Let $\Omega\subset\mathbb{R}^N$, $N\geq 1$, be an open bounded connected set. We consider the indefinite weighted eigenvalue problem $-\Delta u =\lambda m u$ in Ω with $\lambda \in \mathbb{R}$, $m\in L^\infty(\Omega)$ and with homogeneous Neumann boundary conditions. We study weak* continuity, convexity and Gâteaux differentiability of the map $m\mapsto1/\lambda_1(m)$, where $\lambda_1(m)$ is the principal eigenvalue. Then, denoting by $\mathcal{G}(m_0)$ the class of rearrangements of a fixed weight m0, under the assumptions that m0 is positive on a set of positive Lebesgue measure and $\int_\Omega m_0\,dx \lt 0$, we prove the existence and a characterization of minimizers of $\lambda_1(m)$ and the non-existence of maximizers. Finally, we show that, if Ω is a cylinder, then every minimizer is monotone with respect to the direction of the generatrix. In the context of the population dynamics, this kind of problems arise from the question of determining the optimal spatial location of favourable and unfavourable habitats for a population to survive.

In this paper, we investigate a class of McKean–Vlasov stochastic differential equations (SDEs) with Lévy-type perturbations. We first establish the existence and uniqueness theorem for the solutions of the McKean–Vlasov SDEs by utilizing an Eulerlike approximation. Then, under suitable conditions, we demonstrate that the solutions of the McKean–Vlasov SDEs can be approximated by the solutions of the associated averaged McKean–Vlasov SDEs in the sense of mean square convergence. In contrast to existing work, a novel feature of this study is the use of a much weaker condition, locally Lipschitz continuity in the state variables, allowing for possibly superlinearly growing drift, while maintaining linearly growing diffusion and jump coefficients. Therefore, our results apply to a broader class of McKean–Vlasov SDEs.

It was proved in [11, J. Funct. Anal., 2020] that the Cauchy problem for some Oldroyd-B model is well-posed in $\dot{B}^{d/p-1}_{p,1}(\mathbb{R}^d) \times \dot{B}^{d/p}_{p,1}(\mathbb{R}^d)$ with $1\leq p \lt 2d$. In this paper, we prove that the Cauchy problem for the same Oldroyd-B model is ill-posed in $\dot{B}^{d/p-1}_{p,r}(\mathbb{R}^d) \times \dot{B}^{d/p}_{p,r}(\mathbb{R}^d)$ with $1\leq p\leq \infty$ and $1 \lt r\leq\infty$ due to the lack of continuous dependence of the solution.

In this article, we introduce a new logarithmic Q-type space $Q_{\ln ,\lambda }^{p,l,k}(\mathbb R^{n})$ to study the well-posedness of the classical/fractional Naiver–Stokes equations. We show that $\nabla \cdot (Q_{\ln ,\lambda }^{p,l,k}(\mathbb R^{n}))^{n}$ covers the well-known critical spaces $BMO^{-1}(\mathbb R^{n}), Q_{\alpha }^{-1}(\mathbb R^{n})$ and $\mathcal {Q}_{0}^{-1}(\mathbb R^{n})$ for the classical Naiver–Stokes equations. Moreover, it covers the fractional counterparts $BMO^{-(2\beta -1)}(\mathbb R^{n}), Q_{\alpha }^{\beta ,-1}(\mathbb R^{n})$ and even the largest critical space $\dot {B}^{-(2\beta -1)}_{\infty ,\infty }(\mathbb R^{n}).$ In doing so, we first establish some basic properties of $Q_{\ln ,\lambda }^{p,l,k}(\mathbb {R}^{n}).$ Then, via the fractional heat semigroups, we characterize the extension of $Q_{\ln ,\lambda }^{p,l,k}(\mathbb R^{n})$ to $\mathscr H_{K_{\ln }^{(l,k)}}^{p,\lambda }(\mathbb R_+^{n+1})$ which is a function space related to the weight function $K_{\ln }^{(l,k)}(\cdot )$. This extension provides a semigroup characterization of $Q_{\ln ,\lambda }^{p,l,k}(\mathbb R^{n})$. With this in hand, we establish the well-posedness of mild solutions to fractional Naiver–Stokes equations and fractional magneto-hydrodynamic equations, respectively, with small data in $\nabla \cdot \left (Q_{\ln ,\frac {4(1-\beta )}{n}}^{2,k,l+2(1-\beta )}(\mathbb {R}^{n})\right )^{n}$ for $k\in \mathbb {N}$ and $l>n+2\beta -4.$

This overview discusses the inverse scattering theory for the Kadomtsev–Petviashvili II equation, focusing on the inverse problem for perturbed multi-line solitons. Despite the introduction of new techniques to handle singularities, the theory remains consistent across various backgrounds, including the vacuum, 1-line and multi-line solitons.

Numerous evolution equations with nonlocal convolution-type interactions have been proposed. In some cases, a convolution was imposed as the velocity in the advection term. Motivated by analysing these equations, we approximate advective nonlocal interactions as local ones, thereby converting the effect of nonlocality. In this study, we investigate whether the solution to the nonlocal Fokker–Planck equation can be approximated using the Keller–Segel system. By singular limit analysis, we show that this approximation is feasible for the Fokker–Planck equation with any potential and that the convergence rate is specified. Moreover, we provide an explicit formula for determining the coefficient of the Lagrange interpolation polynomial with Chebyshev nodes. Using this formula, the Keller–Segel system parameters for the approximation are explicitly specified by the shape of the potential in the Fokker–Planck equation. Consequently, we demonstrate the relationship between advective nonlocal interactions and a local dynamical system.

In this paper, we deal with the following nonlinear time-harmonic Maxwell’s equations

\begin{equation*}\nonumber\begin{cases} \nabla \times(\nabla \times u_1)=\mu_1|u_1|^4u_1+\beta|u_1||u_2|^3u_1 & \text {in } \mathbb{R}^3, \\ \nabla \times(\nabla \times u_2)=\mu_2|u_2|^4u_2+\beta|u_1|^3|u_2|u_2 & \text {in } \mathbb{R}^3, \end{cases}\end{equation*}

$\nabla\times$

$\mathbb{R}^3$

$\mu_1,\mu_2 \gt 0$

$\beta\in\mathbb{R}\backslash\{0\}$

$\beta\in\mathbb{R}\backslash\{0\}$

We consider the two-dimensional nonlinear Schrödinger equation with point interaction and we establish a local well-posedness theory, including blow-up alternative and continuous dependence on the initial data in the energy space. We provide proof by employing Kato’s method along with Hardy inequalities with logarithmic correction. Moreover, we establish finite time blow-up for solutions with positive energy and infinite variance.

In this article, we prove the local-in-time existence of regular solutions to dissipative Aw–Rascle system with the offset equal to gradient of some increasing and regular function of density. It is a mixed degenerate parabolic-hyperbolic hydrodynamic model, and we extend the techniques previously developed for compressible Navier–Stokes equations to show the well-posedness of the system in the $L_2-L_2$ setting. We also discuss relevant existence results for offset involving singular or non-local functions of density.

We revisit the time evolution of initially trapped Bose-Einstein condensates in the Gross-Pitaevskii regime. We show that the system continues to exhibit BEC once the trap has been released and that the dynamics of the condensate is described by the time-dependent Gross-Pitaevskii equation. Like the recent work [15], we obtain optimal bounds on the number of excitations orthogonal to the condensate state. In contrast to [15], however, whose main strategy consists of controlling the number of excitations with regard to a suitable fluctuation dynamics $t\mapsto e^{-B_t} e^{-iH_Nt}$ with renormalized generator, our proof is based on controlling renormalized excitation number operators directly with regards to the Schrödinger dynamics $t\mapsto e^{-iH_Nt}$.

This article is concerned with the following chemotaxis-growth system

\begin{equation*}\begin{cases}u_t = \nabla\cdot \left(\nabla u - u\bf{w} \right) + \rho u - \mu u^\alpha ,&x \in \Omega ,\ t \gt 0,\\v_{1,t} = \Delta v_1 - v_1 + u,&x \in \Omega ,\ t \gt 0,\\v_{2,t} = \Delta v_2 - v_2 + v_1,&x \in \Omega ,\ t \gt 0,\\\ldots \\v_{k,t} = \Delta v_k - v_k + v_{k - 1},&x \in \Omega ,\ t \gt 0,\\{\bf{w}}_t = \Delta {\bf{w}} - {\bf{w}} + \nabla v_k,&x \in \Omega ,\ t \gt 0,\end{cases}\end{equation*}

under the homogeneous Neumann boundary condition for u, vi and the homogeneous Dirichlet boundary condition for $\bf{w}$ in a smooth bounded domain $\Omega \subset {\mathbb{R}^n}\left( {n \geqslant 1} \right),$ where ρ > 0, µ > 0, α > 1 and $i=1,\ldots,k$. We reveal that when the index α, the spatial variable n, and the number of equations k satisfy certain relationships, the global solution of the system exists and converges to the constant equilibrium state in the form of exponential convergence.

We prove that a solution to the 3D Navier–Stokes or magneto-hydrodynamics equations does not blow up at t = T provided $\displaystyle \limsup_{q \to \infty} \int_{\mathcal{T}_q}^T \|\Delta_q(\nabla \times u)\|_\infty \, dt$ is small enough, where u is the velocity, $\Delta_q$ is the Littlewood–Paley projection and $\mathcal T_q$ is a certain sequence such that $\mathcal T_q \to T$ as $q \to \infty$. This improves many existing regularity criteria.

We present a novel multiscale mathematical model of espresso brewing. The model captures liquid infiltration and flow through a packed bed of ground coffee, as well as coffee solubles transport (both in the grains and in the liquid) and solubles dissolution. During infiltration, a sharp interface separates the dry and wet regions of the bed. A matched asymptotic analysis (based on fast dissolution rates) reveals that the bed can be described by four asymptotic regions: a dry region yet to be infiltrated by the liquid, a region in which the liquid is saturated with solubles and very little dissolution occurs, a slender region in which solubles are rapidly extracted from the smallest grains, and region in which slower extraction occurs from larger grains. The position and extent of each of these regions move with time (one being an intrinsic moving internal boundary layer) making the asymptotic analysis intriguing in its own right. The analysis yields a reduced model that elucidates the rate-limiting physical processes. Numerical solutions of the reduced model are compared to those to the full model, demonstrating that the reduced model is both accurate and significantly cheaper to solve.

We consider local and nonlocal Cahn–Hilliard equations with constant mobility and singular potentials including, e.g., the Flory–Huggins potential, subject to no-flux (or periodic) boundary conditions. The main goal is to show that the presence of a suitable class of reaction terms allows to establish the existence of a weak solution to the corresponding initial and boundary value problem even though the initial condition is a pure state. This fact was already observed by the authors in a previous contribution devoted to a specific biological model. In this context, we examine the essential assumptions required for the reaction term to ensure the existence of a weak solution. Also, we explore the scenario involving the nonlocal Cahn–Hilliard equation and provide some illustrative examples that contextualize within our abstract framework.

We study the long-time asymptotics for the solution of the modified Camassa–Holm (mCH) equation with step-like initial data.

\begin{align*} &m_{t}+\left (m\left (u^{2}-u_{x}^{2}\right )\right )_{x}=0, \quad m=u-u_{xx}, \\[3pt] & {u(x,0)=u_0(x)\to \left \{ \begin{array}{l@{\quad}l} 1/c_+, &\ x\to +\infty, \\[3pt] 1/c_-, &\ x\to -\infty, \end{array}\right .} \end{align*}

$c_+$

$c_-$

$(y,t)$

$(\xi, c)$

$c=c_+/c_-$

$\{ (\xi, c)\,:\, \xi \in \mathbb{R}, \ c\gt 0, \ \xi =y/t\}$

$u(y,t)$

$\mathcal{O}(t^{-2})$

$\mathcal{O}(t^{-1})$

Optimal transport tasks naturally arise in gas networks, which include a variety of constraints such as physical plausibility of the transport and the avoidance of extreme pressure fluctuations. To define feasible optimal transport plans, we utilize a $p$-Wasserstein metric and similar dynamic formulations minimizing the kinetic energy necessary for moving gas through the network, which we combine with suitable versions of Kirchhoff’s law as the coupling condition at nodes. In contrast to existing literature, we especially focus on the non-standard case $p \neq 2$ to derive an overdamped isothermal model for gases through $p$-Wasserstein gradient flows in order to uncover and analyze underlying dynamics. We introduce different options for modelling the gas network as an oriented graph including the possibility to store gas at interior vertices and to put in or take out gas at boundary vertices.

We consider the Cauchy problem of the non-linear Schrödinger equation with the modulated dispersion and power type non-linearities in any spatial dimensions. We adapt the Young integral theory developed by Chouk–Gubinelli [7] and multilinear estimates which are based on divisor counting and show the local well-posedness. This generalizes the result by Chouk–Gubinelli [7] in terms of the dimension and the order of the non-linearity.

This work investigates the online machine learning problem of prediction with expert advice in an adversarial setting through numerical analysis of, and experiments with, a related partial differential equation. The problem is a repeated two-person game involving decision-making at each step informed by $n$ experts in an adversarial environment. The continuum limit of this game over a large number of steps is a degenerate elliptic equation whose solution encodes the optimal strategies for both players. We develop numerical methods for approximating the solution of this equation in relatively high dimensions ($n\leq 10$) by exploiting symmetries in the equation and the solution to drastically reduce the size of the computational domain. Based on our numerical results we make a number of conjectures about the optimality of various adversarial strategies, in particular about the non-optimality of the COMB strategy.

In this article, we investigate the following non-linear Schrödinger (NLS) equation with Neumann boundary conditions:

\begin{equation*}\begin{cases}-\Delta u+ \lambda u= f(u) & {\mathrm{in}} \,~ \Omega,\\\displaystyle\frac{\partial u}{\partial \nu}=0 \, &{\mathrm{on}}\,~\partial \Omega\end{cases}\end{equation*}

coupled with a constraint condition:

\begin{equation*}\int_{\Omega}|u|^2 dx=c,\end{equation*}

where $\Omega\subset \mathbb{R}^N(N\ge3)$ denotes a smooth bounded domain, ν represents the unit outer normal vector to $\partial \Omega$, c is a positive constant, and λ acts as a Lagrange multiplier. When the non-linearity f exhibits a general mass supercritical growth at infinity, we establish the existence of normalized solutions, which are not necessarily positive solutions and can be characterized as mountain pass type critical points of the associated constraint functional. Our approach provides a uniform treatment of various non-linearities, including cases such as $f(u)=|u|^{p-2}u$, $|u|^{q-2}u+ |u|^{p-2}u$, and $-|u|^{q-2}u+|u|^{p-2}u$, where $2 \lt q \lt 2+\frac{4}{N} \lt p \lt 2^*$. The result is obtained through a combination of a minimax principle with Morse index information for constrained functionals and a novel blow-up analysis for the NLS equation under Neumann boundary conditions.

In this paper, we consider the defocusing nonlinear wave equation $-\partial _t^2u+\Delta u=|u|^{p-1}u$ in $\mathbb {R}\times \mathbb {R}^d$. Building on our companion work (Self-similar imploding solutions of the relativistic Euler equations, arXiv:2403.11471), we prove that for $d=4, p\geq 29$ and $d\geq 5, p\geq 17$, there exists a smooth complex-valued solution that blows up in finite time.