110 results
Search Results
Now showing 1 - 10 of 110
- ItemExistence, numerical convergence, and evolutionary relaxation for a rate-independent phase-transformation model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Heinz, Sebastian; Mielke, AlexanderWe revisit the two-well model for phase transformation in a linearly elastic body introduced and studied in [MTL02]. This energetic rate-independent model is posed in terms of the elastic displacement and an internal variable that gives the phase portion of the second phase. We use a new approach based on mutual recovery sequences, which are adjusted to a suitable energy increment plus the associated dissipated energy and, thus, enable us to pass to the limit in the construction of energetic solutions. We give three distinct constructions of mutual recovery sequences which allow us (i) to generalize the existence result in [MTL02], (ii) to establish the convergence of suitable the evolutionary relaxation from the pure-state model to the relaxed mixture model. All these results rely on weak converge and involve the H-measure as an essential tool.
- ItemOptimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Liero, Matthias; Mielke, Alexander; Savaré, GiuseppeWe develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. They arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a couple of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, that quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger-Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger-Kakutani and Kantorovich-Wasserstein distances.
- ItemBlow-up versus boundedness in a nonlocal and nonlinear Fokker-Planck equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Dreyer, Wolfgang; Huth, Robert; Mielke, Alexander; Rehberg, Joachim; Winkler, MichaelLiteraturverz.
- ItemGlobal existence analysis of energy-reaction-diffusion systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Fischer, Julian; Hopf, Katharina; Kniely, Michael; Mielke, AlexanderWe establish global-in-time existence results for thermodynamically consistent reaction-(cross-)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model species-dependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the non-isothermal case lies in the intrinsic presence of cross-diffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where non-integrable diffusion fluxes or reaction terms appear.
- ItemDifferential, energetic, and metric formulations for rate-independent processes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Mielke, AlexanderWe consider different solution concepts for rate-independent systems. This includes energetic solutions in the topological setting and differentiable, local, parametrized and BV solutions in the Banach-space setting. The latter two solution concepts rely on the method of vanishing viscosity, in which solutions of the rate-independent system are defined as limits of solutions of systems with small viscosity. Finally, we also show how the theory of metric evolutionary systems can be used to define parametrized and BV solutions in metric spaces.
- ItemGlobal existence for rate-independent gradient plasticity at finite strain(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Mainik, Andreas; Mielke, AlexanderWe provide a global existence result for the time-continuous elastoplasticity problem using the energetic formulation. For this we show that the geometric nonlinearities via the multiplicative decomposition of the strain can be controlled via polyconvexity and a priori stress bounds in terms of the energy density. While temporal oscillations are controlled via the energy dissipation the spatial compactness is obtain via the regularizing terms involving gradients of the internal variables.
- ItemBV solutions and viscosity approximations of rate-independent systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Mielke, Alexander; Rossi, Riccarda; Savaré, GiuseppeIn the nonconvex case solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential which is a viscous regularization of a given rate-independent dissipation potential. The resulting definition of `BV solutions' involves, in a nontrivial way, both the rate-independent and the viscous dissipation potential, which play a crucial role in the description of the associated jump trajectories. We shall prove a general convergence result for the time-continuous and for the time-discretized viscous approximations and establish various properties of the limiting $BV$ solutions. In particular, we shall provide a careful description of the jumps and compare the new notion of solutions with the related concepts of energetic and local solutions to rate-independent systems.
- ItemNonsmooth analysis of doubly nonlinear evolution equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Mielke, Alexander; Rossi, Riccarda; Savaré, GiuseppeIn this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy functional, for existence of solutions to the related Cauchy problem. We prove our main existence result by passing to the limit in a time-discretization scheme with variational techniques. Finally, we discuss an application to a material model in finite-strain elasticity.
- ItemBalanced viscosity (BV) solutions to infinite-dimensional rate-independent systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Mielke, Alexander; Rossi, Riccarda; Savaré, GiuseppeBalanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-independent flows by adding a superlinear vanishing-viscosity dissipation. We address the main issue of proving the existence of such limits for infinite-dimensional systems and of characterizing them by a couple of variational properties that combine a local stability condition and a balanced energy-dissipation identity. A careful description of the jump behavior of the solutions, of their differentiability properties, and of their equivalent representation by time rescaling is also presented. Our techniques rely on a suitable chain-rule inequality for functions of bounded variation in Banach spaces, on refined lower semicontinuity-compactness arguments, and on new BV-estimates that are of independent interest.
- ItemMulti-pulse evolution and space-time chaos in dissipative systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Zelik, Sergey; Mielke, AlexanderWe study semilinear parabolic systems on the full space Rn that admit a family of exponentially decaying pulse-like steady states obtained via translations. The multi-pulse solutions under consideration look like the sum of infinitely many such pulses which are well separated. We prove a global center-manifold reduction theorem for the temporal evolution of such multi-pulse solutions and show that the dynamics of these solutions can be described by an infinite systems of ODEs for the positions of the pulses. As an application of the developed theory, we verify the existence of Sinai-Bunimovich space-time chaos in 1D space-time periodically forced Swift-Hohenberg equation.