The isothermal Navier–Stokes–Korteweg system is used to model dynamics of a compressiblefluid exhibiting phase transitions between a liquid and a vapor phase in the presence ofcapillarity effects close to phase boundaries. Standard numerical discretizations areknown to violate discrete versions of inherent energy inequalities, thus leading tospurious dynamics of computed solutions close to static equilibria (e.g.,parasitic currents). In this work, we propose a time-implicit discretization of theproblem, and use piecewise linear (or bilinear), globally continuous finite element spacesfor both, velocity and density fields, and two regularizing terms where correspondingparameters tend to zero as the mesh-size h > 0 tends to zero.Solvability, non-negativity of computed densities, as well as conservation of mass, and adiscrete energy law to control dynamics are shown. Computational experiments are providedto study interesting regimes of coefficients for viscosity and capillarity.

In this paper, we first construct a model for free surface flows that takes into accountthe air entrainment by a system of four partial differential equations. We derive it bytaking averaged values of gas and fluid velocities on the cross surface flow in the Eulerequations (incompressible for the fluid and compressible for the gas). The obtained systemis conditionally hyperbolic. Then, we propose a mathematical kinetic interpretation ofthis system to finally construct a two-layer kinetic scheme in which a special treatmentfor the “missing” boundary condition is performed. Several numerical tests on closed waterpipes are performed and the impact of the loss of hyperbolicity is discussed andillustrated. Finally, we make a numerical study of the order of the kinetic method in thecase where the system is mainly non hyperbolic. This provides a useful stability resultwhen the spatial mesh size goes to zero.

A weak solution of the coupling of time-dependent incompressible Navier–Stokes equationswith Darcy equations is defined. The interface conditions include theBeavers–Joseph–Saffman condition. Existence and uniqueness of the weak solution areobtained by a constructive approach. The analysis is valid for weak regularityinterfaces.

A corrector theory for the strong approximation of gradient fields inside periodiccomposites made from two materials with different power law behavior is provided. Eachmaterial component has a distinctly different exponent appearing in the constitutive lawrelating gradient to flux. The correctors are used to develop bounds on the localsingularity strength for gradient fields inside micro-structured media. The bounds aremulti-scale in nature and can be used to measure the amplification of applied macroscopicfields by the microstructure. The results in this paper are developed for materials havingpower law exponents strictly between  −1 and zero.

Mathematical models for option pricing often result in partial differential equations.Recent enhancements are models driven by Lévy processes, which lead to a partialdifferential equation with an additional integral term. In the context of modelcalibration, these partial integro differential equations need to be solved quitefrequently. To reduce the computational cost the implementation of a reduced order modelhas shown to be very successful numerically. In this paper we give a priorierror estimates for the use of the proper orthogonal decomposition technique inthe context of option pricing models.

Nowadays, the Coupled Cluster (CC) method is the probably most widely used high precisionmethod for the solution of the main equation of electronic structure calculation, thestationary electronic Schrödinger equation. Traditionally, theequations of CC are formulated as a nonlinear approximation of a Galerkin solution of theelectronic Schrödinger equation, i.e. within a given discrete subspace.Unfortunately, this concept prohibits the direct application of concepts of nonlinearnumerical analysis to obtain e.g. existence and uniqueness results orestimates on the convergence of discrete solutions to the full solution. Here, thisshortcoming is approached by showing that based on the choice of anN-dimensional reference subspace R of H1(ℝ3 ×{± 1/2}), the original, continuous electronic Schrödingerequation can be reformulated equivalently as a root equation for an infinite-dimensionalnonlinear Coupled Cluster operator. The canonical projected CC equations may then beunderstood as discretizations of this operator. As the main step, continuity properties ofthe cluster operator S and its adjoint S† asmappings on the antisymmetric energy space H1 are established.

We present a reduced basis offline/online procedure for viscous Burgers initial boundaryvalue problem, enabling efficient approximate computation of the solutions of thisequation for parametrized viscosity and initial and boundary value data. This procedurecomes with a fast-evaluated rigorous error bound certifying the approximation procedure.Our numerical experiments show significant computational savings, as well as efficiency ofthe error bound.

Resistance to chemotherapies, particularly to anticancer treatments, is an increasingmedical concern. Among the many mechanisms at work in cancers, one of the most importantis the selection of tumor cells expressing resistance genes or phenotypes. Motivated bythe theory of mutation-selection in adaptive evolution, we propose a model based on acontinuous variable that represents the expression level of a resistance gene (or genes,yielding a phenotype) influencing in healthy and tumor cells birth/death rates, effects ofchemotherapies (both cytotoxic and cytostatic) and mutations. We extend previous work bydemonstrating how qualitatively different actions of chemotherapeutic and cytostatictreatments may induce different levels of resistance. The mathematical interest of ourstudy is in the formalism of constrained Hamilton–Jacobi equations in the framework ofviscosity solutions. We derive the long-term temporal dynamics of the fittest traits inthe regime of small mutations. In the context of adaptive cancer management, we alsoanalyse whether an optimal drug level is better than the maximal tolerated dose.

The least Steklov eigenvalue d1 for the biharmonic operatorin bounded domains gives a bound for the positivity preserving property for the hingedplate problem, appears as a norm of a suitable trace operator, and gives the optimalconstant to estimate the L2-norm of harmonic functions. Theseapplications suggest to address the problem of minimizing d1in suitable classes of domains. We survey the existing results and conjectures about thistopic; in particular, the existence of a convex domain of fixed measure minimizingd1 is known, although the optimal shape is still unknown. Weperform several numerical experiments which strongly suggest that the optimal planar shapeis the regular pentagon. We prove the existence of a domain minimizingd1 also among convex domains having fixed perimeter andpresent some numerical results supporting the conjecture that, among planar domains, thedisk is the minimizer.

In this paper, we study a class of Initial-Boundary Value Problems proposed by Colin andGhidaglia for the Korteweg-de Vries equation posed on a bounded domain(0,L). We show that this class of Initial-Boundary Value Problems islocally well-posed in the classical Sobolev spaceHs(0,L) for s > -3/4, which provides a positive answer to one of the openquestions of Colin and Ghidaglia [Adv. Differ. Equ. 6 (2001)1463–1492].

Dynamics of the nonlinear Schrödinger equation in the presence of a constant electricfield is studied. Both discrete and continuous limits of the model are considered. For thediscrete limit, a probabilistic description of subdiffusion is suggested and asubdiffusive spreading of a wave packet is explained in the framework of a continuous timerandom walk. In the continuous limit, the biased nonlinear Schrödinger equation is shownto be integrable, and solutions in the form of the Painlevé transcendents are obtained.

An overview of recently obtained authors’ results on traveling wave solutions of someclasses of PDEs is presented. The main aim is to describe all possible travelling wavesolutions of the equations. The analysis was conducted using the methods of qualitativeand bifurcation analysis in order to study the phase-parameter space of the correspondingwave systems of ODEs. In the first part we analyze the wave dynamic modes of populationsdescribed by the “growth - taxis - diffusion" polynomial models. It is shown that“suitable" nonlinear taxis can affect the wave front sets and generate non-monotone waves,such as trains and pulses, which represent the exact solutions of the model system.Parametric critical points whose neighborhood displays the full spectrum of possible modelwave regimes are identified; the wave mode systematization is given in the form ofbifurcation diagrams. In the second part we study a modified version of theFitzHugh-Nagumo equations, which model the spatial propagation of neuron firing. We assumethat this propagation is (at least, partially) caused by the cross-diffusion connectionbetween the potential and recovery variables. We show that the cross-diffusion version ofthe model, besides giving rise to the typical fast travelling wave solution exhibited inthe original “diffusion" FitzHugh-Nagumo equations, additionally gives rise to a slowtraveling wave solution. We analyze all possible traveling wave solutions of the model andshow that there exists a threshold of the cross-diffusion coefficient (for a given speedof propagation), which bounds the area where “normal" impulse propagation is possible. Inthe third part we describe all possible wave solutions for a class of PDEs withcross-diffusion, which fall in a general class of the classical Keller-Segel modelsdescribing chemotaxis. Conditions for existence of front-impulse, impulse-front, andfront-front traveling wave solutions are formulated. In particular, we show that anon-isolated singular point in the ODE wave system implies existence of free-boundaryfronts.

Stockwell transforms as hybrids of Gabor transforms and wavelet transforms have beenstudied extensively. We introduce in this paper multi-dimensional Stockwell transformsthat include multi-dimensional Gabor transforms as special cases. Continuous inversionformulas for multi-dimensional Stockwell transforms are proved.

One of the central results in Einstein’s theory of Brownian motion is that the meansquare displacement of a randomly moving Brownian particle scales linearly with time. Overthe past few decades sophisticated experiments and data collection in numerous biological,physical and financial systems have revealed anomalous sub-diffusion in which the meansquare displacement grows slower than linearly with time. A major theoretical challengehas been to derive the appropriate evolution equation for the probability density functionof sub-diffusion taking into account further complications from force fields andreactions. Here we present a derivation of the generalised master equation for an ensembleof particles undergoing reactions whilst being subject to an external force field. Fromthis general equation we show reductions to a range of well known special cases, includingthe fractional reaction diffusion equation and the fractional Fokker-Planck equation.

New conservation laws bifurcating from the classical form of conservation laws are constructed to the nonlinear Boussinesq model describing internal Kelvin waves propagating in a cylindrical wave field of an uniformly stratified water affected by the earth’s rotation. The obtained conservation laws are different from the well known energy conservation law for internal waves and they are associated with symmetries of the Boussinesq model. Particularly, it is shown that application of Lie group analysis provide three infinite sets of nontrivial integral conservation laws depending on two arbitrary functions, namely a(t, θ), b(t, r) and an arbitrary function c(t, θ, r) which is given implicitly as a nontrivial solution of a partial differential equation involving a(t, θ) and b(t, r).

We review recent stability and separation results in volume comparison problems and usethem to prove several hyperplane inequalities for intersection and projection bodies.

The stability of a one-spike solution to a general class of reaction-diffusion (RD)system with both regular and anomalous diffusion is analyzed. The method of matchedasymptotic expansions is used to construct a one-spike equilibrium solution and to derivea nonlocal eigenvalue problem (NLEP) that determines the stability of this solution on an O(1) time-scale. For a particular sub-class of the reactionkinetics, it is shown that the discrete spectrum of this NLEP is determined in terms ofthe roots of certain simple transcendental equations that involve two key parametersrelated to the choice of the nonlinear kinetics. From a rigorous analysis of thesetranscendental equations by using a winding number approach and explicit calculations,sufficient conditions are given to predict the occurrence of Hopf bifurcations of theone-spike solution. Our analysis determines explicitly the number of possible Hopfbifurcation points as well as providing analytical formulae for them. The analysis isimplemented for the shadow limit of the RD system defined on a finite domain and for aone-spike solution of the RD system on the infinite line. The theory is illustrated fortwo specific RD systems. Finally, in parameter ranges for which the Hopf bifurcation isunique, it is shown that the effect of sub-diffusion is to delay the onset of the Hopfbifurcation.

We study origin, parameter optimization, and thermodynamic efficiency of isothermalrocking ratchets based on fractional subdiffusion within a generalized non-MarkovianLangevin equation approach. A corresponding multi-dimensional Markovian embedding dynamicsis realized using a set of auxiliary Brownian particles elastically coupled to the centralBrownian particle (see video on the journal web site). We show that anomalous subdiffusivetransport emerges due to an interplay of nonlinear response and viscoelastic effects forfractional Brownian motion in periodic potentials with broken space-inversion symmetry anddriven by a time-periodic field. The anomalous transport becomes optimal for asubthreshold driving when the driving period matches a characteristic time scale ofinterwell transitions. It can also be optimized by varying temperature, amplitude ofperiodic potential and driving strength. The useful work done against a load shows aparabolic dependence on the load strength. It grows sublinearly with time and thecorresponding thermodynamic efficiency decays algebraically in time because the energysupplied by the driving field scales with time linearly. However, it compares well withthe efficiency of normal diffusion rocking ratchets on an appreciably long time scale.