Local well‐posedness and ill‐posedness for the fractal Burgers equation in homogeneous Sobolev spaces

In this paper, we consider local well‐posedness and ill‐posedness questions for the fractal Burgers equation. First, we obtain the well‐posedness result in the critical Sobolev space. We also present an unconditional uniqueness result. Second, we show the ill‐posedness from the point of view of the Picard iterations in the supercritical space. Copyright © 2008 John Wiley & Sons, Ltd.     

On integro‐differential inclusions with operator‐valued kernels

We study integro‐differential inclusions in Hilbert spaces with operator‐valued kernels and give sufficient conditions for the well‐posedness. We show that several types of integro‐differential equations and inclusions are covered by the class of evolutionary inclusions, and we therefore give criteria for the well‐posedness within this framework. As an example, we apply our results to the equations of visco‐elasticity and to a class of nonlinear integro‐differential inclusions describing phase transition phenomena in materials with memory. Copyright © 2014 John Wiley & Sons, Ltd.     

On evolutionary equations with material laws containing fractional integrals

A well‐posedness result for a time‐shift invariant class of evolutionary operator equations involving material laws with fractional time‐integrals of order α ? ]0, 1[ is considered. The fractional derivatives are defined via a function calculus for the (time‐)derivative established as a normal operator in a suitable L² type space. Employing causality, we show that the fractional derivatives thus obtained coincide with the Riemann‐Liouville fractional derivative. We exemplify our results by applications to a fractional Fokker‐Planck equation, equations describing super‐diffusion and sub‐diffusion processes, and a Kelvin‐Voigt type model in fractional visco‐elasticity. Moreover, we elaborate a suitable perspective to deal with initial boundary value problems. Copyright © 2014 John Wiley & Sons, Ltd.     

Exact boundary controllability for non‐autonomous quasilinear wave equations

In this paper, we first show that quite different from the autonomous case, the exact boundary controllability for non‐autonomous wave equations possesses various possibilities. Then we adopt a constructive method to establish the exact boundary controllability for one‐dimensional non‐autonomous quasilinear wave equations with various types of boundary conditions. Finally, we apply the results to multi‐dimensional quasilinear wave equation with rotation invariance. Copyright © 2007 John Wiley & Sons, Ltd.     

Transcendental lattices of some K 3‐surfaces

In a previous paper, [12], we described six families of K 3‐surfaces (over ?) with Picard‐number 19, and we identified surfaces with Picard‐number 20. In these notes we classify some of the surfaces by computing their transcendental lattices. Moreover, we show that the surfaces with Picard‐number 19 are birational to a Kummer surface which is the quotient of a non‐product type abelian surface by an involution. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

High‐order split‐step theta methods for non‐autonomous stochastic differential equations with non‐globally Lipschitz continuous coefficients

In this paper, we first propose the so‐called improved split‐step theta methods for non‐autonomous stochastic differential equations driven by non‐commutative noise. Then, we prove that the improved split‐step theta method is convergent with strong order of one for stochastic differential equations with the drift coefficient satisfying a superlinearly growing condition and a one‐sided Lipschitz continuous condition. Finally, the obtained results are verified by numerical experiments. Copyright © 2016 John Wiley & Sons, Ltd.     

Well‐posedness of time‐space fractional stochastic evolution equations driven by α‐stable noise

This paper is devoted to the well‐posedness for time‐space fractional Ginzburg‐Landau equation and time‐space fractional Navier‐Stokes equations by α‐stable noise. The spatial regularity and the temporal regularity of the nonlocal stochastic convolution are firstly established, and then the existence and uniqueness of the global mild solution are obtained by the Banach fixed point theorem and Mittag‐Leffler functions, respectively. Numerical simulations for time‐space fractional Ginzburg‐Landau equation are provided to verify the analysis results.     

Local well‐posedness and blow‐up criteria of solutions for a rod equation

In this paper we consider a new rod equation derived recently by Dai [Acta Mech. 127 No. 1–4, 193–207 (1998)] for a compressible hyperelastic material. We establish local well‐posedness for regular initial data and explore various sufficient conditions of the initial data which guarantee the blow‐up in finite time both for periodic and non‐periodic case. Moreover, the blow‐up time and blow‐up rate are given explicitly. Some interesting examples are given also. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of solutions to a phase‐field model for the isothermal solidification process of a binary alloy

We investigate the well‐posedness of a phase‐field model for the isothermal solidification of a binary alloy due to Warren–Boettinger [12]. Existence of weak solution as well as regularity and uniqueness results are established under Lipschitz and boundedness assumptions for the non‐linearities. A maximum principle holds that guarantees the existence of a solution under physical assumptions on the non‐linearities. Copyright © 2000 John Wiley & Sons, Ltd.     

On the well‐posedness of a mathematical model of quorum‐sensing in patchy biofilm communities

We analyze a system of reaction–diffusion equations that models quorum‐sensing in a growing biofilm. The model comprises two nonlinear diffusion effects: a porous medium‐type degeneracy and super diffusion. We prove the well‐posedness of the model. In particular, we present for the first time a uniqueness result for this type of problem. Moreover, we illustrate the behavior of model solutions in numerical simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

Well‐posedness of non‐autonomous linear evolution equations in uniformly convex spaces

This paper addresses the problem of well‐posedness of non‐autonomous linear evolution equations in uniformly convex Banach spaces. We assume that for each t is the generator of a quasi‐contractive, strongly continuous group, where the domain D and the growth exponent are independent of t . Well‐posedness holds provided that is Lipschitz for all . Hölder continuity of degree is not sufficient and the assumption of uniform convexity cannot be dropped.     

Local well‐posedness and blow‐up criterion for a compressible Navier‐Stokes‐Fourier‐P1 approximate model arising in radiation hydrodynamics

We establish a local well‐posedness and a blow‐up criterion of strong solutions for the compressible Navier‐Stokes‐Fourier‐P1 approximate model arising in radiation hydrodynamics. For the local well‐posedness result, we do not need the assumption on the positivity of the initial density and it may vanish in an open subset of the domain.     

Oseen coupling method for the exterior flow. Part II: Well‐posedness analysis

In this paper, we recall the Oseen coupling method for solving the exterior unsteady Navier–Stokes equations with the non‐homogeneous boundary conditions. Moreover, we derive the coupling variational formulation of the Oseen coupling problem by using of the integral representations of the solution of the Oseen equations at an infinity domain. Finally, we provide some properties of the integral operators over the artificial boundary and the well‐posedness of the coupling variational formulation. Copyright © 2004 John Wiley & Sons, Ltd.     

Analysis and numerics for an age‐ and sex‐structured population model

We study a linear model of McKendrick‐von Foerster‐Keyfitz type for the temporal development of the age structure of a two‐sex human population. For the underlying system of partial integro‐differential equations, we exploit the semigroup theory to show the classical well‐posedness and asymptotic stability in a Hilbert space framework under appropriate conditions on the age‐specific mortality and fertility moduli. Finally, we propose an implicit finite difference scheme to numerically solve this problem and prove its convergence under minimal regularity assumptions. A real data application is also given. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 706–736, 2016     

On the global well‐posedness for the two‐dimensional Boussinesq equations with horizontal dissipation

In this paper, we study the global well‐posedness for the two‐dimensional nonlinear Boussinesq equations with horizontal dissipation. The method we adopted is the smoothing effect in horizontal direction and the low‐high decomposition technique.     

Well‐posedness for fractional Navier–Stokes equations in the largest critical spaces

This note studies the well‐posedness of the fractional Navier–Stokes equations in some supercritical Besov spaces as well as in the largest critical spaces for β ∈ (1/2,1). Meanwhile, the well‐posedness for fractional magnetohydrodynamics equations in these Besov spaces is also studied. Copyright © 2012 John Wiley & Sons, Ltd.     

Pullback attractors of 2D incompressible Navier‐Stokes‐Voight equations with delay

In this paper, the 2D Navier‐Stokes‐Voight equations with 3 delays in is considered. By using the Faedo‐Galerkin method, Lions‐Aubin lemma, and Arzelà‐Ascoli theorem, we establish the global well‐posedness of solutions and the existence of pullback attractors in H¹.     

Global well‐posedness for free boundary problem of the Oldroyd‐B model fluid flow

In this paper, we prove the global well‐posedness of non‐Newtonian viscous fluid flow of the Oldroyd‐B model with free surface in a bounded domain of N‐dimensional Euclidean space . The assumption of the problem is that the initial data are small enough and orthogonal to rigid motions. Copyright © 2015 John Wiley & Sons, Ltd.     

Web‐spline‐based finite element approximation of some quasi‐newtonian flows: Existence‐uniqueness and error bound

This article deals with the web‐spline‐based finite element approximation of quasi‐Newtonian flows. First, we consider the scalar elliptic p‐Laplace problem. Then, we consider quasi‐Newtonian flows where viscosity obeys power law or Carreau law. We prove well‐posedness at the continuous as well as the discrete level. We give some error bounds for the solution of quasi‐Newtonian flow problem based on the web‐spline method. Finally, we provide the numerical results for the p‐Laplace problem. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq31: 54–77, 2015     

On well‐posedness for some thermo‐piezoelectric coupling models

There is an increasing reliance on mathematical modelling to assist in the design of piezoelectric ultrasonic transducers since this provides a cost‐effective and quick way to arrive at a first prototype. Given a desired operating envelope for the sensor, the inverse problem of obtaining the associated design parameters within the model can be considered. It is therefore of practical interest to examine the well‐posedness of such models. There is a need to extend the use of such sensors into high‐temperature environments, and so this paper shows, for a broad class of models, the well‐posedness of the magneto‐electro‐thermo‐elastic problem. Because of its widespread use in the literature, we also show the well‐posedness of the quasi‐electrostatic case. Copyright © 2016 John Wiley & Sons, Ltd.