Following pioneering work by Fan and Slemrod, who studied the effect of artificial viscosity terms, we consider the system of conservation laws arising in liquid–vapour phase dynamics with physical viscosity and capillarity effects taken into account. Following Dafermos, we consider self-similar solutions to the Riemann problem and establish uniform total variation bounds, allowing us to deduce new existence results. Our analysis covers both the hyperbolic and the hyperbolic–elliptic regimes and apply to arbitrarily large Riemann data.

The proofs rely on a new technique of reduction to two coupled scalar equations associated with the two wave fans of the system. Strong $L^1$ convergence to a weak solution of bounded variation is established in the hyperbolic regime, while in the hyperbolic–elliptic regime a stationary singularity near the axis separating the two wave fans, or more generally an almost-stationary oscillating wave pattern (of thickness depending upon the capillarity–viscosity ratio), is observed and the solution may not have globally bounded variation.

We consider the following equation:

where d(x) = d(x, ∂Ω), θ > –2 and Ω is a half-space. The existence and non-existence of several kinds of positive solutions to this equation when , f(u) = up (p > 1) and Ω is a bounded smooth domain were studied by Bandle, Moroz and Reichel in 2008. Here, we study exact the behaviour of positive solutions to this equation as d(x) → 0+ and d(x) → ∞, respectively, and the symmetry of positive solutions when , Ω is a half-space and f(u) is a more general nonlinearity term than up . Under suitable conditions for f, we show that the equation has a unique positive solution W, which is a function of x 1 only, and W satisfies

We consider elliptic systems with discontinuous coefficients and prove that if the known term belongs to the Morrey space Lp,λ, then the highest-order derivatives of the local solution belong to the same space. We also obtain local Hölder continuity for lower-order derivatives.

We prove nonlinear stability of planar shock fronts for certain relaxation systems in two spatial dimensions. If the subcharacteristic condition is assumed and the initial perturbation is sufficiently small and the mass carried by the perturbations is not necessarily finite, then the solution converges to a shifted planar shock front solution as time t ↑ ∞. The asymptotic phase shift of shock fronts is, in general, non-zero and governed by a similarity solution to the heat equation. The asymptotic decay rate to the shock front is proved to be t−1/4 in L∞(R2) without imposing extra decay rates in space for the initial perturbations. The proofs are based on an elementary weighted energy analysis to the error equation.

We continue the research on reiteration results between interpolation methods associated to polygons and the real method. Applications are given to N-tuples of function spaces, of spaces of bounded linear operators and Banach algebras.

Let G be a finite group, and let cs(G) be the set of conjugacy class sizes of G. Recalling that an element g of G is called a vanishing element if there exists an irreducible character of G taking the value 0 on g, we consider one particular subset of cs(G), namely, the set vcs(G) whose elements are the conjugacy class sizes of the vanishing elements of G. Motivated by the results inBianchi et al. (2020, J. Group Theory, 23, 79–83), we describe the class of the finite groups G such that vcs(G) consists of a single element under the assumption that G is supersolvable or G has a normal Sylow 2-subgroup (in particular, groups of odd order are covered). As a particular case, we also get a characterization of finite groups having a single vanishing conjugacy class size which is either a prime power or square-free.

This paper is concerned with the global regularity problem on the micropolar Rayleigh-Bénard problem with only velocity dissipation in $\mathbb {R}^{d}$ with $d=2\ or\ 3$. By fully exploiting the special structure of the system, introducing two combined quantities and using the technique of Littlewood-Paley decomposition, we establish the global regularity of solutions to this system in $\mathbb {R}^{2}$. Moreover, we obtain the global regularity for fractional hyperviscosity case in $\mathbb {R}^{3}$ by employing various techniques including energy methods, the regularization of generalized heat operators on the Fourier frequency localized functions and logarithmic Sobolev interpolation inequalities.

We consider reaction–diffusion equations under nonlinear boundary conditions where the nonlinearities are asymptotically linear at infinity and depend on a parameter. We prove that, as the parameter crosses some critical values, a resonance-type phenomenon provides solutions that bifurcate from infinity. We characterize the bifurcated branches when they are sub- or supercritical. We obtain both Landesman–Lazer-type conditions that guarantee the existence of solutions in the resonant case and an anti-maximum principle.

We prove that the two-dimensional Brown–Ravenhall operator is bounded from below when the coupling constant is below a specified critical value—a property also referred to as stability. As a consequence, the operator is then self-adjoint. The proof is based on the strategy followed by Evans et al. and Lieb and Yau, with some relevant changes characteristic of the dimension. Our analysis also yields a sharp Kato inequality.

Let 𝕀 denote an imaginary quadratic field or the field ℚ of rational numbers and let ℤ𝕀 denote its ring of integers. We shall prove a new explicit Baker-type lower bound for a ℤ𝕀-linear form in the numbers 1, eα1 , . . . , eαm, m ⩾ 2, where α0 = 0, α1, . . . , αm are m + 1 different numbers from the field 𝕀. Our work gives substantial improvements on the existing explicit versions of Baker’s work about exponential values at rational points. In particular, dependencies on m are improved.

The stationary Gross–Pitaevskii equation in one dimension is considered with a complex periodic potential satisfying the conditions of the 𝒫𝒯 (parity-time reversal) symmetry. Under rather general assumptions on the potentials, we prove bifurcations of 𝒫𝒯-symmetric nonlinear bound states from the end points of a real interval in the spectrum of the non-selfadjoint linear Schrödinger operator with a complex 𝒫𝒯-symmetric periodic potential. The nonlinear bound states are approximated by the effective amplitude equation, which bears the form of the cubic nonlinear Schrödinger equation. In addition, we provide sufficient conditions for the appearance of complex spectral bands when the complex 𝒫𝒯-symmetric potential has an asymptotically small imaginary part.

Using a dual variational approach, we obtain nontrivial real-valued solutions of the critical nonlinear Helmholtz equation

$$-\Delta u-k^2u = Q(x) \vert u \vert ^{2^*-2}u,\quad u\in W^{2,2^*}({\open R}^{N})$$

$Q \in L^{\infty }({\open R}^{N}){\setminus }\{0\}$

This paper is concerned with the asymptotic behaviour of solutions to quasilinear hyperbolic equations with nonlinear damping on the quarter-plane (x, t) ∈ ℝ+ x ∈ ℝ+. We obtain the Lp (1 ≤ p ≤ +∞) convergence rates of the solution to the quasilinear hyperbolic equations without the additional technical assumptions for the nonlinear damping f(v) given by Li and Saxton.

The main aim of this paper is to introduce an appropriate dyadic one-sided maximal operator , smaller than the one-sided Hardy–Littlewood maximal operator M+ but such that it controls M+ in a similar way to how the usual dyadic maximal operator controls the Hardy-Littlewood maximal operator. We characterize the weighted inequalities for this dyadic one-sided maximal operator.

The interpretation of optical interference experiments at low light intensities and with independent light sources is discussed. A new experiment on interference between modulated light beams is described briefly, and its significance is indicated. There are two appendices, one a fuller discussion of the conditions under which interference and intensity correlation may be produced, the other a comment on one aspect of Jánossy and Nagy's discussion (1956) of the feasibility of the alternating slit interference experiment.

The fundamental impulse determining the migration of birds is connected with the reproductive cycle; but when that has been recognised it must also be admitted that there are secondary conditions of environment which regulate the actual movements of which a migration is compounded. Of these weather has long been looked upon as one of the most important. There is still, however, much divergence of opinion as to the extent and nature of the influence of weather, and indeed in recent years statistical analysis has led workers, such as Bretscher (1915), to doubt whether any satisfactory connection between meteorological conditions and migration can be demonstrated, and Lincoln to announce boldly that “the state of the weather at any point has little if anything to do with the time of arrival of migratory birds” (1935, p. 59).

In this paper it is proposed to study the problem of the vibrations of a uniformly-stretched membrane whose superficial density σ depends upon the distance r from a fixed point in the membrane in accordance with the law

in which C and a are constants. The constant a will be known as the critical radius and the circle r = a as the critical circle.

In this paper we discuss the problem

We show that for c > 0 there exists a positive constant μ* such that (*)μ possesses at least one solution if μ ∈ (0, μ*) and no solutions if μ > μ*. Furthermore, there exists a positive constant μ**≦μ* such that (*)μ possesses at least two solutions if μ∈(0, μ**), 2<N<6. For N≧6, μ∈(0,μ**), we show that problem (*)μ possesses a unique solution if f(x) is radial with f′(r) < 0(r = |x|).

Assume that the local compliance tensor of an elastic composite in two space dimensions is equal to a rank-one tensor plus a null-Lagrangian (there is only one symmetric one in two dimensions). The purpose of this paper is to prove that the effective compliance tensor has the same representation: rank-one plus the null-Lagrangian. This statement generalises the wellknown result of Hill that a composite of isotropic phases with a common shear modulus is necessarily elastically isotropic and shares the same shear modulus. It also generalises the surprising discovery of Avellaneda et al. that under a certain condition on the pure crystal moduli the shear modulus of an isotropic polycrystal is uniquely determined. The present paper sheds light on this effect by placing it in a more general framework and using some elliptic PDE theory rather than the translation method. Our results allow us to calculate the polycrystalline G-closure of the special class of crystals under consideration. Our analysis is contrasted with a two-dimensional model problem for shape-memory polycrystals. We show that the two problems can be thought of as ‘elastic percolation’ problems, one elliptic, one hyperbolic.

In this paper we introduce the concept of fractional powers of a pair of operators between two Banach spaces. The operators need not be closed, but form a closed pair. The properties of the fractional powers are studied. An application of the theory is briefly discussed.