where he has made significant contributions.

We study some inverse problems in a frequency-averaged model for quantitative photoacoustic tomography, in which the optical parameters are modeled by a finite number of spectral functions and spatial coefficients, and the measurements are given by internal observables averaged over a spectral bandwidth. Using multiple measurements generated by suitable complex geometrical optics solutions, we establish uniqueness and Lipschitz-type stability for the associated linearized inverse problem for diffusion and absorption recovery. This linearized stability further yields local Hölder-type stability for the nonlinear inverse problem via an abstract linear-to-nonlinear argument.

We study regularity, existence, and uniqueness of solutions to vector fields on the two-dimensional torus in Sobolev spaces of dominating mixed smoothness, which measure regularity separately in the two variables. For constant-coefficient vector fields, we obtain families of mixed smoothness estimates describing how the gain or loss of regularity can be distributed between the two variables. In the nonreal case, a gain of one derivative can be distributed between the two directions, whereas for real irrational coefficients the loss is governed by the irrationality measure of the coefficient. We also establish sharpness below the corresponding arithmetic threshold and describe the rational and Liouville obstructions. For real-valued variable coefficients, direct estimates for the periodic Fourier-mode equations yield mixed smoothness regularity results and their isotropic and classical consequences. We then use a periodic conjugation to the averaged constant-coefficient normal form. Although this conjugation introduces an additional loss in the mixed smoothness scale, it preserves isotropic Sobolev orders and therefore transfers the sharp constant-coefficient isotropic theory to the variable-coefficient setting. In the nonresonant regimes, we also obtain existence and uniqueness under the natural zero-mean compatibility condition.

We study quantitative unique continuation for solutions of the Schrö\-din\-ger equation with complex-valued singular potentials \(V\in L^t\), \(t>d/2\). We obtain a family of scale-invariant \(L^p\to L^q\) Carleman inequalities for the Laplacian and use them to prove the doubling inequalities with explicit dependence on the \(L^t\)-norm of the potential. These estimates are combined with multiscale arguments to obtain propagation of smallness from arbitrary measurable sets of positive measure. As an application of the main results, we prove an upper bound for the BMO norm for the logarithms of Dirichlet-Laplace eigenfunctions in simply connected planar Lipschitz domains.

We prove the Lipschitz regularity of solutions for a wide class of generalized Bernoulli free boundary problems, in vectorial, higher order, and non-minimizing settings. Our method does not rely on notions of viscosity solutions or comparison methods, which allows us to reach the optimal regularity for Bernoulli-type problems associated to elliptic operators which do not satisfy any maximum principle, namely the elasticity, biharmonic and Stokes equation. With a similar method we obtain the optimal Lipschitz regularity for stationary, non-minimizing solutions of the standard Bernoulli problem in two dimensions.

Let $1 < p < \infty$. We identify the optimal decay exponent of the $p$-harmonic measure $\omega_p ( B( \xi, t) \cap \partial \mathbb{B}, 0, \mathbb{B}) $ for small $t$, where $\mathbb{B}$ is the unit ball in $\mathbb{R}^n$, $\xi \in \partial \mathbb{B}$ and $B(\xi, t)$ denotes the euclidean ball centered at $\xi$ of radius $t$. As a consequence and, under suitable geometrical restrictions on a domain $\Omega \subset \mathbb{R}^n $, we identify the optimal decay exponent of $\omega_p ( B( \xi, t) \cap \partial \Omega, a, \Omega)$ for small $t$, where $a \in \Omega$ and $\xi \in \partial \Omega$. When restricted to the range $1 < p < n$ we provide new estimates complementing previous results of different authors in the unit ball, the unit disc or the upper half-space.

Outliers are important for stress-testing algorithms and understanding system behaviour under rare conditions. Despite being commonly described as low-likelihood events, existing generative approaches rarely control likelihood explicitly. In this work, we introduce a measure-theoretic notion of outliers based on the distribution of log-likelihood values, which is guaranteed to assign higher probability mass to low-likelihood events with a specifiable magnitude. Building on this formulation, we derive how likelihood reweighting modifies the diffusion score and use this relation to motivate a controlled modification of the reverse-time dynamics. In particular, likelihood reweighting implies a scaling of the score function with a control term derived from the Radon-Nikodym derivative of the likelihood distributions. Correspondingly, the updated score function can be obtained with no retraining of the diffusion model. We exploit the Ornstein-Uhlenbeck semigroup underlying diffusion models to motivate an exponentially interpolated controller which approximates the true control. Experiments demonstrate controlled generation of low-likelihood samples while remaining consistent with the data geometry.

We consider the spectrum of the Orr-Sommerfeld operator, which is obtained from the linearized Navier-Stokes (LNS) operator near a laminar flow in an infinite two-dimensional channel in the large Reynolds number limit. For a rather general class of laminar flows, which behave locally near the boundary like a Couette flow and are bounded from below (above) by a linear increasing (decreasing) function, we show that there exists an eigenvalue of the LNS operator which coincides, to leading order, with the eigenvalue of LNS near Couette flow, which was formally obtained by Wasow in 1953.

We establish an a priori bound for the dynamical parabolic $\Phi_3^4$ model with harmonic potential. This bound yields the global well-posedness of the equation and, via the Krylov-Bogoliubov method, the existence of an invariant measure, shown to be non-Gaussian. The argument builds on the strategy developed by Mourrat and Weber for the periodic $\Phi_3^4$ model, with substantial modifications to handle the non-compact geometry of $\mathbb{R}^3$ and the spectral framework imposed by the harmonic oscillator. We further prove that this measure is unique in the small-coupling regime.

In this paper, we prove that continuous $2$-convex solutions of the $\sigma_{2}$ equation with positive continuous density on bounded strictly mean-convex $C^{3}$ domains with $C^{3}$ boundary values belong to $W^{2,p}$ for every $1<p<\infty$ in every dimension. If, in addition, the density is $C^{\alpha}$, $0<\alpha<1$, the solutions belong to $C^{2,\alpha}$. Moreover, two counterexamples show that neither the $C^{3}$ assumption on the domain nor that on the boundary values can in general be weakened to $C^{2,1}$.

We study the nonlocal continuity equation \[ \partial_t\mu_b =\operatorname{div}\!\left( \mu_b\nabla\log\bigl((I-b^2\Delta)^{-1}\mu_b\bigr) \right) \] on a closed connected Riemannian manifold. For smooth strictly positive initial data, we prove that as $b \to 0$, its global solution converges to heat flow $\mu(t)$ at the sharp, uniform-in-time rate \[ \sup_{t\ge0}\|\mu_b(t)-\mu(t)\|_{L^1}\le Cb^2. \] The key estimate is the uniform dissipation of a $b$-weighted higher-order resolvent energy, which yields exponential relaxation despite the absence of a Wasserstein gradient-flow structure. On the circle, we also analyze the corresponding deterministic $N$-particle dynamics. A weak–strong modulated energy argument gives \[ \mathbb E\!\left[ \sup_{t\ge0}W_1(\mu_b^N(t),\mu_b(t)) \right] \le C(Nb)^{-1/2} \] for iid initialization. Consequently, the choice $b\asymp N^{-1/5}$ approximates heat flow uniformly in time at rate $N^{-2/5}$.

We show that scalar local minimizers of the Allen-Cahn functional with the double-well potential and homogeneous Neumann boundary conditions on a bounded Lipschitz domain are isolated in the $L^1$-topology. This affirmatively answers a question raised by Kohn and Sternberg (Local minimisers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989)). The proof implements an analytic Lyapunov-Schmidt reduction inspired by the theory of monotone dynamical systems in works of Jiang-Yu and Hirsch-Smith in an elliptic setting.

We prove the existence of nonnegative radial ground states for a class of nonlinear biharmonic equations in $\mathbb R^N$, covering subcritical and critical nonlinearities. The solutions are obtained as limits of clamped ground states on expanding balls. The Cassani-Tarsia homogeneous maximum principle in [D. Cassani. A. Tarsia, Adv. Nonlinear Anal. 11 (2022)] makes the approximating states positive. A variational decomposition on the radial Nehari manifold then excludes every zero sphere of positive radius, so the limiting ground state is positive on $\mathbb R^N\setminus\{0\}$. When the fourth order operator factorises into two second-order operators with positive resolvents, the conclusion upgrades to $u>0$ throughout $\mathbb R^N$, including at the origin. We also give an explicit counterexample showing that the tempting extension of the homogeneous maximum principle to super-solutions with arbitrary nonhomogeneous boundary data is false, even with an arbitrarily large positive zeroth-order coefficient.

Consider the three-dimensional primitive equations subject to Neumann wind stress driven by finitely many Brownian motions. For arbitrary smooth spatial wind profiles, global pathwise existence and uniqueness without assuming a vertical derivative of the initial datum is proved, extending the local theory of Binz, Hieber, Hussein, and Saal: The primitive equations with stochastic wind driven boundary conditions (J. Math. Pures Appl. (9) (2024), 76–101) for this class of data and wind profiles. The proof combines a local maximal $\rL^2$ regularity construction with weighted estimates for the boundary stochastic convolution.

We study the long-time pointwise behavior of small localized perturbations of a constant state for the one-dimensional compressible isentropic Navier-Stokes equations. After subtracting the two Burgers diffusion waves, the convergent sum of all higher-order diffusion waves, and the cross-family viscous corrections, we prove a cone-preserving pointwise estimate for the exact physical remainder and, in particular, \[ |R_i(x,t)|\leq C E_N\log(2+t)\Psi_i(x,t), \qquad \sup_{x\in\mathbb R}\Psi_i(x,t)\leq C(1+t)^{-1}. \] Here \(E_N\) measures the size of the initial data and \(\Psi_i\) is the cone-resolved weight; both are defined precisely in the main theorem below. Thus $\|R_i(t)\|_{L^\infty}\leq C E_N(1+t)^{-1}\log(2+t)$. The key new idea is to apply a familywise Cole–Hopf transformation to the spatial antiderivative of the remainder, which exactly eliminates the critical same-family first-order feedback. We further construct an approximate Green function adapted to the two characteristic families and combine it with Gaussian-mode extraction and a Kawashima-type energy argument. This yields a cone-preserving estimate at the limiting decay rate, up to a logarithmic loss.

Bressan and Shen (2021) jointly with Bressan and Murray (2020) proposed a simple example of a Cauchy problem on $\mathbb{R}^2$, characterized by two key parameters, intended to yield nonunique weak solutions of the incompressible Euler equations. They suggest that a limiting process applied to two specific approximations of their example initial data yields two distinct solutions. Here we provide evidence coming from a carefully designed numerical procedure to support if, or when, this intriguing claim is valid. Some of the numerical challenges encountered in the present study are: (i) the proposed initial data is unbounded in a neighborhood of the origin, (ii) the proposed problem is posed on all of $\mathbb{R}^2$, (iii) several essential symmetries present in the true solution must be captured, (iv) the proposed example depends on two parameters which may be critically linked. We demonstrate that an extremely large computational domain is mathematically necessary to capture the core symmetries of the exact free-space problem. Accordingly we implement a nested-grid strategy specifically designed for its particular suitability and efficacy for the problem at hand. Our strategy also allows us to achieve extreme grid refinement in the region where solution accuracy is most critical. Within our nested-grid framework, we design an underlying discretization scheme that is high-order, robust, and fast. We consider a broad collection of allowable parameter pairs, but among all of them we find only one that may satisfy the suggested claim.

We introduce a microlocal formulation of the Schrödinger-Klein-Gordon system describing the interaction between a non-relativistic quantum particle and a Klein-Gordon field through Yukawa coupling. Instead of working directly with the Schrödinger wave function, we represent the quantum component by its Fourier-Wigner transform, and we rewrite the Klein-Gordon equation in terms of a complex Fourier variable. Eliminating the field variable yields a closed nonlinear Fourier-Moyal equation on phase space whose unknown is the Fourier-Wigner distribution associated with the quantum component. The resulting formulation provides a refined description of the particle-field interaction at the microlocal level. In particular, the original cubic coupling is transformed into a quadratic self-interaction governed by an explicit bilinear operator. This new representation permits the use of techniques from time-frequency analysis. Building on this, we develop a well-posedness theory for the Fourier-Moyal equation in anisotropic spaces involving Wiener-type norms and prescribed moduli of continuity. Local existence and uniqueness are established under general assumptions on the ultraviolet cutoff, together with propagation of microlocal regularity. We then introduce a new construction of weak L 2 -solutions by exploiting compactness properties of Fourier-Wigner transforms and obtain global existence for prepared data. The aim is to provide an alternative analytical framework for the study of Yukawa-type interactions and to establish a bridge between nonlinear dispersive equations and phase-space methods.

We classify entire stable solutions with quadratic energy growth in the $4$-dimensional abelian Higgs model. More precisely, we prove that such solutions are holomorphic with respect to some complex structure, namely they satisfy the first order equations introduced by Bradlow, and consequently their nodal sets are holomorphic curves. Conversely, we construct solutions whose nodal set is any prescribed holomorphic curve with quadratic area growth. As a corollary, we show that all stable solutions with linear energy growth in $\mathbb{R}^3$ are two-dimensional Bogomolnyi vortices.

We investigate the maximization problem for the family of mixed-norm Strichartz inequalities for the Schrödinger equation, $\|e^{-it\Delta/2}f\|_{L_t^qL_{\boldsymbol{x}}^r(\mathbb{R}^{1+d})}\le C_{q,r}\lVert f\rVert_{L^2(\mathbb{R}^d)}$, with $2/q+d/r=d/2$, $q,r\geq 2$, and thus $r\leq 2d/(d-2)$ if $d\geq 3$. We show that, in low dimensions $1\leq d\leq 5$, the thresholds $\rho_1=10$, $\rho_2=6$, $\rho_3=4\sqrt{7}-6\approx 4.583$, $\rho_4=2\sqrt{15}-4\approx 3.746$, and $\rho_5=10/3\approx 3.333$ are such that gaussians are stable local maximizers for $2<r<\rho_d$, and fail to be local maximizers for $\rho_d<r\leq 2d/(d-2)$ (with the conventions there is no upper bound on $r$ when $d\in\{1,2\}$ and that $r=\infty$ is excluded when $d=2$). In the cases $(q,r,d)\in\{(6,6,1),(8,4,1),(4,4,2)\}$, we establish global stability inequalities with effective stability constants. Both proofs hinge on spectral gaps which we compute exactly.