Papers

In this paper, we introduce a nonlocal, variational model for thin films. We consider convolution-type functionals defined on a thin domain whose thickness is of order $\gamma$, where the effective interactions range between points is of order $\varepsilon$. We study the $\Gamma$-convergence of these energies, as both parameters vanish, to a local integral functional defined on a lower-dimensional domain. In the periodic homogenization setting, the limit energy density is characterized by an asymptotic formula that depends on the interplay between $\varepsilon$ and $\gamma$. Under suitable assumptions, this formula exhibits a separation of scales effect, namely, the limit energy can be obtained by performing two successive $\Gamma$-limits, first letting one parameter tend to zero while keeping the other fixed.

We consider a shallow water model in a homogenization framework. For periodic topographies, Craig, Lannes and Sulem have established a consistency result under some non-resonance conditions. In the present contribution, we significantly relax the periodicity condition and treat general topographies under minimal assumptions.

The paper concerns the problem for the ultrahyperbolic equation in the Euclidean space with data on a characteristic hyperplane. Smoothness and asymptotics of the solution along characteristic lines transversal to the initial hyperplane are investigated.

This paper is concerned with the global solvability of a chemotaxis-Euler system in bounded domains of $\mathbb{R}^2$. Completing the system with physical boundary conditions, we firstly prove local well-posedness of the corresponding initial-boundary value problem and then present some blow-up criteria. We also show that the local solutions can be extended globally in time under suitable smallness assumptions on initial and boundary data.

In this paper, we study the family of inhomogeneous discounted Hamilton-Jacobi equations \begin{equation}\label{hjs1} \lambda(x)u+h(x,d_x u)=c \quad \tag{$\ast$} \end{equation} on a closed manifold $M$ with a non-identically vanishing discount factor $\lambda(x)$. There is a critical value $c_0\in[-\infty,\infty)$ such that \eqref{hjs1} admits a viscosity solution if $c>c_0$ and no solution if $c<c_0$. Inspired by the recent development \cite{RWY} on the stability theory of viscosity solutions, we show that the equation admits an asymptotically stable solution if and only if $c>c_0$. In this case, we determine the basin of the stable solution and investigate the long-time behavior of the solution semigroup associated with \eqref{hjs1}. In particular, we obtain a formula relating the lowest convergence rate of the solution semigroup near a stable solution to a minimizing problem concerning the integral of $\lambda$ over Mather measures supporting on the $1$-graph of the stable solution. This formula has two direct consequences: 1) it yields a description of their asymptotic behavior as $c$ tends to infinity; 2) it helps to classify the ergodic Mather measures and locate their distribution in the phase space. The second consequence leads to a dynamical necessary condition for $c=c_0$.

In this paper, we derive new commutator estimates in the Triebel-Lizorkin spaces by employing Bony's para-product decomposition, the Nikol'skij representation, and the Fefferman-Stein vector-valued maximal function. These estimates are then applied to develop a general theory for transport equations. Although analogous results are already available in the setting of Besov spaces, the methods developed there do not carry over directly to the Triebel-Lizorkin case. Our approach works for Triebel-Lizorkin spaces and, as a byproduct, also yields the corresponding results in Besov spaces. All proofs are presented in a unified manner that applies to both scales of function spaces, thereby extending and sharpening previous results on transport equations in these frameworks. Furthermore, the general theory we obtain is widely applicable to evolution equations, including incompressible and compressible ideal fluid flows, shallow water waves, and related models. As an illustration, we consider the two-component Euler-Poincaré system. Using the theoretical framework developed herein, we establish its local well-posedness and a blow-up criterion in both sub-critical and critical Triebel-Lizorkin spaces.

We study Liouville-type results for the stationary Navier–Stokes equations in $\mathbb{R}^3$. We prove that any $\dot{H}^1(\mathbb{R}^3)$ solution is trivial under an integrability condition imposed only on the radial component of the velocity, namely $u_\rho(x) \in L^p(\mathbb{R}^3)$ with $3/2 < p \leq 3$. We also establish a uniqueness result in a variable-exponent setting, where an $L^6$-type condition is required only on a bounded region, while the exponent approaches the critical value $3$ at infinity. Our analysis reveals that the rigidity of the stationary Navier–Stokes system can be driven by localized and radial integrability properties, rather than uniform global conditions.

In this note, we want to establish several formulas about functionals along harmonic Ricci flow on surface with boundary

In this work, we propose an easy-to-implement fixed-point algorithm for reconstructing a space-time dependent source in a subdiffusion model from lateral boundary measurements. The numerical scheme combines a Galerkin finite element method for spatial discretization with a finite difference method for temporal discretization. We establish the linear convergence of the fixed-point iteration and derive an error bound that depends explicitly on the discretization parameters and the noise level. The error analysis relies on stability properties of the continuous inverse problem and technical estimates for the associated direct problem with limited-regularity data. Numerical experiments are presented to support and complement the theoretical analysis.

In \cite{CV23}, Chamorro and Vergara-Hermosilla established several Liouville-type theorems to the Navier-Stokes equations in the framework of the variable Lebesgue spaces. These results may allow the variable exponent $p(\cdot)$ beyond the range of $[3,\frac{9}{2}]$ in some non-negligible regions in $\mathbb{R}^3$. In this paper we find two new non-negligible regions, in which the Liouville-type theorems still hold under some assumptions imposed on $p(\cdot)$ in these regions. Our results can be regarded as the generalization of the results in \cite{CV23}.

In this paper we establish Hölder continuity estimates for viscosity solutions to first order Hamilton-Jacobi equations linked to linear control systems satisfying the Kalman rank condition. Our model Hamiltonians are non-convex in the generalised momentum variable and - more importantly - they lack coercivity in certain directions. Therefore, all previously available results from the literature cannot be applied to these degenerate settings. In order to overcome these obstructions, we design a geometric argument, dictated by the linear control system. As a result of this, the obtained Hölder estimates are quantified in an anisotropic way within this geometric framework. The estimates hold true for unbounded source terms, for which one part of our analysis is inspired by a recent result on De Giorgi type methods for hypoelliptic operators.

On metric measure spaces with sub-Gaussian heat kernel behavior in small time, we obtain a sufficient condition to solve Wick renormalized stochastic quantization equations with polynomial interaction. Given the power of the nonlinearity, the local solution condition depends on the Hausdorff dimension $d_h$, the walk dimension $d_w$, and the maximal spatial Hölder regularity of the heat kernel $\Theta$. A slightly more restrictive condition based on the same parameters is required for a global solution. For all global solutions, we construct an invariant measure for the Markov process defined by the solution. Our results apply to many rough spaces such as Barlow–Kigami type fractals as well as their Cartesian products and open up the possibility of making rigorous various structures in quantum field theory and statistical mechanics in non-integer dimensions. In the process, we build entirely from the short-time heat semigroup the necessary analytic framework that accommodates the issues which come with allowing rough local geometry.

We study a local trace estimate for the microlocal angular square function \[ G_R f := \left(\sum_\Theta |f_\Theta|^2\right)^{1/2} \] associated with a parabolic decomposition of the frequency annulus of radius $R$ in $\mathbb{R}^3$. The measure under consideration is \[ \mu_Q=\chi\, H^2\lfloor S_Q, \] where $\chi\in L^\infty(S_Q)$ is a measurable nonnegative density compactly supported in the patch, and \[ S_Q=\{(u_1,u_2,Q(u_1,u_2)):u\in U\}, \qquad Q(u_1,u_2)=\frac12(\lambda_1u_1^2+\lambda_2u_2^2), \qquad \lambda_1\lambda_2 >0. \] Writing $\rho=R^{-1/2}$, we prove \[ \| G_R f\|_{L^2(\mathrm d\mu_Q)} \lesssim R^{1/8}\|f\|_{L^2(\mathbb R^3)}. \] Under local positivity of the density near the tangency point, the factor $R^{1/8}$ is attained by a tangent wave packet test and hence cannot be improved within this elliptic quadratic model, at this parabolic scale and for this angular square function. In particular, it measures the failure of a trace bound uniform in $R$ within this class. Its source is the extreme tangential interaction between a tube of radius $\rho$ and $S_Q$: the relevant surface measure is $\sim\rho^{3/2}$, whereas an $L^2$-normalized wave packet has quadratic size $\sim\rho^{-2}$. Thus the optimal quadratic cost is $\rho^{-1/2}$, producing the norm factor $\rho^{-1/4}=R^{1/8}$.

In this paper we revisit the $\sigma_k$-Yamabe problem on $M^n$, namely, finding a conformal metric with constant $\sigma_k$-scalar curvature. We prove that on a closed manifold $\left(M,\left[g_0\right]\right)$ with positive Yamabe constant $Y_1\left(M,\left[g_0\right]\right)>0$, the $\sigma_2$-Yamabe constant $$ Y_2\left(M,\left[g_0\right]\right):=\inf _{g \in\left[g_0\right], R_g>0} \frac{\int_M \sigma_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}} $$ is achieved by a conformal metric $g \in\left[g_0\right]$, which in particular solves the $\sigma_2$-Yamabe problem, assuming $Y_2\left(M,\left[g_0\right]\right)>0$. As a consequence, for any $\left(M, g_0\right)$ with $Y_1\left(M,\left[g_0\right]\right)>$ 0 and $Y_2\left(M,\left[g_0\right]\right)>0$ one has $$ \inf _{g \in\left[g_0\right], R_g>0} \frac{\int_M \sigma_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}}=\inf _{g \in\left[g_0\right], R_g>0, \sigma_2(g)>0} \frac{\int_M \sigma_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}} . $$ We also show that these conclusions can fail if the condition $R_g>0$ is removed.

In this work, we significantly extend the results of D. Houpa, 2006 on the Goursat problem for second-order semi-linear hyperbolic systems to the broader framwork of second-order hyper-quasilinear hyperbolic systems of Goursat type, in which the coefficients of the second-order derivatives depend linearly on the unknown. By adapting techniques inspired by Y. Foures (Choquet)- Bruhat, Acta Mathematica, 1952. we show that in the Sobolev type spaces for the Goursat problem quasilinear hyperbolic of the second order considered, the solution exists and is defined in the vicinity of the meeting characteristic hypersurfaces which carry the initial data. As an application, in harmonic gauge, we derive a semi-global existence and uniqueness result for the vacuum Einstein equations.

In this note, we report five mathematical discoveries made in collaboration with Grok, all of which have been subsequently verified by the authors. These include an improved lower bound on the maximal Gaussian perimeter of convex sets in $\mathbb{R}^n$, sharper $L_2$-$L_1$ moment comparison inequalities on the Hamming cube $\{-1,1\}^n$, a strengthened autoconvolution inequality, improved asymptotic bounds on the size of the largest $g$-Sidon sets in $\{1,\dots,n\}$, and an optimal balanced Szarek's inequality.

In this work we consider the well posed version of the Kaup-Broer-Kuperschmidt system in two dimensions. We numerically construct soliton type solutions and show that they are unstable both against dispersion and singularity formation. Further, we study line solitons and their stability, as well as generally localised initial data. In either case we fail to find stable structures.

The Velocity-Vorticity (VV) formulation of the incompressible Navier-Stokes equations has become popular in recent years, especially in numerical studies, due to its structural advantages. Recently, with L. Rebholz, we introduced a Voigt regularization to the momentum equation in this formulation, establishing global well-posedness of the regularized system in 3D, along with convergence results and a blow-up criterion. In the present work, we extend these ideas to the 3D magnetohydrodynamics (MHD) equations. While it may seem that a “VV-type” split on the magnetic equation is required, we show that no such modification is necessary, and global well-posedness holds with a Voigt regularization only on the momentum equation, preserving the structure of both the vorticity and magnetic equations. We also prove that the regularized system converges to the original system, up to a possible blow-up time, and we establish a blow-up criterion for solutions to the original 3D MHD system.

We consider the stabilization of Vlasov–Poisson plasma dynamics, a central control problem in nuclear fusion. Our focus is the gap between what an ideal controller would use and what experiments can actually observe: while optimal policy may rely on the full phase-space state, practical feedback is typically limited to sparse macroscopic diagnostics. We therefore study imitation learning methods that distill a fully observed expert policy into controllers operating only on macroscopic measurements. We show the stability guarantees of the learned policy, where the error floor depends on the minimal behavior cloning loss achievable under the observation constraints. We further characterize this minimal loss in terms of a notion of entropy that quantifies the complexity of the initial distribution. Our results demonstrates the theoretical feasibility of learning stabilizing feedback policies for kinetic plasma dynamics from macroscopic observations, and exhibits the adaptivity of the learning approach to low-complexity structures. Through extensive numerical experiments, we validate our theory and show that the learned policies can stabilize the system using only macroscopic observations, within a significantly longer time horizon than non-adaptive baseline controllers.

We show that if $M^n$ is a properly immersed, two-sided, stable minimal hypersurface in $B^{n+1}_1(0)\setminus S$, where $S$ is closed with $\mathcal{H}^{n-2}(S)=0$, then $\text{dim}_{\mathcal{H}}\text{sing}(M)\leq n-7$, namely $\overline{M}\cap B^{n+1}_1(0)$ is represented by a smooth minimal immersion outside a closed set of generally unavoidable singularities which has Hausdorff dimension at most $n-7$. This provides the optimal a priori size assumption on the non-immersed singular set in order to guarantee optimal regularity. Consequently, such objects form a compact class under mass upper bounds.

This article proves norm inflation in the critical Sobolev space $H^{3/2}(\mathbb{R})$ for the $b$-Novikov equation, which is a $1$-parameter family of Camassa-Holm-type equations with cubic nonlinearities. This result completes the well-posedness theory for this equation, which was previously known to be locally well-posed in $H^{s}(\mathbb{R})$ for $s>3/2$ and ill-posed in $H^{s}(\mathbb{R})$ for $s<3/2$.

This paper establishes the existence and uniqueness of mild solutions to stationary Hamilton-Jacobi-Bellman (HJB) equations associated with infinite-horizon stochastic optimal control problems in separable Hilbert spaces. Our framework includes settings with a lack of global smoothing properties of the transition semigroup, singular dynamics involving unbounded control operators, and state-dependent running costs. We overcome these challenges by lifting the state space using the Projected Evolutionary Lifting technique. This work is an extension of G. Bolli and F. Gozzi, Lifting and partial smoothing for stationary HJB equations and related control problems in infinite dimensions, 2025, in which existence and uniqueness is proved via a contraction mapping argument and is consequently restricted to sufficiently large discount factors. We remove this restriction, proving existence and uniqueness for any discount rate $\lambda > 0$ using tools from the theory of maximally monotone operators.

We generalize recent results on the monotonicity method, for inclusion detection in the partial data anisotropic Calderón problem, to very general non-self-adjoint perturbations. This involves a forward model that accounts for both the anisotropic real conductivity and the anisotropic permittivity, and the results hold in any spatial dimension $d \geq 2$. We assume that the inclusion boundaries can be reached from the domain boundary via a set on which the background conductivity is self-adjoint, and that a definiteness condition holds near the inclusion boundaries. Away from the inclusion boundaries we allow general $L^\infty$ non-self-adjoint perturbations. We only require unique continuation based on the self-adjoint part of the background conductivity, thus making the methods compatible with generic unique continuation results.

In this paper, we consider the upper and lower bounds of the lifespan of classical solutions of the Cauchy problem for the one-dimensional quasilinear wave equation $u_{tt}-c(u_x)^2u_{xx}=0$ where the derivative of $c(\theta)$ tends to $0$ near the origin. In particular, our result shows that the lifespan of the solution extends algebraically depending on the smallness of the initial data. Furthermore, we also show that when $c(\theta)$ is flat at the origin ($c'(\theta)$ and any higher order derivatives vanish at the origin), the lifespan extends exponentially depending on the smallness of the initial data. Our proof is based on the method of Lax's characteristics and Riemann invariants.

Strichartz inequalities are a cornerstone of the modern theory of dispersive PDEs, but their extremizers are known explicitly only in a handful of sharp cases. The non-convexity of the underlying functional makes the problem hard, and to our knowledge no systematic numerical attack has been attempted. We propose a simple neural-network-based pipeline that searches for extremizers as critical points of the Strichartz ratio, and apply it in three settings. First, on the Schrödinger group we recover the Gaussian extremizers of Foschi and Hundertmark–Zharnitsky in dimensions $d=1,2$ to within $10^{-3}$ relative error, with no analytical prior. Second, on $59$ further admissible pairs in $d=1$ where the answer is conjectural, the method consistently finds Gaussians, supporting the conjecture that Gaussians are the universal extremizers in the admissible range. Third, on the critical Airy–Strichartz inequality at $\gamma=1/q$, where existence is open, the optimization does not converge to any $L^2$ profile: instead, the iterates organize themselves as mKdV breathers $B(0,\cdot;\alpha,1,0,0)$ with growing internal frequency $\alpha$, and the discovered ratio approaches the Frank–Sabin universal lower bound $\widetilde A_{q,r}$ from below with a power-law gap $\sim\alpha^{-0.9}$. We confirm the same picture with an independent Hermite-basis ansatz. We propose a precise conjecture: the supremum equals $\widetilde A_{q,r}$ and is approached, but not attained, along the breather family. The pipeline thus serves both as a validator on known cases and as a discovery tool when no extremizer exists.

In this paper, we study detonation wave solutions to one-dimensional piston problem for the Zeldovich-von Neumann-Döring (ZND) combustion model with a one-step exothermic chemical reaction. As a special type of shock wave, the position of the detonation wave is unknown, which make our model to be a free boundary problem.~The global existence of detonation wave solutions to this free boundary problem is proved. Compared with previous result~\cite{Lai}, we do not impose any dissipation conditions on the equations and the boundaries.

In this paper, we consider the following problem: \[ \begin{cases} -\nabla\cdot A(x,u,\nabla u) + H(x,u,\nabla u) = f(x), & x \in \Omega, u = 0, & x \in \partial \Omega, \end{cases} \] in a bounded open set \( \Omega \subset \mathbb{R}^N \). We have established certain gradient estimates and proved the existence of a renormalized solution for the equation.

Let $f(t,x),M(t,x)\in C([0,1]^2)$ with $M(t,x)>0$. We consider differential equations of the form \[ \frac{\partial f}{\partial t}(t,x)=\frac{M(t,x)f(t,x)-M(t,0)f(t,0)}{x},\quad x>0. \] For a fixed positive weight $M$, we ask whether the condition $f(0,x)=0$ forces $f\equiv 0$. We show the answer is negative for smooth functions: there exist $f(t,x),M(t,x)\in C^{\infty}([0,1]^2)$ with $f(0,x)=0$, $f(t,0)\not\equiv 0$, and $M(t,x)>0$ satisfying the above equation. However, we show that for a Borel-Laplace class of $M(t,x)$, the equation does have uniqueness. We relate this to uniqueness/non-uniqueness theorems for weighted Laplace transforms. A key example originated in an output by ChatGPT-5.5-Pro.

This paper is devoted to the study of normalized solutions to the Kirchhoff type equation with a logarithmic perturbation\[-\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2 \,\mathrm{d}x \right) \Delta u=\lambda u+|u|^{p-2}u+u\log u^2,\quad x \in\mathbb{R}^3, \]under the normalized constraint $\int_{\mathbb{R}^3} u^2 \,\mathrm{d}x = c^2$, where $a,b>0$, $2<p\leq 6$, $c>0$ is a constant, and $\lambda\in\mathbb{R}$ emerges as a Lagrange multiplier which is not a priori known. A unified variational framework is developed based on Orlicz spaces together with the Pohozaev constraint method and refined fiber map analysis. For $2<p<\frac{14}{3}$ or $p=\frac{14}{3}$ with small mass, the energy functional is bounded from below and admits a positive radial ground state minimizer. For $\frac{14}{3}<p<6$, where the energy functional is unbounded from below, we establish the existence of two normalized solutions for small mass: a ground state $u_c^+$ obtained via local minimization, and a second solution $u_c^-$ obtained via minimization on the negative component of the Pohozaev manifold. For the Sobolev critical case $p=6$, we construct a ground state solution and, under a technical condition on the parameters, a second solution by introducing a proper auxiliary functional and precise energy estimates with Aubin-Talenti bubbles. Asymptotically as $c\to0^+$, the $L^{2}$ norm of the gradient of ground state solution vanishes for $2<p\le6$. Surprisingly, for $\frac{14}{3}<p<6$, the $L^{2}$ norm of the gradient of the second solution diverges to infinity as $c\to 0^+$, while for $p=6$ it concentrates around the Aubin-Talenti bubble with energy converging to the energy level of the corresponding critical Kirchhoff equation.

In this article, we study threshold phenomena for the semilinear double-power elliptic equation $$-\Delta_{\mathbb{B}^N} u - \lambda u = |u|^{p-1}u - |u|^{q-1}u, \quad u \in H^1(\mathbb{B}^N),$$ on the hyperbolic space $\mathbb{B}^N$ for $N \ge 3$. For parameters $1 < p \le 2^*-1$ (though we occasionally allow for supercritical exponents) and $q > 0$, we seek to identify the optimal spectral regimes for $\lambda \in \mathbb{R}$ that delineate the existence and non-existence of positive-energy solutions. We achieve a complete resolution of these thresholds across all exponent configurations: $p < q$, $0 < q < 1 < p$, and $1 < q < p$. Our results demonstrate that the boundary separating these regimes is governed by an explicit critical spectral parameter, which depends on $p$, $q$, and $N$ in the regime where $p < q$, but depends solely on $N$ in the remaining cases.