The present article is devoted to the study of global solution and large time behaviour of solution for the isentropic compressible Euler system with source terms in $\mathbb {R}^d$, $d\geq 1$, which extends and improves the results obtained by Sideris et al. in ‘T.C. Sideris, B. Thomases, D.H. Wang, Long time behavior of solutions to the 3D compressible Euler equations with damping, Comm. Partial Differential Equations 28 (2003) 795–816’. We first establish the existence and uniqueness of global smooth solution provided the initial datum is sufficiently small, which tells us that the damping terms can prevent the development of singularity in small amplitude. Next, under the additional smallness assumption, the large time behaviour of solution is investigated, we only obtain the algebra decay of solution besides the $L^2$-norm of $\nabla u$ is exponential decay.

This paper focuses on a 2D magnetohydrodynamic system with only horizontal dissipation in the domain $\Omega = \mathbb {T}\times \mathbb {R}$ with $\mathbb {T}=[0,\,1]$ being a periodic box. The goal here is to understand the stability problem on perturbations near the background magnetic field $(1,\,0)$. Due to the lack of vertical dissipation, this stability problem is difficult. This paper solves the desired stability problem by simultaneously exploiting two smoothing and stabilizing mechanisms: the enhanced dissipation due to the coupling between the velocity and the magnetic fields, and the strong Poincaré type inequalities for the oscillation part of the solution, namely the difference between the solution and its horizontal average. In addition, the oscillation part of the solution is shown to converge exponentially to zero in $H^{1}$ as $t\to \infty$. As a consequence, the solution converges to its horizontal average asymptotically.

Steady-state diffusion in long axisymmetric structures is considered. The goal is to assess one-dimensional approximations by comparing them with axisymmetric eigenfunction expansions. Two problems are considered in detail: a finite tube with one end that is partly absorbing and partly reflecting; and two finite coaxial tubes with different cross-sectional radii joined together abruptly. Both problems may be modelled using effective boundary conditions, containing a parameter known as the trapping rate. We show that trapping rates depend on the lengths of the finite tubes (and that they decay slowly as these lengths increase) and we show how trapping rates are related to blockage coefficients, which are well known in the context of potential flow along tubes of infinite length.

We initiate the study of C*-algebras and groupoids arising from left regular representations of Garside categories, a notion which originated from the study of Braid groups. Every higher rank graph is a Garside category in a natural way. We develop a general classification result for closed invariant subspaces of our groupoids as well as criteria for topological freeness and local contractiveness, properties which are relevant for the structure of the corresponding C*-algebras. Our results provide a conceptual explanation for previous results on gauge-invariant ideals of higher rank graph C*-algebras. As another application, we give a complete analysis of the ideal structures of C*-algebras generated by left regular representations of Artin–Tits monoids.

In this paper, we consider the following two-component Schrödiner system

\[ \begin{cases} -\Delta u_1 +V_1(x) u_1= \mu_1 u_1^{3} +\beta u_1u_2^{2} & \text{in}\;\mathbb{R}^{3},\\ -\Delta u_2+V_2(x) u_2= \mu_2 u_2^{3} +\beta u_1^{2}u_2 & \text{in}\;\mathbb{R}^{3}. \end{cases} \]

Relations between the Atiyah–Patodi–Singer rho invariant and signatures of links have been known for a long time, but they were only partially investigated. In order to explore them further, we develop a versatile cut-and-paste formula for the rho invariant, which allows us to manipulate manifolds in a convenient way. With the help of this tool, we give a description of the multivariable signature of a link $L$ as the rho invariant of some closed three-manifold $Y_L$ intrinsically associated with $L$. We study then the rho invariant of the manifolds obtained by the Dehn surgery on $L$ along integer and rational framings. Inspired by the results of Casson and Gordon and Cimasoni and Florens, we give formulas expressing this value as a sum of the multivariable signature of $L$ and some easy-to-compute extra terms.

There is a problem with the proofs of [1], Lemma 4.4 and the related Theorems 4.5, 4.8 and 4.12 regarding the computation of zero-divisor cup-length of real Grassmann manifolds ${G_k({{\mathbb {R}}}^{n})}$. The correct statements and improved estimates of the topological complexity of ${G_k({{\mathbb {R}}}^{n})}$ will appear in a separate paper by M. Radovanović [2].

In 1991, Chicone and Jacobs showed the equivalence between the computation of the first-order Taylor developments of the Lyapunov constants and the developments of the first Melnikov function near a non-degenerate monodromic equilibrium point, in the study of limit cycles of small-amplitude bifurcating from a quadratic centre. We show that their proof is also valid for polynomial vector fields of any degree. This equivalence is used to provide a new lower bound for the local cyclicity of degree six polynomial vector fields, so $\mathcal {M}(6) \geq 44$. Moreover, we extend this equivalence to the piecewise polynomial class. Finally, we prove that $\mathcal {M}^{c}_{p}(4) \geq 43$ and $\mathcal {M}^{c}_{p}(5) \geq 65.$

Hilbert schemes are an object arising from geometry and are closely related to physics and modular forms. Recently, there have been investigations from number theorists about the Betti numbers and Hodge numbers of the Hilbert schemes of points of an algebraic surface. In this paper, we prove that Göttsche's generating function of the Hodge numbers of Hilbert schemes of $n$ points of an algebraic surface is algebraic at a CM point $\tau$ and rational numbers $z_1$ and $z_2$. Our result gives a refinement of the algebraicity on Betti numbers.

This paper considers a model for oncolytic virotherapy given by the doubly haptotactic cross-diffusion system

\[ \left\{\begin{array}{@{}ll} u_t=D_u\Delta u-\xi_u\nabla\cdot(u\nabla v)+\mu_u u(1-u)-\rho uz,\\ v_t={-} (\alpha_u u+\alpha_w w)v,\\ w_t=D_w\Delta w-\xi_w\nabla\cdot(w\nabla v)- w+\rho uz,\\ z_t=D_z\Delta z-\delta_z z- \rho uz+\beta w, \end{array}\right. \]

$D_u,D_w,D_z,\xi _u,\xi _w,\delta _z,\rho$

$\alpha _u,\alpha _w,\mu _u,\beta$

$\Omega \subset {\mathbb {R}}^{2}$

$\beta <1$

$(u,v,w,z)$

$(1, 0, 0, 0)$

$(L^{\infty }(\Omega ))^{4}$

$t\rightarrow \infty$

We study a variant of the mean curvature flow for closed, convex hypersurfaces where the normal velocity is a non-homogeneous function of the principal curvatures. We show that if the initial hypersurface satisfies a certain pinching condition, then this is preserved and the flow converges to a sphere under rescaling.

Ganea proved that the loop space of $\mathbb{C} P^n$ is homotopy commutative if and only if $n=3$. We generalize this result to that the loop spaces of all irreducible Hermitian symmetric spaces but $\mathbb{C} P^3$ are not homotopy commutative. The computation also applies to determining the homotopy nilpotency class of the loop spaces of generalized flag manifolds $G/T$ for a maximal torus T of a compact, connected Lie group G.

We prove the existence of a solution for a class of activator–inhibitor system of type $- \Delta u +u = f(u) -v$, $-\Delta v+ v=u$ in $\mathbb{R}^{N}$. The function f is a general nonlinearity which can grow polynomially in dimension $N\geq 3$ or exponentiallly if $N=2$. We are able to treat f when it has critical growth corresponding to the Sobolev space we work with. We transform the system into an equation with a nonlocal term. We find a critical point of the corresponding energy functional defined in the space of functions with norm endowed by a scalar product that takes into account such nonlocal term. For that matter, and due to the lack of compactness, we deal with weak convergent minimizing sequences and sequences of Lagrange multipliers of an action minima problem.

We introduce and study a fermionisation procedure for the cohomological Hall algebra $\mathcal{H}_{\Pi_Q}$ of representations of a preprojective algebra, that selectively switches the cohomological parity of the BPS Lie algebra from even to odd. We do so by determining the cohomological Donaldson–Thomas invariants of central extensions of preprojective algebras studied in the work of Etingof and Rains, via deformed dimensional reduction. Via the same techniques, we determine the Borel–Moore homology of the stack of representations of the $\unicode{x03BC}$-deformed preprojective algebra introduced by Crawley–Boevey and Holland, for all dimension vectors. This provides a common generalisation of the results of Crawley-Boevey and Van den Bergh on the cohomology of smooth moduli schemes of representations of deformed preprojective algebras and my earlier results on the Borel–Moore homology of the stack of representations of the undeformed preprojective algebra.

We compute the Lie symmetries of characteristic function (CF) hierarchy of compressible turbulence, ignoring the effects of viscosity and heat conductivity. In the probability density function (PDF) hierarchy, a typical non-local nature is observed, which is naturally eliminated in the CF hierarchy. We observe that the CF hierarchy retains all the symmetries satisfied by compressible Euler equations. Broadly speaking, four types of symmetries can be discerned in the CF hierarchy: (i) symmetries corresponding to coordinate system invariance, (ii) scaling/dilation groups, (iii) projective groups and (iv) statistical symmetries, where the latter define measures of intermittency and non-gaussianity. As the multi-point CFs need to satisfy additional constraints such as the reduction condition, the projective symmetries are only valid for monatomic gases, that is, the specific heat ratio, $\gamma = 5/3$. The linearity of the CF hierarchy results in the statistical symmetries due to the superposition principle. For all of the symmetries, the global transformations of the CF and various key compressible statistics are also presented.

We deal with analytic three-dimensional symmetric systems whose origin is a Hopf-zero singularity. Once it is not completely analytically integrable, we provide criteria on the existence of at least one functionally independent analytic first integral. In the generic case, we characterize the analytic partially integrable systems by using orbitally equivalent normal forms. We also solve the problem through the existence of a class of formal inverse Jacobi multiplier of the system.

Let p be a rational prime. Let F be a totally real number field such that F is unramified over p and the residue degree of any prime ideal of F dividing p is $\leq 2$. In this paper, we show that the eigenvariety for $\mathrm {Res}_{F/\mathbb {Q}}(\mathit {GL}_{2})$, constructed by Andreatta, Iovita, and Pilloni, is proper at integral weights for $p\geq 3$. We also prove a weaker result for $p=2$.