Papers

We introduce the calibration energy for oriented immersions into Euclidean space, quantifying the deviation from calibrated geometry. A key property is that this energy may remain finite for infinite-volume immersions, while a null-Lagrangian structure ensures that it has the same first variation as the volume functional. We establish an exact dissipation identity for the calibration energy along proper oriented mean curvature flows in arbitrary dimensions and codimensions, under a mild local-volume bound. In fact, our result covers a class of singular calibrations and singular initial data. Even in the smooth setting, this provides a new finite variational framework for mean curvature flow beyond the finite-volume regime. As a main application, we establish a general dynamical rigidity theorem for calibrated cones in arbitrary codimension: no singular calibrated cone can be desingularized by a proper oriented mean curvature flow. Our framework further yields novel rigidity theorems for solitons and convergence for two-dimensional immortal flows.

We show that H-convergence for multiplication type operators as envisioned by Murat and Tartar in the 1970's always implies nonlocal H-convergence as introduced in 2018 in Calc.~Var.~PDE 57(6):159. In contrast to earlier findings, the results presented here work for arbitrary space dimensions, are not bound to a certain geometry of the underlying domain, and do not explicitly require an underlying Hilbert complex for the application of any particular version of the div-curl lemma. We extend classical theory and the main results to more general differential operators with different boundary conditions and orders. Furthermore, the present results confirm homogenisation formulas used in the literature of which we failed to find an explicit proof. As a consequence, H-convergence for multiplication operators in divergence form problems will always imply H-type convergence for a different variational problem for free.

For any $N\geq 2$, we construct a global solution of the energy-critical focusing wave equation in dimension four which blows up in infinite time at $N$ prescribed points $z_1,\ldots,z_N\in \mathbb R^4$, provided that the points form one orbit under a finite group of orthogonal symmetries. We denote by $c:=2\sum_{j\ne k}|z_j-z_k|^{-2}>0$ the corresponding interaction coefficient, which is independent of $k$. The common concentration scale satisfies \[ \log\frac{1}{\lambda(t)} = \left(\frac{9c}{4}\right)^{1/3}t^{2/3}+O(t^{1/3}) \qquad \text{as } t\to+\infty . \] This concentration rate comes from a genuinely four-dimensional effect: the borderline decay of the ground state makes the interaction between different bubbles enter the leading order parameter dynamics.

We extend the exponential stability theorem for the three-dimensional incompressible Navier-Stokes equations from hyperbolic 3-space $\HH^3$ (established in a companion paper) to complete simply connected Riemannian 3-manifolds $(M^3, g)$ with pinched negative sectional curvature $-b^2 \leq K \leq -a^2 < 0$ and bounded geometry (including a strictly positive injectivity radius). The deformation Laplacian $\Delta_\Def = \Delta_B + \Ric$ remains the viscous operator, selected by Lagrangian kinematics. We prove that the {exact} system admits a unique global mild solution for small $L^3$ data, with exponential decay at a rate determined by the spectral gap of the Stokes operator. The extension overcomes three obstacles absent on $\HH^3$: (i) the semigroup factorisation $e^{t\Delta_\Def} = e^{-2t}e^{t\Delta_B}$ fails because $\Ric$ is not a scalar multiple of the metric; (ii) the Leray projector no longer commutes with $\Delta_\Def$; (iii) the exact spectral gap is unknown. We resolve (i) unconditionally, without any curvature restriction, by observing that the Ricci perturbation $V = \Ric + 2a^2 g$ is negative semi-definite and applying a Trotter product bound with the diamagnetic inequality. We resolve (ii) by an algebraic reduction of the commutator $[\PP, \Delta_\Def]$ to the complementary projector $(I-\PP)$ applied to the shifted Ricci endomorphism, giving a clean zeroth-order bound proportional to the curvature variation $b^2 - a^2$. This is the sole source of a curvature pinching constraint. We resolve (iii) via McKean's theorem, the diamagnetic inequality, and the Weitzenböck identity. The Fujita-Kato temporal singularity exponent $1/2 - 3/(2p)$ is unchanged from the $\HH^3$ case, confirming that the ultraviolet scaling obstruction is local and geometry-independent, driven fundamentally by an unresolvable temporal scaling mismatch.

We establish the well-posedness of the 3D rotating incompressible Navier-Stokes equations with critical initial data $u_{0,\Omega}\in X_{0,q,p}^{\Omega}$ for $p<5$, where $X_{0,q,p}^{\Omega}$ is defined by the norm \begin{equation*} \begin{aligned} &\|u_{0,\Omega}\|_{X_{0,q,p}^{\Omega}}:= \Omega^{3- \frac{6}{q}}\|u_{0,\Omega}\|_{\dot{B}_{q,\infty}^{-7+\frac{15}{q}}}^{\ell_\Omega} +\|u_{0,\Omega}\|_{\dot{B}_{p,\infty}^{-1+\frac{3}{p}}}^{h_\Omega}. \end{aligned} \end{equation*} This extends the previous results by Chen, Miao, and Zhang (\cite{CMZ2013}). The main ingredients are a new global-in-time dissipative-dispersive estimate for the Stokes–Coriolis semigroup and corresponding bilinear estimates. Furthermore, we establish the vanishing rotational limit for the 3D rotating Navier-Stokes equations as $\Omega\rightarrow 0^{+}$.

We prove a comparison principle for a class of second-order Hamilton–Jacobi–Bellman equations on the Wasserstein space whose second-order term is generated by a general common-noise Hessian. The main difficulty is that the relevant second-order direction is induced by a state- and measure-dependent coefficient, so the associated perturbation of the measure is no longer a translation or a fixed state-dependent transformation. We introduce a nonlinear flow of measures and use it to transform the Wasserstein-space equation into an augmented equation on $[0,T]\times \mathcal P_2(\mathbb R)\times\mathbb R$, where the general Hessian becomes an ordinary second derivative in the auxiliary variable. The construction may be viewed as a measure-dependent Lamperti transform: it removes the common-noise direction at the level of the equation, but unlike the classical one-dimensional Lamperti transform it permits degeneracy of the coefficient and dependence on the conditional law. We establish the spatial, measure-derivative, and negative-Sobolev estimates for this flow that are needed in the viscosity argument. Under structural assumptions on the transformed Hamiltonian, these estimates yield a Crandall–Ishii type comparison theorem for semicontinuous viscosity sub- and supersolutions. This gives, to the best of our knowledge, the first viscosity comparison framework of this kind for the filtering-driven equations considered here, and opens a new class of second-order PDEs on spaces of measures with state- and law-dependent common-noise directions. As an application, we identify the value function of a controlled stochastic filtering problem with state- and law-dependent common noise as the unique viscosity solution of its dynamic programming equation. We also explain how the same change-of-variable viewpoint applies to Zakai-type Kolmogorov equations on spaces of finite positive measures.

In this paper, we study the robustness of exponential stability for semigroups generated by linear operators under perturbations. Extending a classical result of Gibson's Stability Theorem, we show that if the generator of an analytic exponentially stable semigroup is perturbed by a class of relatively bounded operators satisfying certain assumptions, then exponential stability is preserved, provided the perturbed semigroup is strongly stable. We also show that, for a restricted class of perturbations, the analyticity requirement can be relaxed to Gevrey regularity. Moreover, we present applications to uniformly parabolic equations, degenerate/singular parabolic equations, coupled hyperbolic plate systems, and generalized coupled systems of Kirchhoff-Love plates and a membrane-like electric network.

The Cattaneo-Vernotte model and the relativistic Schrödinger operator represent two fundamental frameworks for the relativistic modifications of normal diffusion, leading to the telegraph equation and a class of spatially nonlocal diffusion equations, respectively. This paper investigates the intrinsic connection between these two relativistic diffusion models. While it is well-established that spatially non-local diffusion arises from subordinating normal diffusion, we reveal a novel reciprocal mechanism: the normal diffusion process can be recovered by subordinating the telegraph process via the corresponding inverse subordinator. Consequently, leveraging the duality between the subordinator and the inverse subordinator, we establish a distinct duality between the telegraph equation and the spatially nonlocal diffusion equation. Furthermore, this specific duality, centered on normal diffusion, is generalized to a broader class of space-time dual operator families anchored on $C_0$-semigroups.

The Bevilacqua-Galeão Model of Anomalous Diffusion introduces two fluxes: a primary flux that follows Fick's law of diffusion, representing the fraction of particles undergoing classical diffusion, and a secondary flux modeled by a fourth-order differential term, which accounts for retention phenomena. We investigate the “analytic emergence” of this secondary flux by treating the model as a (singular) perturbation of the heat equation, which describes the classical diffusion. Rather than applying traditional perturbation methods, or a straightforward Fourier transformation, we employ the powerful framework of enhanced modern group analysis to study this problem. Specifically, by utilizing approximate symmetries we derive an analytic expression for the emergent secondary diffusion as a perturbation of the classical diffusion process, the approximate fundamental solution and the approximate solution of the related Cauchy problem.

We answer in the affirmative a question posed by V. Maz'ya of whether one can continue as a meromorphic function of $t$ the series representation of the fundamental solution of a certain nonlocal parabolic equation associated to a logarithmic Laplacian on the circle, which arises in the study of boundary value problems associated to the ordinary Laplacian on domains with thin cavities. The $a\ln(n) +O(1)$ growth of the eigenvalues of the integral operator, together with explicit formulas for the eigenfunctions and the subleading asymptotic behavior of the eigenvalues, allows us to show that the fundamental solution is reminiscent of a sum of shifted Riemann zeta functions or polylogarithms, depending on the spatial variable. We show an analogous result for an operator related to a different logarithmic Laplacian on the interval, whose structure is similar. Along the way we are led to prove and to conjecture a number of curious identities involving Bell polynomials and Bernoulli numbers related to the exponential of the digamma function which are of independent interest.

For the 2D Navier-Stokes equations with no-slip boundary condition, we consider the issue of whether anomalous dissipation away from the boundary vanishes. In particular, we show that such vanishing occurs if $u^{\nu}$ is uniformily bounded in the Onsager supercritical space $L^{1+}_{t}L^{\infty}_{x,loc}$ with appropriate bounds on the initial conditions. Our method involves arguments from \cite{AD23} and \cite{CW23} involving localization via modulation, together with vorticity energy type estimates inspired by \cite{CFLS16} and estimates involving $L^2$-based structure functions inspired by \cite{DP25dissconc}. Next we show that the aforementioned setting produces convergence to an Euler solution with its large scale approximation satisfying a local energy balance equation. Notably, we do not assume any uniform-in-viscosity bounds on the pressure. The large scale approximation has been introduced in \cite{PGLR18} in the context of partial regularity of the 3D Navier-Stokes equations, yet to the best of our knowledge this is the first time it has been considered in the context of inviscid limits.

Let $(\mathcal{C}, J_{\mathcal{C}}, \omega_{\mathcal{C}}, g_{\mathcal{C}})$ be a Calabi-Yau cone and $(M, J, \omega, g)$ an open Ricci-flat Kähler manifold. We show that, if there exists a diffeomorphism $\Phi: \mathcal{C} \setminus \overline{B_1(o)} \rightarrow M \setminus K$, for some compact $K \subset M$, such that $\Phi^{*}J$ is asymptotic to $J_{\mathcal{C}}$ and $C^{-1} \omega_{\mathcal{C}} \leq \Phi^{*} \omega \leq C \omega_{\mathcal{C}}$ for some $C \geq 1$, then $(M, g)$ is asymptotically conical (AC) with tangent cone at infinity given by $(\mathcal{C}, d_{g_{\mathcal{C}}})$. As a consequence, we obtain that any Ricci-flat Kähler metric on $T^{*}S^n$ which is quasi-isometric to the Stenzel metric must be equal to the Stenzel metric up to scaling and diffeomorphism. Similarly, any Ricci-flat Kähler metric on $\mathcal{O}_{\mathbb{P}^1}(-1)^{\oplus2}$ which is quasi-isometric to the Candelas-De la Ossa metric must be equal to the Candelas-De la Ossa metric up to scaling and diffeomorphism. This provides new examples of complete Calabi-Yau manifolds for which a Liouville-type theroem holds.

We study reaction-diffusion equations with finite signed measure delays and nonlinear, possibly nonlocal feedback. We treat abstract reactions that are Lipschitz on bounded $L^2$-balls with quadratic coercivity, and pointwise cubic polynomials with positive leading coefficient in spatial dimensions at most three. For affine-growth feedback and time-independent forcing in $H^{-1}(\Omega)$, both classes generate global weak solution semiflows on $X=C([-r,0];L^2(\Omega))$. Besides total-variation Lipschitz stability, we obtain quantitative weak-star stability with modulus $\varpi(d)=d\log(e+d^{-1})$ for the abstract class and $\varpi(d)^{1/2}$ for cubic reactions, where $d$ is the bounded-Lipschitz distance between delay measures. The estimates are uniform on bounded sets of continuous histories and yield concentration and atomic-quadrature rates. A damping condition gives common compact absorption for the abstract class. Cubic coercivity removes that smallness condition on every fixed total-variation-bounded kernel class. In both cases the global attractors are upper semicontinuous in $X$ and in $C([-r,0];H_0^1(\Omega))$.

We give a proof of the cosmological splitting conjecture of Robert Bartnik from 1988, which expresses the rigidity of the cosmological Hawking–Penrose singularity theorem. It states that a timelike geodesically complete, globally hyperbolic spacetime which has compact Cauchy surfaces and satisfies the strong energy condition must split isometrically as a Lorentzian product. Our methods combine the construction of global viscosity solutions to the Lorentzian eikonal equation by Zhu–Wu–Cui with our recently developed elliptic approach to the proof of Lorentzian splitting theorems in joint work with Braun, Gigli and Sämann, where we make use of the $p$-d'Alembertian operator for $p < 1$.

In this paper, we study the non-symmetric system of semilinear damped $\sigma$-evolution equations with different damping types, where two power exponents of nonlinearities belong to the critical curve, by using moduli of continuity in nonlinear terms. Our goal is to determine the sharp conditions on these moduli of continuity that guarantee the global (in time) existence of Sobolev solutions or, conversely, lead to finite-time blow-up. Furthermore, by employing the analysis introduced in the proof of our blow-up result, we provide a positive answer to an open problem for the symmetric models posed in the literature.

Carlen, Frank and Lieb studied stability estimates for the lowest eigenvalue of a Schrödinger operator by decomposing the problem into a stability estimate for Hölder's inequality and a stability estimate for a Gagliardo–Nirenberg–Sobolev inequality. In this note we point out that, if the Hölder step is replaced by the optimal $L^1$-stability theorem of Leng and Lu in probabilistic form, then one obtains a density-distance version of the Carlen–Frank–Lieb stability theorem. The new formulation measures the $L^1$ distance between the normalized density $V_-^s/\int V_-^s$ induced by the negative part of the potential and the corresponding density induced by an optimal potential, where $s=\gamma+d/2$. As a geometric application of the same idea, we also derive a density-stability version of the $L_p$ mixed volume inequality. In the case where one of the two convex bodies is centrally symmetric and both bodies are trapped between two concentric Euclidean balls, this gives an averaged stability estimate for the non-evenness of the support function.

We consider the compressible time-dependent Navier-Stokes equations in a bounded perforated domain in dimensions two and three. Provided the perforations are small enough, we show that the limiting equations do not change their form when the perforation size goes to zero while their number increases to infinity. The novelty of this result is the form of the pressure: we consider a hard-sphere pressure law, giving an a priori bound for the density while, compared to the barotropic case, having worse regularity for the pressure, therefore causing significant problems in the homogenization procedure. To the best of our knowledge, the homogenization for this kind of pressures has not been addressed in the literature yet.

We prove that, on a bounded open convex domain $\Omega\subset\mathbb{R}^n$, the first Dirichlet eigenfunction of the Laplacian or the Ornstein–Uhlenbeck operator is $\alpha$-logconcave for every $\alpha\in(0,1/2]$. This extends the recent $1/2$-logconcavity theorem of Crasta–Fragalà for the Laplacian to the weighted Gaussian setting and, simultaneously, to a broader range of exponents. More precisely, if $u$ denotes the first eigenfunction normalized by $\|u\|_\infty=1$, then for every $\alpha\in(0,1/2]$, the function $-\bigl(-\log(\kappa u(x))\bigr)^{\alpha}$ is concave in $\Omega$ provided the scaling parameter $\kappa$ lies below an explicit threshold $\kappa_\alpha(\Omega)\in(0,1)$, which depends on the first Dirichlet eigenvalue and on the diameter of~$\Omega$. For the Ornstein–Uhlenbeck operator, $\kappa_\alpha(\Omega)$ also depends on the distance between $\Omega$ and the origin. Moreover, we establish a local counterpart: for every $\kappa\in(0,1)$, the function $\bigl(-\log(\kappa u)\bigr)^{\alpha}$ is convex on a convex neighborhood $\Omega_\kappa$ of the unique maximum point of~$u$. We also provide counterexamples showing that unscaled $1/2$-logconcavity may fail for the first Dirichlet eigenfunction of a Schrödinger operator with a smooth convex potential, and for the first Dirichlet eigenfunction of a weighted Laplacian associated with an affine log-concave weight.

This paper studies the global regularity problem for the two-dimensional incompressible Boussinesq equations with fractional dissipation given by $(-\Delta)^{\frac\alpha2}u$ and $(-\Delta)^{\frac\beta2} \theta$. Attention is focused on the subcritical regime where $\alpha+ \beta>1$. The case $\alpha >\frac23$ was recently settled in a joint work of the authors [Math. Ann., 391 (2025), 5965-6012], which established global regularity under this condition. This paper addresses the remaining case $\alpha \leq \frac23$. We obtain the sharpest regularity result by minimizing assumptions on $\alpha$ and $\beta$. We derive nonlinear lower bounds for the fractional Laplacian operator and implement an iterative procedure.

In this paper we prove that an $(n+1)$-manifold, compactly $n$-enlargeable, where $3\leq (n+1)\leq 7$, has connected, immersed Morse index one, closed minimal hypersurfaces with unbounded volumes for bumpy metrics. We prove that in the three-dimensional case the hypersurfaces are geometrically distinct using cyclic coverings of manifolds with boundary. The proof extends to $(n+1)$-fiberings. We prove a scalar curvature rigidity theorem for area-nonincreasing maps of three-dimensional manifolds. The case of stable surfaces is also discussed by using cohomology classes and incompressible surfaces.

We study selection problems for the vanishing discount limit of first-order stationary mean field games with local coupling. The associated ergodic problem may admit several value functions for the same density and ergodic constant. We decompose the state space into the positive-density region, the essential zero-density interior, and a residual set, and show that possible non-uniqueness of selected value functions can occur only on the gap part of the residual set. If the gradients of selected value functions coincide on this gap residual set, then the selected value function is unique up to additive constants; under compactness and stability assumptions, this yields convergence of the whole normalized discounted family. We show that the gap residual set is null for one-dimensional problems and for a specific Hamiltonian in the multidimensional setting, and hence obtain convergence results in these cases.

In this paper, we study the incompressible, viscous and resistive Hall-magnetohydrodynamic (Hall-MHD) system. We first prove that every two-dimensional Leray-Hopf weak solution becomes smooth after a finite time. In three dimensions, where eventual smoothness for arbitrary Leray-Hopf weak solutions is not known, we construct Leray-Hopf weak solutions for which the magneto-vorticity field $B+\nabla\times u$ eventually gains additional regularity. Finally, under suitable low-frequency pseudomeasure assumptions on initial data, we establish decoupled algebraic decay rates for the velocity and magnetic fields by combining a generalized Fourier splitting method with the eventual smoothness in two dimensions and strong regularity in three dimensions.

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

In this paper, we study the noncommutative fractional symmetric Sobolev spaces on noncommutative torus. We prove noncommutative distributional fractional Sobolev inequality and as its application, we obtain Sobolev embeddings. In order to obtain these results, we first prove a noncommutative version of the famous O'Neil inequality for the convolution. As a first application of our main results, we obtain a Cwikel-Solomyak-type estimate. As an another application, we show a $L_2$-time decay for the mild solution of the Cauchy problem for the diffusion equation in this noncommutative setting. When $\theta=0,$ our results recover many known results on Sobolev embedding on the torus.

The stochastic incompressible Navier-Stokes equations on $\TT^3$, completed by the fluctuation-dissipation noise, have a Fokker-Planck generator that decomposes into a self-adjoint Ornstein-Uhlenbeck (dissipative) part and an antisymmetric (convective) part. We prove two results about this generator. First, the logarithmic Sobolev inequality holds with the same optimal constant as the pure Ornstein-Uhlenbeck operator, $c_\mathrm{LSI} = \nu\lambda_1$ (where $\nu$ is the viscosity and $\lambda_1$ is the smallest nonzero eigenvalue of the Laplacian on $\TT^3$), independent of the number of retained Fourier modes. Second, the full semigroup is hypercontractive with the same rate as the Ornstein-Uhlenbeck semigroup. Both results follow from a single structural property: the convective generator is antisymmetric in $L^2(P_\mathrm{eq})$ (where $P_\mathrm{eq}$ is the Gibbs measure), and therefore contributes nothing to the Dirichlet form or the $L^q$ norm evolution. The antisymmetry is a consequence of two properties of the incompressible Navier-Stokes nonlinearity: energy conservation and phase-space volume preservation (the Liouville property). These are the same properties that underpin the fluctuation-dissipation theorem for the nonlinear Navier-Stokes equations.

We investigate the convergence properties of the family of operators $$ \mathcal{T}_{\bf N} f(x)=\sum_{j=N_1}^{N_2} v_j\Bigl( e^{-a_{j+1}(-\Delta_k)^s}f(x) - e^{-a_j(-\Delta_k)^s}f(x) \Bigr), \qquad x\in\mathbb{R}^d, $$ where $\{e^{-t(-\Delta_k)^s}\}_{t>0}$ denotes the fractional heat semigroup generated by the Dunkl Laplacian $\Delta_k$. Here ${\bf N}=(N_1,N_2)\in\mathbb{Z}^2$, $N_1<N_2$, the coefficients $\{v_j\}_{j\in\mathbb{Z}}$ form a bounded sequence of real numbers, and $\{a_j\}_{j\in\mathbb{Z}}$ is a monotone increasing sequence of nonnegative reals. The primary objective of this work is to establish boundedness results for these differential transform operators on weighted $L^p(\mathbb{R}^d,d\mu_k)$ spaces as well as on Dunkl $\mathrm{BMO}(\mathbb{R}^d)$ spaces. We also establish analogous boundedness properties for the associated maximal operator $$ \mathcal{T}^* f(x)=\sup_{\bf N} |\mathcal{T}_{\bf N} f(x)| $$ and study the pointwise convergence of the corresponding series. In addition, we prove that, for compactly supported functions, the maximal differential transform operator $\mathcal{T}^*$ exhibits local behaviour comparable to that of classical singular integral operators.

In this paper, we establish new connections between lattice graphs and metric grids, providing a unified framework for the study of singular limit problems and Gagliardo–Nirenberg type inequalities on lattice graphs. The main technical ingredients are restriction and extension estimates, which enable us to compare variational problems posed on lattice graphs, metric grids and \(\mathbb R^d\). As applications, we first prove that extensions of action ($2<p<2^*$) and energy ($2<p<2+\frac{4}{d}$) ground states of the nonlinear Schrödinger (NLS) equation on $d$-dimensional lattice graphs converge strongly in $H^1(\R^d)$ to the corresponding ground states on $\R^d$ as the edge length tends to zero. As a by-product of the arguments developed for the singular limit problem on lattice graphs, we obtain multiplicity results for fixed-mass critical points of the energy functional on lattice graphs. Furthermore, employing a strategy analogous to that used in the singular limit analysis, we investigate the optimal constants of Gagliardo-Nirenberg type inequalities on lattice graphs for $2<p<2^*$. Beyond the classical subcritical framework, we also study the singular limit of action ground states in the Sobolev supercritical regime ($d \geq 3$ and $p>2^*$), the singular limit of energy ground states in the mass-supercritical regime ($p>2+\frac{4}{d}$) on lattice graphs, and the optimal constants in Gagliardo-Nirenberg type inequalities in the Sobolev critical case $d \geq 3$ and $p=2^*$ on lattice graphs. Notably, we settle an open problem posed by Dovetta [Adv. Math. 444 (2024), 109633] by establishing a new Gagliardo-Nirenberg type inequality.

In this paper, we study the initial-boundary value problem for the nonlinear Schrödinger equation in $\mathbb{R}^{n}_{+}$ \begin{equation*} i\partial_{t}u+\Delta u+\lambda |u|^pu=0, \qquad (x, t) \in \mathbb{R}_{+}^{n} \times \mathbb{R}_{+},\ \ p\in\mathbb{R}_{+} \end{equation*} with nonhomogeneous Dirichlet boundary conditions. For the corresponding linear problem, endpoint Strichartz estimates are derived. For the nonlinear problem, we prove local well-posedness in $H^{s}(\mathbb{R}^{n}_{+})$ with $s\in[0,\frac{5}{2})$ and $p<\frac{4}{n-2s}$. Moreover, global well-posedness is established in the same regularity range. For $s\in[1,\frac{5}{2})$, the one-dimensional global theory of \cite{figment} in $H^{s}(\mathbb{R}_{+})$ is extended to $H^{s}(\mathbb{R}^{n}_{+})$. Additionally, we obtain global solutions in the lower regularity setting $s\in[0,1)$ for the first time. It is noteworthy that for $s=0$, we overcome the lack of mass conservation resulting from the nonzero boundary data and derive the pivotal $L^{2}(\mathbb{R}^{n}_{+})$ a priori estimates.

We obtain the leading term in the solution of the Cauchy problem for the Benjamin-Ono equation in the limit $t\to+\infty$ with $x=O(t^{1/2})$. We show that the rate of decay exceeds that of self-similar solutions and obtain an explicit universal profile for the decaying solution, relating it to the linearization of the profile equation for self-similar solutions. The proof assumes a class of rational initial data $u_0$ in $L^2(\mathbb{R})\cap L^1(\mathbb{R})$ that exhibit generic behavior of the reflection coefficient at the origin.

In this work, we introduce a new macroscopic model for crowd motion inspired by the celebrated Hughes' model \cite{Hughes2002, Hughes2003}, which couples a nonlinear conservation law for the pedestrian density with an Eikonal equation describing the shortest path to the target. Our approach can be viewed both as a modification of Hughes' original formulation and as a refinement of the prediction-correction framework proposed in the recent work \cite{ennaji2023prediction}. The resulting model incorporates anticipatory behavior and dynamic route adjustment, offering a more realistic representation of crowd dynamics in complex environments. We present the mathematical formulation of the model, discuss its well-posedness properties, and illustrate its qualitative behavior through numerical simulations. Ultimately, we show, at least from a numerical perspective, that this variant provides a promising avenue towards establishing the well-posedness of the classical Hughes' model, which has remained a challenging open problem for a long time.