An analytic solution is developed for the one-dimensional dissipational slip gradient equation first described by Gurtin [“On the plasticity of single crystals: free energy, microforces, plastic strain-gradients”, J. Mech. Phys. Solids48 (2000) 989–1036] and then investigated numerically by Anand et al. [“A one-dimensional theory of strain-gradient plasticity: formulation, analysis, numerical results”, J. Mech. Phys. Solids53 (2005) 1798–1826]. However we find that the analytic solution is incompatible with the zero-sliprate boundary condition (“clamped boundary condition”) postulated by these authors, and is in fact excluded by the theory. As a consequence the analytic solution agrees with the numerical results except near the boundary. The equation also admits a series of higher mode solutions where the numerical result corresponds to (a particular case of) the fundamental mode. Anand et al. also established that the one-dimensional dissipational gradients strengthen the material, but this proposition only holds if zero-sliprate boundary conditions can be imposed, which we have shown cannot be done. Hence the possibility remains open that dissipational gradient weakening may also occur.

A three-layer compartmental model of the geological structure in the Taupo Volcanic Zone of New Zealand is developed, based on the assumptions of isostasy (constant geostatic pressure at 25 km depth) and a constant rate of volcanism. The upper layer consists of volcanic infill to a depth of about 2500 m, then a middle layer of greywacke-like material to a depth of about 15 km, and a lower layer of andesitic-like material to a depth of 25 km. Our model assumptions predict that the area of each layer increases at a constant rate; that there is a constant ratio between the rate of energy production from volcanic activity and geothermal convection; and that there is the possibility of an abrupt change from rhyolitic to basaltic volcanism, if the middle layer becomes sufficiently thin. Two models are considered: a rifting and a spreading model. Both models predict the lower layer has an andesitic-like density. The spreading model has difficulty matching heat output with observed extension rates. The rifting model predicts the observed extension rates, but requires very deep circulation of groundwater to be consistent with observed chemical and isotopic properties of geothermal fluids.

We consider a model for the control of a linear network flow system with unknown but bounded demandand polytopic bounds on controlled flows. We are interested in the problem of finding a suitable objective functionthat makes robust optimal the policy represented by the so-called linear saturated feedback control. We regard the problem as a suitable differential game with switching cost and study it in the framework of the viscosity solutions theory for Bellman and Isaacs equations.

We investigate a global-in-time variational approach to abstract evolution by means of the weighted energy-dissipation functionals proposed by Mielke and Ortiz [ESAIM: COCV14 (2008) 494–516]. In particular, we focus on gradient flows in Hilbert spaces. The main result is the convergence of minimizers and approximate minimizers of these functionals to the unique solution of the gradient flow. Sharp convergence rates are provided and the convergence analysis is combined with time-discretization. Applications of the theory to various classes of parabolic PDE problems are presented. In particular, we focus on two examples of microstructure evolution from [S. Conti and M. Ortiz, J. Mech. Phys. Solids56 (2008) 1885–1904.].

We study the stabilization of global solutions of theKawahara (K) equation in a bounded interval, under the effect ofa localized damping mechanism. The Kawahara equation is a modelfor small amplitude long waves. Using multiplier techniques andcompactness arguments we prove the exponential decay of the solutions of the (K) model. The proofrequires of a unique continuation theorem and the smoothing effectof the (K) equation on the real line, which are proved in this work.

One shows that the linearized Navier-Stokes equation in ${\mathcal{O}}{\subset} R^d,\;d \ge 2$ , around an unstable equilibriumsolution is exponentially stabilizable in probability by aninternal noise controller $V(t,\xi)=\displaystyle\sum\limits_{i=1}^{N} V_i(t)\psi_i(\xi)\dot\beta_i(t)$ , $\xi\in{\mathcal{O}}$ , where $\{\beta_i\}^N_{i=1}$ areindependent Brownian motions in a probability space and $\{\psi_i\}^N_{i=1}$ is a system of functions on ${\mathcal{O}}$ withsupport in an arbitrary open subset ${\mathcal{O}}_0\subset {\mathcal{O}}$ . Thestochastic control input $\{V_i\}^N_{i=1}$ is found in feedbackform. One constructs also a tangential boundary noise controllerwhich exponentially stabilizes in probability the equilibriumsolution.

We investigate the evolution of an almost flat membrane driven by competition of the homogeneous, Frank, and bending energies as wellas the coupling of the local order of the constituent molecules of the membrane to its curvature.We propose an alternative to the model in [J.B. Fournier and P. Galatoa, J. Phys. II7 (1997) 1509–1520; N. Uchida, Phys. Rev. E66 (2002) 040902] which replacesa Ginzburg-Landau penalization for the length of theorder parameter by a rigid constraint.We introduce a fully discrete scheme, consisting of piecewise linearfinite elements, show that it is unconditionally stable for a large range of the elastic moduli in the model, and prove its convergence(up to subsequences) thereby proving the existence of a weak solutionto the continuous model. Numerical simulations are included that examine typical model situations, confirm our theory, and explore numerical predictions beyond that theory.

We provide an approximation of Mather variational problem by finite dimensional minimization problems in the framework of Γ-convergence. By a linear programming interpretation as done in [Evans and Gomes, ESAIM: COCV 8 (2002) 693–702] we state a duality theorem for the Mather problem, as well a finite dimensional approximation for the dual problem.

We provide a detailed analysis of the minimizers of the functional $u \mapsto \int_{\Bbb R^n} |\nabla u|^2 + D \int_{\Bbb R^n} |u|^{\gamma}$ , $\gamma \in (0, 2)$ , subject to the constraint $\|u\|_{L^2} = 1$ . This problem, e.g., describes the long-time behavior of the parabolic Anderson in probability theory or ground state solutions of a nonlinear Schrödinger equation. While existence can be proved with standard methods, we show that the usual uniqueness results obtained with PDE-methods can be considerably simplified by additional variational arguments. In addition, we investigate qualitative properties of the minimizers and also study their behavior near the critical exponent 2.

A new approach to irreversible quasistatic fracture growth is given, by means of Young measures.The study concerns a cohesive zone model with prescribed crack path, when the material gives different responses to loading and unloading phases.In the particular situation of constant unloading response,the result contained in [G. Dal Maso and C. Zanini,Proc. Roy. Soc. Edinburgh Sect. A137 (2007) 253–279] is recovered.In this case, the convergence of the discrete time approximationsis improved.

For a class of anisotropic integrodifferential operators ${\cal B}$ arising as semigroup generators of Markov processes, we present a sparse tensor product wavelet compression scheme for the Galerkin finite element discretization of the corresponding integrodifferential equations ${\cal B}$ u = f on [0,1]n with possibly large n. Under certain conditions on ${\cal B}$ , the scheme is of essentially optimal and dimension independent complexity $\mathcal{O}$ (h -1| log h |2(n-1)) without corrupting the convergence or smoothness requirements of the original sparse tensor finite element scheme. If the conditions on ${\cal B}$ are not satisfied, thecomplexity can be bounded by $\mathcal{O}$ (h -(1+ε)), whereε $\ll 1$ tends to zero with increasing number of the wavelets' vanishing moments. Here h denotes the width of the corresponding finite element mesh. The operators under consideration are assumed to be of non-negative (anisotropic) order and admit a non-standard kernel κ $(\cdot,\cdot)$ that can be singular on all secondary diagonals. Practical examples of such operators from Mathematical Finance are given and some numerical results are presented.

This paper is concerned with the numerical approximation of the solutions of a two-fluid two-pressure model used in the modelling of two-phase flows.We present a relaxation strategy for easily dealing with both the nonlinearities associated with the pressure laws and the nonconservative termsthat are inherently present in the set of convective equations and that couple the two phases.In particular, the proposed approximate Riemann solver is given by explicit formulas, preservesthe natural phase space, and exactly captures the coupling waves between the two phases.Numerical evidences are given to corroborate the validity of our approach.

We study a class of hyperbolic partial differential equations on a one dimensional spatial domain with control and observation at the boundary. Using the idea of feedback we show these systems are well-posed in the sense of Weiss and Salamon if and only if the state operator generates a C 0-semigroup. Furthermore, we show that the corresponding transfer function is regular, i.e., has a limit for s going to infinity.

We consider a variational formulation of the three-dimensional Navier–Stokes equations with mixed boundary conditions and prove that the variational problem admits a solution provided that the domain satisfies a suitable regularity assumption. Next, we propose a finite element discretization relying on the Galerkin method and establish a priori and a posteriori error estimates.


In a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations, the construction of lower bounds for the coercivity and inf-supstability constants is essential. In [Huynh et al., C. R. Acad.Sci. Paris Ser. I Math.345 (2007) 473–478], the authors presented an efficientmethod, compatible with an off-line/on-line strategy, where the on-line computation is reduced tominimizing a linear functional under a few linear constraints. These constraints depend on nested sets of parameters obtained iteratively using a greedy algorithm. We improve here this method so that it becomes more efficient and robust due to two related properties: (i) the lower bound isobtained by a monotonic process with respect to the size of the nested sets; (ii) less eigen-problems need to be solved. This improved evaluation of the inf-sup constant is then used to consider a reduced basis approximation of a parameter dependent electromagnetic cavity problem both for the greedy construction of the elements of the basis and the subsequent validation of the reduced basis approximation. The problem we consider has resonance features for some choices of the parameters that are well captured by the methodology.