In this paper the free boundary problem for groundwater phreatic surface is represented in the form of a variational principle. It is proved that the flow domain Ω that solves the problem is a minimizer of some functional Λ(Ω). Weak solutions are introduced as minimizers of the lower semi-continuous regularization of Λ(⋅). Within this approach the existence of weak solutions is proved for a wide class of input data.

The Polubarinova–Galin equation describes a parametric conformal map for the zero-surface-tension Hele-Shaw flow driven by injection or suction. In this paper, we classify degree three polynomial solutions to the Polubarinova–Galin equation into three categories: global solutions, solutions which can be continued after blow-up and solutions which cannot be continued after blow-up. The coefficient region of the initial functions in each category is obtained.

An enhanced understanding of the microstructure of oxide ceramics will help scientists and engineers improve their efficiency and design. A phase-field model for the composition and phase distribution of the oxide ceramic components is studied. The model, which includes an obstacle in the phase portion of the energy potential, results in a minimisation problem that characterises the distribution of the bulk phases. The transition region between them is studied in several mathematically plausible asymptotic limits. The behaviour of the system in these limits provides insights into the applicability of the model and indicates appropriate parameter regimes.

We use the two-dimensional heat equation as an illustrative example to show that the unified transform is capable of constructing analytical solutions for linear evolution partial differential equations (PDEs) in two spatial dimensions involving non-separable boundary conditions. Such non-separable boundary value problems apparently cannot be solved by the usual transforms. We note that the unified transform always yields integral expressions which, in contrast to the expressions obtained by the usual transforms, have the advantage that are uniformly convergent at the boundary. Thus, even for the cases of separable boundary value problems where the usual transforms can be implemented, the unified transform provides alternative solution expressions which have advantages for both numerical and asymptotic considerations. The former advantage is illustrated by providing the numerical evaluation of a typical boundary value problem, by extending the approach of Flyer and Fokas (2008Proc. R. Soc.464, 1823–1849). This work is the two-dimensional continuation of the heat equation with oblique Robin boundary conditions which was analysed in Mantzavinos and Fokas (2013Eur. J. Appl. Math.24(6), 857–886.

Analysis ℓ1-recovery refers to a technique of recovering a signal that is sparse in some transform domain from incomplete corrupted measurements. This includes total variation minimization as an important special case when the transform domain is generated by a difference operator. In the present paper, we provide a bound on the number of Gaussian measurements required for successful recovery for total variation and for the case that the analysis operator is a frame. The bounds are particularly suitable when the sparsity of the analysis representation of the signal is not very small.

In this paper, an initial-value problem for the modified Korteweg-de Vries (mKdV) equation is addressed. Previous numerical simulations of the solution of

\[ u_{t} - 6u^{2} u_{x}+u_{xxx}=0, \quad -\infty<x<\infty, \quad t>0, \]

\[ u(x,0)=\tanh ( Cx ), \quad -\infty<x<\infty, \]

When faced with slowly depleting resources (such as decrease in precipitation due to climate change), complex ecological systems are prone to sudden irreversible changes (such as desertification) as the resource level dips below a tipping point of the system. A possible coping mechanism is the formation of spatial patterns, which allows for concentration of sparse resources and the survival of the species within “ecological niches” even below the tipping point of the homogeneous vegetation state. However, if the change in resource availability is too sudden, the system may not have time to transition to the patterned state and will pass through the tipping point instead, leading to extinction. We argue that the deciding factors are the speed of resource depletion and the amount of the background noise (seasonal climate changes) in the system. We illustrate this phenomenon on a model of patterned vegetation. Our analysis underscores the importance of, and the interplay between, the speed of climate change, heterogeneity of the environment, and the amount of seasonal variability.