104,780 research outputs found
Global regularity and asymptotic stabilization for the incompressible Navier–Stokes-Cahn–Hilliard model with unmatched densities
No full textWe study an initial-boundary value problem for the incompressible Navier–Stokes–Cahn–Hilliard system with non-constant density proposed by Abels, Garcke and Grün in 2012. This model arises in the diffuse interface theory for binary mixtures of viscous incompressible fluids. This system is a generalization of the well-known model H in the case of fluids with unmatched densities. In three dimensions, we prove that any global weak solution (for which uniqueness is not known) exhibits a propagation of regularity in time and stabilizes towards an equilibrium state as t→∞. More precisely, the concentration function φ is a strong solution of the Cahn–Hilliard equation for (arbitrary) positive times, whereas the velocity field u becomes a strong solution of the momentum equation for large times. Our analysis hinges upon the following key points: a novel global regularity result (with explicit bounds) for the Cahn–Hilliard equation with divergence-free velocity belonging only to L^2(0,∞;H0,σ^1(Ω)), the energy dissipation of the system, the separation property for large times, a weak-strong uniqueness type result, and the Lojasiewicz–Simon inequality. Additionally, in two dimensions, we show the existence and uniqueness of global strong solutions for the full system. Finally, we discuss the existence of global weak solutions for the case of the double obstacle potential
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No full text"Mit vier Ansichten"Vorlageform der Veröffentlichungsangabe: "Merseburg, Louis Garcke. 1848." - Kolophon: "Halle, Druck von H. W. Schmidt
On a phase field model of Cahn–Hilliard type for tumour growth with mechanical effects
No full textMechanical effects have mostly been neglected so far in phase field tumour models that are based on a Cahn–Hilliard approach. In this paper we study a macroscopic mechanical model for tumour growth in which cell–cell adhesion effects are taken into account with the help of a Ginzburg–Landau type energy. In the overall model an equation of Cahn–Hilliard type is coupled to the system of linear elasticity and a reaction–diffusion equation for a nutrient concentration. The highly non-linear coupling between a fourth-order Cahn–Hilliard equation and the quasi-static elasticity system lead to new challenges which cannot be dealt within a gradient flow setting which was the method of choice for other elastic Cahn–Hilliard systems. We show existence, uniqueness and regularity results. In addition, several continuous dependence results with respect to different topologies are shown. Some of these results give uniqueness for weak solutions and other results will be helpful for optimal control problems
Willmore flow of planar networks
No full textGeometric gradient flows for elastic energies of Willmore type play an important role in mathematics and in many applications. The evolution of elastic curves has been studied in detail both for closed as well as for open curves. Although elastic flows for networks also have many interesting features, they have not been studied so far from the point of view of mathematical analysis. So far it was not even clear what are appropriate boundary conditions at junctions. In this paper we give a well-posedness result for Willmore flow of networks in different geometric settings and hence lay a foundation for further mathematical analysis. A main point in the proof is to check whether different proposed boundary conditions lead to a well posed problem. In this context one has to check the Lopatinskii–Shapiro condition in order to apply the Solonnikov theory for linear parabolic systems in Hölder spaces which is needed in a fixed point argument. We also show that the solution we get is unique in a purely geometric sense
On a fourth-order degenerate parabolic equation: global entropy estimates, existence, and qualitative behavior of solutions
No full textBy means of energy and entropy estimates, we prove existence and positivity results in higher space dimensions for degenerate parabolic equations of fourth order with nonnegative initial values. We discuss their asymptotic behavior for t --> infinity and give a counterexample to uniqueness
Sparse optimal control of a phase field tumor model with mechanical effects
No full textIn this paper, we study an optimal control problem for a macroscopic mechanical tumor model based on the phase field approach. The model couples a Cahn-Hilliard-type equation to a system of linear elasticity and a reaction-diffusion equation for a nutrient concentration. By taking advantage of previous analytical well-posedness results established by the authors, we seek optimal controls in the form of a boundary nutrient supply as well as concentrations of cytotoxic and antiangiogenic drugs that minimize a cost functional involving mechanical stresses. Special attention is given to sparsity effects, where with the inclusion of convex nondifferentiable regularization terms to the cost functional, we can infer from the first-order optimality conditions that the optimal drug concentrations can vanish on certain time intervals
Long time existence of solutions to an elastic flow of networks
No full textThe L2-gradient flow of the elastic energy of networks leads to a Willmore type evolution law with non-trivial nonlinear boundary conditions. We show local in time existence and uniqueness for this elastic flow of networks in a Sobolev space setting under natural boundary conditions. In addition, we show a regularisation property and geometric existence and uniqueness. The main result is a long time existence result using energy methods
A dimension adaptive sparse grid combination technique for machine learning
No full textWe introduce a dimension adaptive sparse grid combination technique for the machine learning problems of classification and regression. A function over a -dimensional space, which assumedly describes the relationship between the features and the response variable, is reconstructed using a linear combination of partial functions that possibly depend only on a subset of all features. The partial functions are adaptively chosen during the computational procedure. This approach (approximately) identifies the \textsc{anova} decomposition of the underlying problem. Experiments on synthetic data, where the structure is known, show the advantages of a dimension adaptive combination technique in run time behaviour, approximation errors, and interpretability.
References M. Griebel, M. Schneider, and C. Zenger. A combination technique for the solution of sparse grid problems. In P. de Groen and R. Beauwens, editors, Iterative Methods in Linear Algebra, pages 263--281. IMACS, Elsevier, North Holland, 1992. http://wissrech.ins.uni-bonn.de/research/pub/griebel/griesiam.ps.gz. M. Hegland. Adaptive sparse grids. In K. Burrage and Roger B. Sidje, editors, Proc. of 10th Computational Techniques and Applications Conference CTAC-2001, volume 44 of ANZIAM J., pages C335--C353, 2003. http://anziamj.austms.org.au/V44/CTAC2001/Hegl. M. Hegland, J. Garcke, and V. Challis. The combination technique and some generalisations. Linear Algebra and its Applications, 420(2--3):249--275, 2007. doi:10.1016/j.laa.2006.07.014. Ian H. Sloan, Xiaoqun Wang, and Henryk Wozniakowski. Finite-order weights imply tractability of multivariate integration. Journal of Complexity, 20:46--74, 2004. doi:10.1016/j.jco.2003.11.003. Jerome H. Friedman. Multivariate adaptive regression splines. Ann. Statist., 19(1):1--141, 1991. http://projecteuclid.org/euclid.aos/1176347963. J. Garcke. Regression with the optimised combination technique. In W. Cohen and A. Moore, editors, Proceedings of the 23rd ICML, pages 321--328, 2006. doi:10.1007/s006070170007. J. Garcke, M. Griebel, and M. Thess. Data mining with sparse grids. Computing, 67(3):225--253, 2001. doi:10.1007/s006070170007. T. Gerstner and M. Griebel. Dimension-Adaptive Tensor-Product Quadrature. Computing, 71(1):65--87, 2003. doi:10.1007/s00607-003-0015-5
A Hele–Shaw–Cahn–Hilliard Model for Incompressible Two-Phase Flows with Different Densities
No full textTopology changes in multi-phase fluid flows are difficult to model within a traditional sharp interface theory. Diffuse interface models turn out to be an attractive alternative to model two-phase flows. Based on a Cahn-Hilliard-Navier-Stokes model introduced by Abels et al. (Math Models Methods Appl Sci 22(3):1150013, 2012), which uses a volume-averaged velocity, we derive a diffuse interface model in a Hele-Shaw geometry, which in the case of non-matched densities, simplifies an earlier model of Lee et al. (Phys Fluids 14(2):514-545, 2002). We recover the classical Hele-Shaw model as a sharp interface limit of the diffuse interface model. Furthermore, we show the existence of weak solutions and present several numerical computations including situations with rising bubbles and fingering instabilities
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