Adaptive finite element methods for Cahn–Hilliard equations

We develop a method for adaptive mesh refinement for steady state problems that arise in the numerical solution of Cahn–Hilliard equations with an obstacle free energy. The problem is discretized in time by the backward-Euler method and in space by linear finite elements. The adaptive mesh refinement is performed using residual based a posteriori estimates; the time step is adapted using a heuristic criterion. We describe the space–time adaptive algorithm and present numerical experiments in two and three space dimensions that demonstrate the usefulness of our approach.     

A blow-up criterion for 3D compressible viscoelastic flow with large initial data

In this paper, we consider a Cauchy problem for the three-dimensional compressible viscoelastic flow with large initial data. We establish a blow-up criterion for the strong solutions in terms of the gradient of velocity only, which is similar to the Beale-Kato-Majda criterion for ideal incompressible flow (cf. Beale et al. (1984) [20]) and the blow-up criterion for the compressible Navier-Stokes equations (cf. Huang et al. (2011) [21]).     

On Gevrey solutions of threefold singular nonlinear partial differential equations

We study Gevrey asymptotics of the solutions to a family of threefold singular nonlinear partial differential equations in the complex domain. We deal with both Fuchsian and irregular singularities, and allow the presence of a singular perturbation parameter. By means of the Borel–Laplace summation method, we construct sectorial actual holomorphic solutions which turn out to share a same formal power series as their Gevrey asymptotic expansion in the perturbation parameter. This result rests on the Malgrange–Sibuya theorem, and it requires to prove that the difference between two neighboring solutions is exponentially small, what in this case involves an asymptotic estimate for a particular Dirichlet-like series.     

Lorentz estimates for degenerate and singular evolutionary systems

We prove estimates of Calderón–Zygmund type for evolutionary p-Laplacian systems in the setting of Lorentz spaces. We suppose the coefficients of the system to satisfy only a VMO condition with respect to the space variable. Our results hold true, mutatis mutandis, also for stationary p-Laplacian systems.     

A free boundary problem for compressible hydrodynamic flow of liquid crystals in one dimension

In this paper, we consider the free boundary problem for a simplified version of Ericksen–Leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals in dimension one. We obtain both existence and uniqueness of global classical solutions provided that the initial density is away from vacuum.     

On the best constant of Hardy–Sobolev inequalities

We obtain the sharp constant for the Hardy-Sobolev inequality involving the distance to the origin. This inequality is equivalent to a limiting Caffarelli–Kohn–Nirenberg inequality. In three dimensions, in certain cases the sharp constant coincides with the best Sobolev constant.     

Anti-self-dual Lagrangians: Variational resolutions of non-self-adjoint equations and dissipative evolutions

We develop the concept and the calculus of anti-self-dual (ASD) Lagrangians and their derived vector fields which seem inherent to many partial differential equations and evolutionary systems. They are natural extensions of gradients of convex functions – hence of self-adjoint positive operators – which usually drive dissipative systems, but also provide representations for the superposition of such gradients with skew-symmetric operators which normally generate unitary flows. They yield variational formulations and resolutions for large classes of non-potential boundary value problems and initial-value parabolic equations. Solutions are minima of newly devised energy functionals, however, and just like the self (and anti-self) dual equations of quantum field theory (e.g. Yang–Mills) the equations associated to such minima are not derived from the fact they are critical points of the functional I, but because they are also zeroes of suitably derived Lagrangians. The approach has many advantages: it solves variationally many equations and systems that cannot be obtained as Euler–Lagrange equations of action functionals, since they can involve non-self-adjoint or other non-potential operators; it also associates variational principles to variational inequalities, and to various dissipative initial-value first order parabolic problems. These equations can therefore be analyzed with the full range of methods – computational or not – that are available for variational settings. Most remarkable are the permanence properties that ASD Lagrangians possess making their calculus relatively manageable and their domain of applications quite broad.     

Improved interpolation inequalities,relative entropy and fast diffusion equations

We consider a family of Gagliardo–Nirenberg–Sobolev interpolation inequalities which interpolate between Sobolev?s inequality and the logarithmic Sobolev inequality, with optimal constants. The difference of the two terms in the interpolation inequalities (written with optimal constant) measures a distance to the manifold of the optimal functions. We give an explicit estimate of the remainder term and establish an improved inequality, with explicit norms and fully detailed constants. Our approach is based on nonlinear evolution equations and improved entropy–entropy production estimates along the associated flow. Optimizing a relative entropy functional with respect to a scaling parameter, or handling properly second moment estimates, turns out to be the central technical issue. This is a new method in the theory of nonlinear evolution equations, which can be interpreted as the best fit of the solution in the asymptotic regime among all asymptotic profiles.     

Uniqueness of weak solutions to a non-linear hyperbolic system in electrohydrodynamics

In this note we prove the uniqueness of weak solutions to a nonlinear hyperbolic system in electrohydrodynamics without the effects of a dissociation–recombination process. It is still open in the presence of a special dissociation–recombination process, although the existence of at least one weak solution was proved via the method of renormalized solutions by Feireisl [E. Feireisl, Weak solutions to a non-linear hyperbolic system arising in the theory of dielectric liquids, Math. Methods Appl. Sci. 18 (1995) 1041–1052] in 1995.     

Some new nonlinear inequalities and applications to boundary value problems

In this paper, we establish some new nonlinear integral inequalities of the Gronwall–Bellman–Ou-Iang-type in two variables. These on the one hand generalizes and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of differential equations. We illustrate this by applying our new results to certain boundary value problem.     

Mathematical models of tumour angiogenesis

We first study a parabolic–ODE system modelling tumour growth proposed by Othmer and Stevens [Aggregation, blowup, and collapse: the ABC's of taxis in reinforced random walks, SIAM J. Appl. Math. 57 (4) (1997) 1044–1081]. According to Levine and Sleeman [A system of reaction and diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math. 57 (3) (1997) 683–730], we reduced it to a hyperbolic equation and showed the existence of collapse in [A. Kubo, T. Suzuki, Asymptotic behavior of the solution to a parabolic ODE system modeling tumour growth, Differential Integral Equations 17 (2004) 721–736]. We also deal with the system in case the reduced equation is elliptic and show the existence of collapse analogously. Next we apply the above result to another model proposed by Anderson and Chaplain arising from tumour angiogenesis and show the existence of collapse. Further we investigate a contact point between these two models and a common property to them.     

Sturm–Liouville problems with parameter dependent potential and boundary conditions

This paper deals with the computation of the eigenvalues of Sturm–Liouville problems with parameter dependent potential and boundary conditions. We shall extend the domain of application of the method based on sampling theory to the case where the classical Whittaker–Shannon–Kotel’nikov theorem is not applicable. A few numerical examples will be presented.     

On the analysis of traveling waves to a nonlinear flux limited reaction–diffusion equation

In this paper we study the existence and qualitative properties of traveling waves associated with a nonlinear flux limited partial differential equation coupled to a Fisher–Kolmogorov–Petrovskii–Piskunov type reaction term. We prove the existence and uniqueness of finite speed moving fronts of C²$C^2$ classical regularity, but also the existence of discontinuous entropy traveling wave solutions.     

Analysis and finite element approximation of a Ladyzhenskaya model for viscous flow in streamfunction form

In this paper we consider a model for the motion of incompressible viscous flows proposed by Ladyzhenskaya. The Ladyzhenskaya model is written in terms of the velocity and pressure while the studied model is written in terms of the streamfunction only. We derived the streamfunction equation of the Ladyzhenskaya model and present a weak formulation and show that this formulation is equivalent to the velocity–pressure formulation. We also present some existence and uniqueness results for the model. Finite element approximation procedures are presented. The discrete problem is proposed to be well posed and stable. Some error estimates are derived. We consider the 2D driven cavity flow problem and provide graphs which illustrate differences between the approximation procedure presented here and the approximation for the streamfunction form of the Navier–Stokes equations. Streamfunction contours are also displayed showing the main features of the flow.     

Global existence for degenerate quadratic reaction–diffusion systems

We consider a class of degenerate reaction–diffusion systems with quadratic nonlinearity and diffusion only in the vertical direction. Such systems can appear in the modeling of photochemical generation and atmospheric dispersion of pollutants. The diffusion coefficients are different for all equations. We study global existence of solutions.     

Existence and nonlinear stability of stationary solutions to the full compressible Navier–Stokes–Korteweg system

This paper is concerned with the existence, uniqueness, and nonlinear stability of stationary solutions to the Cauchy problem of the full compressible Navier–Stokes–Korteweg system effected by the given mass source, the external force of general form, and the energy source in R³${\mathbb{R}}^3$. Based on the weighted L²$L^2$-method and some delicate L^∞$L^{\infty }$ estimates on solutions to the linearized problem, the existence and uniqueness of stationary solution are obtained by the contraction mapping principle. The proof of the stability result is given by an elementary energy method and relies on some intrinsic properties of the full compressible Navier–Stokes–Korteweg system.     

Advection-mediated competition in general environments

We consider a reaction–diffusion–advection system of two competing species with one of the species dispersing by random diffusion as well as a biased movement upward along resource gradient, while the other species by random diffusion only. It has been shown that, under some non-degeneracy conditions on the environment function, the two species always coexist when the advection is strong. In this paper, we show that for general smooth environment function, in contrast to what is known, there can be competitive exclusion when the advection is strong, and, we give a sharp criterion for coexistence that includes all previously considered cases. Moreover, when the domain is one-dimensional, we derive in the strong advection limit a system of two equations defined on different domains. Uniqueness of steady states of this non-standard system is obtained when one of the diffusion rates is large.     

Ill-posedness of nonlocal Burgers equations

Nonlocal generalizations of Burgers equation were derived in earlier work by Hunter [J.K. Hunter, Nonlinear surface waves, in: Current Progress in Hyberbolic Systems: Riemann Problems and Computations, Brunswick, ME, 1988, in: Contemp. Math., vol. 100, Amer. Math. Soc., 1989, pp. 185–202], and more recently by Benzoni-Gavage and Rosini [S. Benzoni-Gavage, M. Rosini, Weakly nonlinear surface waves and subsonic phase boundaries, Comput. Math. Appl. 57 (3–4) (2009) 1463–1484], as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage [S. Benzoni-Gavage, Local well-posedness of nonlocal Burgers equations, Differential Integral Equations 22 (3–4) (2009) 303–320] under an appropriate stability condition originally pointed out by Hunter. In this article, it is shown that the latter condition is not only sufficient for well-posedness in Sobolev spaces but also necessary. The main point of the analysis is to show that when the stability condition is violated, nonlocal Burgers equations reduce to second order elliptic PDEs. The resulting ill-posedness result encompasses various cases previously studied in the literature.     

On the behavior of the solution of the dissipative Camassa–Holm equation with the arbitrary dispersion coefficient

In this paper, we consider the dissipative Camassa–Holm equation with arbitrary dispersion coefficient and compactly supported initial data. We demonstrate the simple conditions on the initial data that lead to finite time blow-up of the solution in finite time or guarantee that the solution exists globally. Also, propagation speed for the equation under consideration is investigated.