We consider the existence of normalized solutions to non-linear Schrödinger equations on non-compact metric graphs in the L2-supercritical regime. For sufficiently small prescribed mass (L2 norm), we prove existence of positive solutions on two classes of graphs: periodic graphs and non-compact graphs with finitely many edges and suitable topological assumptions. Our approach is based on mountain pass techniques. A key point to overcome the serious lack of compactness is to show that all solutions with small mass have positive energy. To complement our analysis, we prove that this is no longer true, in general, for large masses. To the best of our knowledge, these are the first results with an L2-supercritical non-linearity extended on the whole graph and unravelling the role of topology in the existence of solutions.

Substantially extending previous results of the authors for smooth solutions in the viscous case, we develop linear damping estimates for periodic roll-wave solutions of the inviscid Saint-Venant equations and related systems of hyperbolic balance laws. Such damping estimates, consisting of $H^s$ energy estimates yielding exponential slaving of high-derivative to low-derivative norms, have served as crucial ingredients in nonlinear stability analyses of traveling waves in hyperbolic or partially parabolic systems, both in obtaining high-frequency resolvent estimates and in closing a nonlinear iteration for which available linearized stability estimates apparently lose regularity. Here, we establish for systems of size $n\leq 6$ a Lyapunov-type theorem stating that such energy estimates are available whenever strict high-frequency spectral stability holds; for dimensions $7$ and higher, there may be in general a gap between high-frequency spectral stability and existence of the type of energy estimate that we develop here. A key ingredient is a dimension-dependent linear algebraic lemma reminiscent of Lyapunov’s Lemma for ODE that is to our knowledge new.

We extend a classical model of continuous opinion formation to explicitly include an age-structured population. We begin by considering a stochastic differential equation model which incorporates ageing dynamics and birth/death processes, in a bounded confidence type opinion formation model. We then derive and analyse the corresponding mean field partial differential equation and compare the complex dynamics on the microscopic and macroscopic levels using numerical simulations. We rigorously prove the existence of stationary states in the mean field model, but also demonstrate that these stationary states are not necessarily unique. Finally, we establish connections between this and other existing models in various scenarios.

This paper is devoted to a comprehensive analysis of a family of solutions of the focusing nonlinear Schrödinger equation called general rogue waves of infinite order. These solutions have recently been shown to describe various limit processes involving large-amplitude waves, and they have also appeared in some physical models not directly connected with nonlinear Schrödinger equations. We establish the following key property of these solutions: they are all in $L^2(\mathbb{R})$ with respect to the spatial variable but they exhibit anomalously slow temporal decay. In this paper, we define general rogue waves of infinite order, establish their basic exact and asymptotic properties, and provide computational tools for calculating them accurately.

The Cahn–Hilliard model with reaction terms can lead to situations in which no coarsening is taking place and, in contrast, growth and division of droplets occur which all do not grow larger than a certain size. This phenomenon has been suggested as a model for protocells, and a model based on the modified Cahn–Hilliard equation has been formulated. We introduce this equation and show the existence and uniqueness of solutions. Then, formally matched asymptotic expansions are used to identify a sharp interface limit using a scaling of the reaction term, which becomes singular when the interfacial thickness tends to zero. We compute planar solutions and study their stability under non-planar perturbations. Numerical computations for the suggested model are used to validate the sharp interface asymptotics. In addition, the numerical simulations show that the reaction terms lead to diverse phenomena such as growth and division of droplets in the obtained solutions, as well as the formation of shell-like structures.

We consider the Cauchy problem to the 3D fractional Schrödinger equation with quadratic interaction of $u\bar u$ type. We prove the global existence of solutions and scattering properties for small initial data. For the proof, one novelty is that we combine the normal form methods and the space-time resonance methods. Using the normal form transform enables us to have more flexibility in designing the resolution spaces so that we can control various interactions. It is also convenient for the final data problem.

We define a new class of plane billiards – the “pensive billiard” – in which the billiard ball travels along the boundary for some distance depending on the incidence angle before reflecting, while preserving the billiard rule of equality of the angles of incidence and reflection. This generalizes so-called “puck billiards” proposed by M. Bialy, as well as a “vortex billiard,” that is, the motion of a point vortex dipole in two-dimensional hydrodynamics on domains with boundary. We prove the variational origin and invariance of a symplectic structure for pensive billiards, as well as study their properties including conditions for a twist map, the existence of periodic orbits, etc. We also demonstrate the appearance of both the golden and silver ratios in the corresponding hydrodynamical vortex setting. Finally, we introduce and describe basic properties of pensive outer billiards.

We consider the Schrödinger equation on the one dimensional torus with a general odd-power nonlinearity $p \geq 5$, which is known to be globally well-posed in the Sobolev space $H^\sigma(\mathbb{T})$, for every $\sigma \geq 1$, thanks to the conservation and finiteness of the energy. For regularities σ < 1, where this energy is infinite, we explore a globalization argument adapted to random initial data distributed according to the Gaussian measures µs, with covariance operator $(1-\Delta)^s$, for s in a range $(s_p,\frac{3}{2}]$. We combine a deterministic local Cauchy theory with the quasi-invariance of Gaussian measures µs, with additional Lq-bounds on the Radon-Nikodym derivatives, to prove that the Gaussian initial data generate almost surely global solutions. These Lq-bounds are obtained with respect to Gaussian measures accompanied by a cutoff on a renormalization of the energy; the main tools to prove them are the Boué-Dupuis variational formula and a Poincaré-Dulac normal form reduction. This approach is similar in spirit to Bourgain’s invariant argument [7] and to arecent work by Forlano-Tolomeo in [18].

In this article, we consider the global-in-time existence and singularity formation of smooth solutions for the radially symmetric relativistic Euler equations of polytropic gases. We introduce the rarefaction/compression character variables for the supersonic expanding wave with relativity and derive their Riccati type equations to establish a series of priori estimates of solutions by the characteristic method and the invariant domain idea. It is verified that, for rarefactive initial data with vacuum at the origin, smooth solutions will exist globally. On the other hand, the smooth solution develops a singularity in finite time when the initial data are compressed and include strong compression somewhere.

We prove an extension theorem for local solutions of the 3d incompressible Euler equations. More precisely, we show that if a smooth vector field satisfies the Euler equations in a spacetime region $\Omega \times (0,T)$, one can choose an admissible weak solution on $\mathbb R^3\times (0,T)$ of class $C^\beta $ for any $\beta <1/3$ such that both fields coincide on $\Omega \times (0,T)$. Moreover, one controls the spatial support of the global solution. Our proof makes use of a new extension theorem for local subsolutions of the incompressible Euler equations and a $C^{1/3}$ convex integration scheme implemented in the context of weak solutions with compact support in space. We present two nontrivial applications of these ideas. First, we construct infinitely many admissible weak solutions of class $C^\beta _{\text {loc}}$ with the same vortex sheet initial data, which coincide with it at each time t outside a turbulent region of width $O(t)$. Second, given any smooth solution v of the Euler equation on $\mathbb {T}^3\times (0,T)$ and any open set $U\subset \mathbb {T}^3$, we construct admissible weak solutions which coincide with v outside U and are uniformly close to it everywhere at time $0$, yet blow up dramatically on a subset of $U\times (0,T)$ of full Hausdorff dimension. These solutions are of class $C^\beta $ outside their singular set.

For Maxwell’s equations with nonlinear polarization we prove the existence of time-periodic breather solutions travelling along slab or cylindrical waveguides. The solutions are TE-modes which are localized in one (slab case) or both (cylindrical case) space directions orthogonal to the direction of propagation. We assume a magnetically inactive and electrically nonlinear material law with a linear $\chi^{(1)}$- and a cubic $\chi^{(3)}$-contribution to the polarization. The $\chi^{(1)}$-contribution may be retarded in time or instantaneous whereas the $\chi^{(3)}$-contribution is always assumed to be retarded in time. We consider two different cubic nonlinearities which provide a variational structure under suitable assumptions on the retardation kernels, in particular we require that for time-periodic solutions Maxwell’s equations are invariant under time-inversion. By choosing a sufficiently small propagation speed along the waveguide the second order formulation of the Maxwell system becomes essentially elliptic for the E-field so that solutions can be constructed by the mountain pass theorem. The compactness issues arising in the variational method are overcome by either the cylindrical geometry itself or by extra assumptions on the linear and nonlinear parts of the polarization in case of the slab geometry. Our approach to breather solutions in the presence of time-retardation is systematic in the sense that we look for general conditions on the Fourier-coefficients in time of the retardation kernels. Our main existence result is complemented by concrete examples of coefficient functions and retardation kernels.

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.