Papers

Let $H\in C^2(T^*M)$ be a Tonelli Hamiltonian on a closed connected manifold and let $u_\lambda$ solve \[ \lambda u_\lambda+H(x,Du_\lambda)=c(H)\qquad\text{in }M. \] We study the convergence rate of $u_\lambda$ to the selected critical solution $u_0$. Assume that the lifted Aubry set is a finite union $\widetilde{A}=\Gamma_1\sqcup\cdots\sqcup\Gamma_N$, where each $\Gamma_i$ is either a hyperbolic equilibrium or a periodic orbit hyperbolic in the critical energy level. We prove \[ -C\lambda\le u_\lambda-u_0\le C\lambda|\log\lambda|. \] Let $\mu_i$ be the projected Mather measure associated with $\Gamma_i$. If \[ \int_M u_0\,\mathrm d\mu_i=0\qquad \text{for every }i, \] then \[ \|u_\lambda-u_0\|_\infty\le C\lambda. \] In particular, if the lifted Aubry set consists of a single hyperbolic equilibrium or a single hyperbolic periodic orbit, the convergence rate is $O(\lambda)$. We give examples showing that both convergence rates $O(\lambda)$ and $O(\lambda|\log\lambda|)$ are optimal. Without hyperbolicity, finite-order degenerate examples give lower bounds of order $\lambda^{1/(2r-1)}$ with $r\ge2$. We also construct infinite-order degenerate examples with arbitrarily slow convergence. Taken together, these results provide, to our knowledge, the first systematic quantitative theory for the vanishing discount problem in the Tonelli setting.

We construct divergence-free, vector-valued approximation functions on the whole space $\mathbb{R}^d$, $d\geq 2$, as well as on general unbounded domains of uniform $\mathrm{C}^{1,1}$-type. These approximations converge simultaneously in both Sobolev and Lebesgue spaces. In the whole-space setting, we employ Bogovskiĭ\ operators to construct such approximations, thereby extending the approximation theory developed for smooth bounded domains and the simultaneous approximation framework introduced by \emph{Fefferman, Hajduk, and Robinson, {Proc. Lond. Math. Soc.} (3) \ {125} (2022), no.~4, 759-777}. As an application, we establish energy equality for Leray-Hopf weak solutions of the incompressible convective Brinkman-Forchheimer (CBF) equations on $\mathbb{R}^d$, $d\in\{2,3\}$ covering both the critical and supercritical regimes. For general unbounded domains, we employ the resolvent operator associated with the Stokes operator, developed by \emph{Farwig, Kozono and Sohr, {Acta Math.}, {195} (2005), 21-53}, to obtain simultaneous approximation results. We also establish a generalized version of the classical Lions-Magenes lemma, which is of independent interest. Finally, by combining this result with the simultaneous approximation framework for unbounded domains, we establish energy equality for weak solutions of the CBF equations on general unbounded domains.

One of the most common descriptions of solutions of singular SPDEs is their characterisation as the limit of solutions of renormalised smooth random PDEs. We quantify the speed of this convergence in the case of the KPZ equation. In particular, we show that the naïve guess that the rate is given by the distance of the noise regularity from the endpoint regularity for well-posedness is not correct. Instead, we obtain convergence at the larger rate $1/2$ and show that this rate is sharp. This is achieved by considering the equation satisfied by the rescaled error, which is critical for variance blowup, and showing that this equation has a limit given by an affine linear singular SPDE driven by a new, independent noise. This result can be alternatively interpreted as identifying the asymptotic size and law of fluctuations of the solutions of the KPZ equation driven by mollified noise around their singular limit.

We study fixed-temperature logarithmic and Riesz gases after subtracting the leading mean-field contribution relative to a prescribed background law. For repulsive interactions in the Hilbert–Schmidt regime, we prove $N$-uniform bounds and quantitative convergence of the resulting modulated partition function to a normalization expressed by the Carleman–Fredholm determinant of the centered interaction operator. We show that the Hilbert–Schmidt threshold is sharp and obtain explicit lower bounds on the rate of divergence at and above it; these rates are expected to be nonoptimal. The proof combines positive-definite truncations and a low/high-frequency decomposition with exponential inequalities and Gaussian-chaos asymptotics for canonical degree-two $U$-statistics. As consequences, we establish entropic commutator estimates with the sharp $O(N^{-1})$ additive scale in the modulated-free-energy method, a static joint linear-statistics central limit theorem, and a dynamical central limit theorem for joint linear statistics at finitely many times. This extends the logarithmic partition-function estimates of the first two authors to the full Riesz Hilbert–Schmidt range and identifies the limiting determinant normalization. For the attractive logarithmic interaction at sufficiently small inverse temperature, we also prove analogous results.

We consider on a Riemannian manifold $M$ the Leibenson equation \begin{equation*}\label{eqabs}\partial _{t}u=\Delta _{p}u^{q},\end{equation*} where $p>1$ and $q>0$. When $q(p-1)\geq 1$, we prove conservation of mass for solutions of Leibenson's equation assuming only the volume bound $V(x_0, r)\leq \exp\left(C r^{\frac{p}{p-1}}\right)$ for some $x_{0}\in M$ and all large enough $r>0$. When $q(p-1)< 1$, we prove this property assuming $V(x_0, r)\leq Cr^{N}$ and $p>N[1-q(p-1)]$, which matches the threshold in $\mathbb{R}^{n}$ with $N=n$. We also show that solutions on the hyperbolic space $\mathbb{H}^{n}$ have a finite extinction time in the case $q(p-1)< 1$, which implies that the conservation of mass property does not hold. Using the conservation of mass result in the case $q(p-1)=1$, we also prove a $L^{p-1}$- Liouville property, which partially answers a conjecture stated by I. Holopainen \cite{holopainen2000sharp}.

We approximate strongly continuous convex monotone semigroups by learning their Chernoff-type one-step operators with neural operators. First, we introduce the general class of so-called Chernoff-neural operators and show in a universal approximation theorem that they can approximate the Chernoff one-step operators arbitrarily well. By using stability estimates between weighted Hölder spaces, the one-step approximation error can be propagated through the iterations which yields universal approximation of the corresponding semigroup. Second, we introduce the more specialized class of envelope-neural operators for envelope semigroups which allows us to derive quantitative approximation rates. Finally, we illustrate the effectiveness of these neural operators in several numerical examples arising from non-linear partial differential equations, stochastic optimal control and stochastic processes under model uncertainty.

We study insensitizing control problems for a class of coupled linear stochastic parabolic systems. We establish the existence of controls such that a sentinel functional, involving localized observations of the state variables and their spatial gradients, is insensitive to small perturbations of the null initial data. We first reformulate the insensitizing control problem as a null controllability problem for a coupled forward–backward stochastic parabolic system, in which the observation terms induce both zeroth- and second-order coupling terms. By duality, the analysis is reduced to an observability inequality for the corresponding adjoint system. The main analytical contribution is the derivation of new global Carleman estimates for coupled stochastic parabolic systems with zeroth- and second-order coupling terms, under suitable geometric assumptions on the control and observation regions. These estimates yield the required observability inequalities and, consequently, the existence of insensitizing controls. Furthermore, depending on the value of a weighting parameter $\beta\in[0,1]$, which determines the relative contributions of the two state components to the sentinel functional, we consider two cases. If $\beta\in\{0,1\}$, the sentinel functional depends on only one state component, and a single localized control acting in the drift of the first equation is sufficient. In contrast, if $\beta\in(0,1)$, both state components contribute to the sentinel functional, and two localized controls acting in the drift terms of the two equations are sufficient. Moreover, the control strategy in this paper involves two additional controls acting throughout the diffusion terms.

This paper establishes a new comparison principle framework for the subquadratic fractional $p$-Laplacian, i.e.~$1 < p < 2$ under minimal regularity assumptions, that has remained a significant challenging issue due to the singularity of the operator. Our results provide the essential analytical tools required for the moving plane method in this setting. We prove first a weak comparison principle for $(-\Delta)_p^s u = f(u)$ in bounded domains with sufficiently small measure, where only the boundedness of the weak solution is required. More importantly, we establish a strong comparison principle for continuous weak solutions in the parameter range $s \in (0, \frac{1}{2})$ and $\frac{1}{1-s} < p < 2$. Our proof introduces a localized barrier function and does not require any Hölder regularity of the weak solution, nor any smoothness of the domain. This presents a substantial contrast over previous study, which relied heavily on Hölder or even $C^{1,1}$ regularity. As a direct application, we employ these comparison principles to prove the symmetry of weak solutions to $(-\Delta)_p^s u = f(u)$ under mild assumptions, which significantly extend existing symmetry theories for nonlocal quasilinear equations.

In this work, we study nonnegative viscosity solutions to Hénon-type equations driven by the infinity-Laplacian with a degenerate weight, strong absorption and an additional source term. We focus on free boundary points lying in the degeneracy set of the weight. We prove the degeneracy set comparison principle, which yields uniqueness within each slice determined by the value on the degeneracy set of the weight, although global uniqueness is not available in general. We establish sharp improved regularity estimates near free boundary points, with the intrinsic growth rate $r^{\frac{4+\alpha}{3-m}}$. Finally, we obtain a matching non-degeneracy estimate at free boundary points; the property holds in the inhomogeneous case with suitable condition. Consequently, solutions detach from their zero phase at this rate.

This paper investigates the global dynamics of a doubly degenerate nutrient-taxis system with indirect consumption: \begin{equation*} \left\{ \begin{aligned} &u_{t}=\nabla \cdot (uv\nabla u)-\nabla \cdot (u^{2}v\nabla v)+\ell vw,&x\in \Omega,\, t>0,\\ & v_{t}=\Delta v-vw,&x\in \Omega,\, t>0,\\ &w_t=\Delta w-w+u,&x\in\Omega,t>0 \end{aligned} \right. \end{equation*} posed on a smooth bounded domain $\Omega\subset\mathbb{R}^{3}$ with no-flux boundary conditions. It is shown that for suitably regular initial data $(u_0,v_0,w_0)$, the associated initial-boundary value problem admits a global weak solution. Furthermore, in an appropriate topological setting, this solution converges to an equilibrium $(u_\infty, 0,w_\infty)$ as $t\rightarrow \infty$. Notably, when $u_0$ is nonconstant and the mass of $v_0$ is sufficiently small, the limiting profiles $u_{\infty}$ and $w_{\infty}$ are are spatially nonhomogeneous, capturing emergent patterning in nutrient-depleted environments. A cornerstone of our analysis is the introduction of novel functional inequalities, which provide estimates from below for the integral $\int_{\Omega}u^{k}v|\nabla u|^2$ with some $k>-1$.

We resolve the long-standing problem of establishing interior \(C^2\) estimates for admissible solutions of the graphical scalar curvature equation in every dimension \(n\ge 3\). More precisely, we prove interior curvature estimates for admissible solutions to the constant graphical scalar curvature equation. The proof combines Jacobi inequalities with a two-surface maximum principle and a two-surface Pogorelov estimate.

By refining the approach developed in \cite{AGH-2015, GS2026a}, we establish resolvent estimates in $L^p_\sigma (\Omega)$ for the Stokes operator in a bounded Lipschitz domain $\Omega$ in $\R^3$ for any $(3/2)-\e< p\le \infty$, where $\e>0$ depends on $\Omega$. As a consequence, the Stokes operator generates a uniformly bounded analytic semigroup in $L^p_\sigma (\Omega)$. The results are particularly surprising, as it is long believed that the $L^p$ resolvent estimates in three-dimensional Lipschitz domains may fail for large $p$. In the Appendices we prove a theorem on interpolation between $L^2_\sigma(\Omega)$ and $L^\infty_\sigma(\Omega)$. Given $1< p< \infty$ and a bounded Lipschitz domain $\Omega$ in $\R^d, d\ge 2$, we show that the resolvent estimate cannot hold in $L^p(\Omega; \C^d)$ unless the Helmholtz projection is bounded on $L^p(\Omega; \C^d)$. This explains the counter-example constructed in \cite{Deuring-2001}.

We study the Hunter–Saxton equation on the real line and its global dissipative solution in the energy space $ L^\infty(\mathbb R)\cap \dot H^1(\mathbb R). $ We prove strong ill-posedness through instantaneous failure of Sobolev regularity at and below the Lipschitz threshold. More precisely, for every $s\in(1,\nicefrac32]$, we construct $ u_0\in L^\infty(\mathbb R)\cap\dot H^1(\mathbb R)\cap\dot H^s(\mathbb R) $ whose unique global dissipative solution satisfies $u\notin C([0,T]; \dot H^s (\mathbb R))$, for every $T>0$. The constructions differ substantially in the subcritical and critical regimes. For $1<s<\nicefrac32$, we superpose rescaled, localized bubbles with increasingly negative slopes; subcritical scaling preserves $\dot H^s$-summability, while the explicit characteristic formula produces norm inflation. At $s=\nicefrac32$, where scaling yields no smallness, we use logarithmically distributed compactly supported multiscale profiles whose breaking times converge to zero. A one-sided localization principle and an almost-orthogonality estimate then transfer the inflation of individual profiles to the full dissipative solution.

In this paper, we study the torsion problem on bounded, smooth, strictly horo-convex domains in hyperbolic space. We prove that if the diameter of the domain is sufficiently small, then $-\sqrt{u}$ is strictly convex, where $u$ denotes the torsion function. The main ingredients of our proof are the constant rank theorem, a boundary convexity estimate, and a deformation argument for the domain. We also show that the exponent $1/2$ is optimal, even among strictly horo-convex domains of arbitrarily small diameter.

Let $(M^n,g)$ be complete, let $u>0$, and assume $\operatorname{Ric}_g-\alpha\frac{\Delta_g u}{u}g\ge (n-1)\kappa g$. We introduce mixed radial balls associated with the conformal metric $u^{2\alpha}g$. For these mixed radial balls, we obtain a space-form-sharp model comparison and polynomial weighted volume growth without pointwise bounds on $u$. As applications, we give a radial derivation of the spectral Bonnet–Myers and sharp volume theorem of Antonelli–Xu, and a short volume growth derivation of the stable Bernstein theorem in $\mathbb{R}^4$.

We study the Cauchy problem for the Benjamin–Ono–Zakharov–Kuznetsov equation on $\mathbb R^2$. Following the strategy of Kenig and Ziesler, we establish new maximal-function estimates adapted to the BO–ZK equation and use them to implement the Kenig–Koenig method. As a result, we improve the best previously known isotropic result of Nascimento (2020), lowering the local well-posedness threshold from $s>5/4$ to $s>19/16$. At the BO–ZK endpoint, the resulting isotropic data class also contains the anisotropic $E^{5/4+}$ class of the preceding theory. On bounded subsets of $H^s(\mathbb R^2)$, the lifespan may be chosen so that $ T\gtrsim_s \bigl(1+\norm{u_0}_{H^s}\bigr)^{-8}. $ The proof combines a sharp dyadic mixed maximal-function estimate with an anisotropic local-smoothing mechanism that exploits the complementary behavior of the longitudinal and transverse group velocities. In particular, transverse dispersion compensates for the degeneration of longitudinal smoothing near the characteristic region. Together with refined short-time Strichartz estimates and a modified energy argument, these ingredients close the nonlinear estimates at the stated regularity. Existence, uniqueness, and continuous dependence on the initial data are then established in the corresponding solution class. The resulting threshold reflects the present optimization of the method and is not claimed to be sharp.

We consider generalized principal eigenvalues of one-dimensional nonlocal dispersal operators with a drift of fixed sign. A principal difficulty in this non-self-adjoint setting is the absence of a variational characterization of the principal value. In the symmetric drift-free problem, the critical nonlocal-to-local limit can be treated through a quadratic variational structure and Sobolev-seminorm approximation as Berestycki-Coville-Vo \cite{BCV}. The drift destroys this structure and, on a bounded interval, introduces at the same time a one-sided inflow–outflow boundary geometry. Replacing the missing variational argument by estimates which remain stable under singular rescaling is therefore a central technical issue. Our approach is direct : we work with a single generalized principal value and prove the maximum principle, simplicity in the positive cone and the scaling limits from the equation itself, without passing through auxiliary generalized principal eigenvalues analogous to $\lambda_1'$ and $\lambda_1''$ in Berestycki–Rossi \cite{BR}. The replacement mechanisms are directional Harnack inequalities and coefficient barriers, direct–adjoint identities, logarithmic Collatz–Wielandt transforms, Fourier coercivity of zero extensions and scale-dependent localization. At the critical diffusive scale, Fourier compactness and an exact energy–transport identity recover the missing inflow Dirichlet condition and identify the local Dirichlet limit without a Rayleigh quotient. For variable coefficients we determine the complete small-range phase diagram on bounded intervals and on the line, and we obtain sharp large-range three-term asymptotics by a rank-one reduction and a corner Laplace analysis in the advective travel-time variable. In the homogeneous whole-line problem the critical correction is of order $\sigma^2$.

We investigate kinetic Fokker–Planck equations with rough divergence-free drifts in anisotropic mixed Lebesgue spaces. The divergence-free structure yields an energy cancellation that remains effective under kinetic localization. Combining this cancellation with anisotropic regularization estimates and a De Giorgi iteration, we establish local boundedness for weak subsolutions and, consequently, global well-posedness for the Cauchy problem in a scaling-supercritical regime.

We study a parabolic obstacle partial integro-differential equation (PIDE) with a dynamically moving bilateral free boundary. This type of problem arises in the mathematical modeling of speculative asset bubbles with Lévy jump processes. We investigate the existence of viscosity supersolution barriers within the class of functions exhibiting linear asymptotic growth ($O(|g|)$ at infinity) across three distinct parametric regimes. Our intention is to determine when such a barrier can be constructed by analyzing the balance between the stabilizing local drift, defined by the discount rate $r$ and mean-reversion $\rho$, and the non-local jump dispersion, characterized by the large-jump intensity $\lambda$ and Lipschitz constant $L_\gamma$. First, when $r+\rho > \sqrt{\lambda}L_\gamma$, we prove the global existence of non-negative viscosity supersolutions. Second, in the deficit regime ($r+\rho < \sqrt{\lambda}L_\gamma$), we construct non-negative supersolutions for every finite horizon $T>0$. However, by utilizing an asymptotic slope envelope, we prove that these barriers cannot be bounded by a fixed, pre-determined linear growth ceiling $C_{\max}$ across arbitrarily large horizons; rather, the required linear growth constant must inflate exponentially as the horizon length increases. Finally, at the exact critical boundary ($r+\rho = \sqrt{\lambda}L_\gamma$), we show the existence of a supersolution with a uniform spatial growth bound, provided an additional spatial no-crossing condition holds on the negative tail.

In this work we prove a Payne-Weinberger type inequality for quantum dot Dirac operators defined on bounded and simply connected planar domains. This is a sharp upper bound for their first positive eigenvalue depending only on the isoperimetric deficit of the domain. To this end, we use a recently studied connection with the so-called $\overline\partial$-Robin Laplacian and we establish an analogous inequality for its first eigenvalue, relying on the corresponding inequality for the Robin Laplacian.

We extend the global existence results for critical parameters of [Bianchini and Leccese, J. Math. Anal. Appl. (2024)] to the multidimensional problem \begin{equation*} \partial_t u + \text{div } (b(t,x) u^{1+k}) = \Delta u \end{equation*} for $b:\mathbb R\times{\mathbb R}^d\to{\mathbb R}^d$ non-autonomous fields, in the case \begin{equation*} (\text{div } b)_-\in L^\infty_{\mathrm{loc}} \big([0,\infty);L^{p,\infty}({\mathbb R}^d)\big), \qquad p>d. \end{equation*} The proof of global existence follows the parabolic symmetrization approach of [Bandle, J. Analyse Math. (1976)] and is based on the study of the function \begin{equation*} m(t,s)=\int_0^s u^*(t,r)\,dr. \end{equation*}

This article is concerned with generalizations of pseudo-differential operators in $L^2(\mathbb{R}^n)$, $n\geq 1$. The definition of the new calculi depends only on bounded measures on the phase space $\mathbb{R}^{2n}$ and each measure gives rise to a specific calculus. Quantizations of anti-Wick, Weyl, classical and Born-Jordan are particular cases of the general calculi. Classes of symbols in this framework are then studied. The Gevrey class of parameter 1/2 is a class of symbol that is common to all the general calculi, that is, a class of symbols independent on the bounded measures parametrizing the quantizations. Precise additional hypotheses on the measures are necessary in the aim to consider the larger class of symbols $L^{\infty}(\mathbb{R}^{2n})$. This result can be applied for anti-Wick but not for Weyl quantization. Concerning Weyl pseudo-differential calculus, we recover the standard class of Sjöstrand and Gröchening. Then, we prove that probability measures of Lévy processes on the phase space $\mathbb{R}^{2n}$ with diffusion larger than 1/4 are natural examples of measures satisfying the latter additional hypotheses in order to consider $L^{\infty}(\mathbb{R}^{2n})$ symbols. This relation is derived using the Lévy-Khintchine formula. Composition laws in that general context are next investigated. In that purpose, we give a formula for the composition of two symbols in some precise class of symbols valid for all general quantizations. General calculi are relying on Wick quantization which is therefore primarily examined for some precise classes of symbols. Additional results in that context are provided, such as Mizrahi series expansions and Banach algebra isomorphisms between operators and symbol classes.

In this paper, we develop the method of moving planes entirely in the weak formulation for the fractional $p$-Laplacian in the singular range $1<p\leq2$. We first establish a small region principle for antisymmetric functions and apply it to prove radial symmetry and monotonicity of nonnegative weak solutions of fractional $p$-Laplacian equations in a bounded domain. We also consider the nonlocal quasilinear Lane–Emden equation $(-\Delta)^s_pu=u^q$ in $\mathbb R^n$. In the Sobolev critical case, we establish radial symmetry, monotonicity, and precise asymptotic behavior at infinity for finite-energy weak solutions. Under a suitable decay condition, we also obtain radial symmetry for the full range $q>p-1$. Our results complete those of Chen-Li (Adv. Math., 2018: 735-758), where analogous results were obtained for $C^{1,1}$ solutions in the pointwise sense.

We prove the existence and uniqueness of a local-in-time solution to the stochastic Cahn-Hilliard equation in space dimensions 1, 2, 3, 4 and 5. Our method relies on the Da Prato-Debussche trick for $d=4$ and paracontrolled distributions for $d=5.$

Distribution dependence generally prevents McKean-Vlasov state solution maps from satisfying a cocycle identity, while the relevant Sobolev embedding is noncompact on unbounded domains. For a class of stochastic reaction-diffusion equations with a dissipative polynomial reaction, a one-sided monotone state-law coupling and finite-dimensional additive noise, we combine the deterministic law semiflow with the pathwise state evolution to construct a continuous random dynamical system on the product of the state space and its quadratic Wasserstein law space. An abstract product-space criterion, mixed-energy estimates, local regularity and uniform far-field estimates yield a unique pullback random attractor without global law contraction. Its law projection is the global attractor of the law semiflow, and its state fibers need not be singletons. Under an additional strict contraction condition, the law attractor reduces to the unique invariant law and the corresponding state fiber to a self-consistent random equilibrium. We also establish zero-noise upper semicontinuity and, in the contractive regime, quantitative convergence of the invariant laws and random equilibria with bounds linear in the noise amplitude.

We investigate inverse scattering at a fixed energy for semiclassical Schrödinger operators with smooth potentials. For compactly supported potentials satisfying a natural virial condition, we prove that $ \liminf_{h\to0} \|S_1(\lambda,h)-S_2(\lambda,h)\|<\sqrt{2} $ implies $V_1=V_2$, provided the fixed energy $\lambda$ lies above both potentials. We then consider radial short-range repulsive potentials. Under a monotonicity assumption on the radial force, we show that the semiclassical differential cross section at a single fixed energy, up to $o(1)$ as $h\to0$, determines the potential throughout the classically accessible region. Finally, we obtain an analogous rigidity result for nontrapping compactly supported perturbations of the Euclidean metric under a strict convexity assumption.

In this paper, we study normalized solutions to the following $L^2$-supercritical nonlinear Schrödinger equation with a periodic potential \begin{equation*} \begin{dcases} -\Delta u +V (x)u + \lambda u=\chi_{{\Omega}}(x)f(u)\quad \text{in } \mathbb{R}^N, u>0 \quad \text{in} \ \mathbb{R}^N, \int_{\mathbb{R}^N}\abs{u}^2\, dx =\mu. \end{dcases} \end{equation*} Here $N \geq 1$, $\mu>0$ is prescribed, $\lambda \in \mathbb{R}$ is a Lagrange multiplier, $V\in C(\mathbb{R}^N)$ is $1$-periodic in $x_1,...,x_N$, $f \in C^1(\mathbb{R})$ is a nonlinearity having $L^2$-supercritical growth at infinity, $\Omega\subset \mathbb{R}^N$ is either a (nonempty) bounded open set with smooth boundary $\partial \Omega$ or the whole space \(\mathbb{R}^N\), and $\chi_{{\Omega}}$ is the characteristic function of ${\Omega}$. In both cases, we prove the existence of normalized solutions of mountain pass type when $\mu>0$ is small and $f$ behaves like a power function $|t|^{p-2}t$ with $2+4/N<p<2^*$ at infinity. Moreover, if $\Omega$ is bounded, $1 \leq N \leq 4$ and $f$ has $L^2$-supercritical growth near the origin, then the existence of normalized solutions is obtained for all $\mu>0$. On the other hand, when $\Omega=\mathbb{R}^N$, we prove the existence of solutions corresponding to local minimizers when $\mu>0$ is small. When $\Omega$ is bounded, the results are obtained via a monotonicity trick for mountain pass values and blow-up analysis with the Morse index estimates. In the case where $\Omega = \mathbb{R}^N$, we develop the concentration-compactness argument based on the Morse index for the existence of mountain pass type solutions.

This paper is concerned with the time-asymptotic stability of the generic Riemann solution for the one-dimensional system of heat-conductive ideal gas without viscosity, where the generic Riemann solution consists of a shock, a contact discontinuity, and a rarefaction wave. We prove that, as time tends to infinity, the solution of the non-viscous and heat-conductive ideal gas system converges uniformly to a composite wave composed of rarefaction wave, viscous contact wave, and viscous shock wave with a time-dependent shift. Motivated by the recent work of Kang-Vasseur-Wang [Arch. Ration. Mech. Anal. 249: 42 (2025)], we overcome the difficulties arising from the concurrence of shock and rarefaction waves for the partially dissipative hyperbolic-parabolic system with dissipation acting only on a single variable. More notably, the absence of velocity dissipation gives rise to new and intrinsic difficulties when handling the terms associated with the density and velocity. To resolve this, we exploit the precise structure of the governing equations and the additional properties of shock waves. Furthermore, we utilize the wave structure of the system without viscosity and perform separate space-time estimates for the density and velocity.

We prove existence of closed non-planar $p$-elasticae for general exponents $p\in (1,\infty)$. In particular, we show that for any $p \in (1,\infty)$, there exists a countable family of non-planar $p$-elasticae, which are realized as torus knots. This generalizes well-known results of Langer–Singer for the quadratic case $p=2$ to general exponents $p\in (1,\infty)$.

The Fokker-Planck equation admits both a macroscopic Eulerian description through probability densities and a microscopic Lagrangian description through stochastic trajectories. Consequently, optimization problems constrained by the Fokker-Planck equation can be formulated from either perspective. Surprisingly, the corresponding adjoint equations appear to be fundamentally different: the macroscopic adjoint is governed by the backward Kolmogorov equation, whereas the microscopic adjoint evolves pathwise along stochastic trajectories. In this note, we reconcile these two formulations by establishing their correspondence in the continuum setting. We further show that, although their discrete gradients no longer coincide after discretization, both provide consistent numerical approximations of the continuum gradient. Explicit convergence rates are established for both discretization strategies.