We consider a strongly coupled PDE-ODE system that describes the influence of a slow and large vehicle on road traffic. The model consists of a scalar conservation law describing the main traffic evolution and an ODE accounting for the trajectory of the slower vehicle that depends on the downstream traffic density. The moving constraint is operated by an inequality on the flux, which accounts for the bottleneck created on the road by the presence of the slower vehicle. We introduce a conservative scheme for the constrained hyperbolic PDE and a tracking algorithm for the ODE. We show numerical tests and compute numerically the order of convergence.
{"title":"A conservative scheme for non-classical solutions to a strongly coupled PDE-ODE problem","authors":"C. Chalons, M. D. Monache, P. Goatin","doi":"10.4171/IFB/392","DOIUrl":"https://doi.org/10.4171/IFB/392","url":null,"abstract":"We consider a strongly coupled PDE-ODE system that describes the influence of a slow and large vehicle on road traffic. The model consists of a scalar conservation law describing the main traffic evolution and an ODE accounting for the trajectory of the slower vehicle that depends on the downstream traffic density. The moving constraint is operated by an inequality on the flux, which accounts for the bottleneck created on the road by the presence of the slower vehicle. We introduce a conservative scheme for the constrained hyperbolic PDE and a tracking algorithm for the ODE. We show numerical tests and compute numerically the order of convergence.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80755745","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present the analysis of advection-diffusion equations with random coefficients on moving hypersurfaces. We define weak and strong material derivative, that take into account also the spacial movement. Then we define the solution space for these kind of equations, which is the Bochner-type space of random functions defined on moving domain. Under suitable regularity assumptions we prove the existence and uniqueness of solutions of the concerned equation, and also we give some regularity results about the solution.
{"title":"Advection-diffusion equations with random coefficientson evolving hypersurfaces","authors":"A. Djurdjevac","doi":"10.4171/IFB/391","DOIUrl":"https://doi.org/10.4171/IFB/391","url":null,"abstract":"We present the analysis of advection-diffusion equations with random coefficients on moving hypersurfaces. We define weak and strong material derivative, that take into account also the spacial movement. Then we define the solution space for these kind of equations, which is the Bochner-type space of random functions defined on moving domain. Under suitable regularity assumptions we prove the existence and uniqueness of solutions of the concerned equation, and also we give some regularity results about the solution.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89059056","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a discrete-continuum model of a biomembrane with embedded particles. While the membrane is represented by a continuous surface, embedded particles are described by rigid discrete objects which are free to move and rotate in lateral direction. For the membrane we consider a linearized Canham-Helfrich energy functional and height and slope boundary conditions imposed on the particle boundaries resulting in a coupled minimization problem for the membrane shape and particle positions. When considering the energetically optimal membrane shape for each particle position we obtain a reduced energy functional that models the implicitly given interaction potential for the membrane-mediated mechanical particle-particle interactions. We show that this interaction potential is differentiable with respect to the particle positions and orientations. Furthermore we derive a fully practical representation of the derivative only in terms of well defined derivatives of the membrane. This opens the door for the application of minimization algorithms for the computation of minimizers of the coupled system and for further investigation of the interaction potential of membrane-mediated mechanical particle-particle interaction. The results are illustrated with numerical examples comparing the explicit derivative formula with difference quotient approximations. We furthermore demonstrate the application of the derived formula to implement a gradient flow for the approximation of optimal particle configurations.
{"title":"On differentiability of the membrane-mediated mechanical interaction energy of discrete–continuum membrane–particle models","authors":"Tobias Kies, C. Graser","doi":"10.4171/ifb/461","DOIUrl":"https://doi.org/10.4171/ifb/461","url":null,"abstract":"We consider a discrete-continuum model of a biomembrane with embedded particles. While the membrane is represented by a continuous surface, embedded particles are described by rigid discrete objects which are free to move and rotate in lateral direction. For the membrane we consider a linearized Canham-Helfrich energy functional and height and slope boundary conditions imposed on the particle boundaries resulting in a coupled minimization problem for the membrane shape and particle positions. \u0000When considering the energetically optimal membrane shape for each particle position we obtain a reduced energy functional that models the implicitly given interaction potential for the membrane-mediated mechanical particle-particle interactions. We show that this interaction potential is differentiable with respect to the particle positions and orientations. Furthermore we derive a fully practical representation of the derivative only in terms of well defined derivatives of the membrane. This opens the door for the application of minimization algorithms for the computation of minimizers of the coupled system and for further investigation of the interaction potential of membrane-mediated mechanical particle-particle interaction. \u0000The results are illustrated with numerical examples comparing the explicit derivative formula with difference quotient approximations. We furthermore demonstrate the application of the derived formula to implement a gradient flow for the approximation of optimal particle configurations.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2017-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81927369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We analyze the convergence rates to a planar interface in the Mullins-Sekerka model by applying a relaxation method based on relationships among distance, energy, and dissipation. The relaxation method was developed by two of the authors in the context of the 1-d Cahn-Hilliard equation and the current work represents an extension to a higher dimensional problem in which the curvature of the interface plays an important role. The convergence rates obtained are optimal given the assumptions on the initial data.
{"title":"Relaxation to a planar interface in the Mullins–Sekerka problem","authors":"Olga Chugreeva, F. Otto, Maria G. Westdickenberg","doi":"10.4171/IFB/415","DOIUrl":"https://doi.org/10.4171/IFB/415","url":null,"abstract":"We analyze the convergence rates to a planar interface in the Mullins-Sekerka model by applying a relaxation method based on relationships among distance, energy, and dissipation. The relaxation method was developed by two of the authors in the context of the 1-d Cahn-Hilliard equation and the current work represents an extension to a higher dimensional problem in which the curvature of the interface plays an important role. The convergence rates obtained are optimal given the assumptions on the initial data.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2017-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75816591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we established the global existence of supersonic entropy solutions with a strong contact discontinuity over Lipschitz wall governed by the two-dimensional steady exothermically reacting Euler equations, when the total variation of both initial data and the slope of Lipschitz wall is sufficiently small. Local and global estimates are developed and a modified Glimm-type functional is carefully designed. Next the validation of the quasi-one-dimensional approximation in the domain bounded by the wall and the strong contact discontinuity is rigorous justified by proving that the difference between the average of weak solution and the solution of quasi-one-dimensional system can be bounded by the square of the total variation of both initial data and the slope of Lipschitz wall. The methods and techniques developed here is also helpful for other related problems.
{"title":"Two-dimensional steady supersonic exothermically reacting Euler flows with strong contact discontinuity over a Lipschitz wall","authors":"Wei Xiang, Yongqian Zhang, Qin Zhao","doi":"10.4171/IFB/408","DOIUrl":"https://doi.org/10.4171/IFB/408","url":null,"abstract":"In this paper, we established the global existence of supersonic entropy solutions with a strong contact discontinuity over Lipschitz wall governed by the two-dimensional steady exothermically reacting Euler equations, when the total variation of both initial data and the slope of Lipschitz wall is sufficiently small. Local and global estimates are developed and a modified Glimm-type functional is carefully designed. Next the validation of the quasi-one-dimensional approximation in the domain bounded by the wall and the strong contact discontinuity is rigorous justified by proving that the difference between the average of weak solution and the solution of quasi-one-dimensional system can be bounded by the square of the total variation of both initial data and the slope of Lipschitz wall. The methods and techniques developed here is also helpful for other related problems.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2017-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88231470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The optimal insulation of a heat conducting body by a thin film of variable thickness can be formulated as a nondifferentiable, nonlocal eigenvalue problem. The discretization and iterative solution for the reliable computation of corresponding eigenfunctions that determine the optimal layer thickness are addressed. Corresponding numerical experiments confirm the theoretical observation that a symmetry breaking occurs for the case of small available insulation masses and provide insight in the geometry of optimal films. An experimental shape optimization indicates that convex bodies with one axis of symmetry have favorable insulation properties.
{"title":"Numerical solution of a nonlinear eigenvalue problem arising in optimal insulation","authors":"S. Bartels, G. Buttazzo","doi":"10.4171/IFB/414","DOIUrl":"https://doi.org/10.4171/IFB/414","url":null,"abstract":"The optimal insulation of a heat conducting body by a thin film of variable thickness can be formulated as a nondifferentiable, nonlocal eigenvalue problem. The discretization and iterative solution for the reliable computation of corresponding eigenfunctions that determine the optimal layer thickness are addressed. Corresponding numerical experiments confirm the theoretical observation that a symmetry breaking occurs for the case of small available insulation masses and provide insight in the geometry of optimal films. An experimental shape optimization indicates that convex bodies with one axis of symmetry have favorable insulation properties.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2017-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83635856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article we study a normalised double obstacle problem with polynomial obstacles $ p^1leq p^2$ under the assumption that $ p^1(x)=p^2(x)$ iff $ x=0$. In dimension two we give a complete characterisation of blow-up solutions depending on the coefficients of the polynomials $p^1, p^2$. In particular, we see that there exists a new type of blow-ups, that we call double-cone solutions since the coincidence sets ${u=p^1}$ and ${u=p^2}$ are cones with a common vertex. We prove the uniqueness of blow-up limits, and analyse the regularity of the free boundary in dimension two. In particular we show that if the solution to the double obstacle problem has a double-cone blow-up limit at the origin, then locally the free boundary consists of four $C^{1,gamma}$-curves, meeting at the origin. In the end we give an example of a three-dimensional double-cone solution.
{"title":"Analysis of blow-ups for the double obstacle problem in dimension two","authors":"G. Aleksanyan","doi":"10.4171/IFB/419","DOIUrl":"https://doi.org/10.4171/IFB/419","url":null,"abstract":"In this article we study a normalised double obstacle problem with polynomial obstacles $ p^1leq p^2$ under the assumption that $ p^1(x)=p^2(x)$ iff $ x=0$. In dimension two we give a complete characterisation of blow-up solutions depending on the coefficients of the polynomials $p^1, p^2$. In particular, we see that there exists a new type of blow-ups, that we call double-cone solutions since the coincidence sets ${u=p^1}$ and ${u=p^2}$ are cones with a common vertex. We prove the uniqueness of blow-up limits, and analyse the regularity of the free boundary in dimension two. In particular we show that if the solution to the double obstacle problem has a double-cone blow-up limit at the origin, then locally the free boundary consists of four $C^{1,gamma}$-curves, meeting at the origin. In the end we give an example of a three-dimensional double-cone solution.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2017-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75575382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
By using a suitable triple cover we show how to possibly model the construction of a minimal surface with positive genus spanning all six edges of a tetrahedron, working in the space of BV functions and interpreting the film as the boundary of a Caccioppoli set in the covering space. After a question raised by R. Hardt in the late 1980's, it seems common opinion that an area-minimizing surface of this sort does not exist for a regular tetrahedron, although a proof of this fact is still missing. In this paper we show that there exists a surface of positive genus spanning the boundary of an elongated tetrahedron and having area strictly less than the area of the conic surface.
{"title":"Triple covers and a non-simply connected surface spanning an elongated tetrahedron and beating the cone","authors":"G. Bellettini, M. Paolini, F. Pasquarelli","doi":"10.4171/IFB/407","DOIUrl":"https://doi.org/10.4171/IFB/407","url":null,"abstract":"By using a suitable triple cover we show how to possibly model the construction of a minimal surface with positive genus spanning all six edges of a tetrahedron, working in the space of BV functions and interpreting the film as the boundary of a Caccioppoli set in the covering space. After a question raised by R. Hardt in the late 1980's, it seems common opinion that an area-minimizing surface of this sort does not exist for a regular tetrahedron, although a proof of this fact is still missing. In this paper we show that there exists a surface of positive genus spanning the boundary of an elongated tetrahedron and having area strictly less than the area of the conic surface.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2017-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78249470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Harbir Antil, M. Hintermüller, R. Nochetto, T. Surowiec, D. Wegner
A time-discrete spatially-continuous electrowetting on dielectric (EWOD) model with contact line pinning is considered as the state system in an optimal control framework. The pinning model is based on a complementarity condition. In addition to the physical variables describing velocity, pressure, and voltage, the solid-liquid-air interface, i.e., the contact line, arises as a geometric variable that evolves in time. Due to the complementarity condition, the resulting optimal control of a free boundary problem is thus a mathematical program with equilibrium constraints (MPEC) in function space. In order to cope with the geometric variable, a finite horizon model predictive control approach is proposed. Dual stationarity conditions are derived by applying a regularization procedure, exploiting techniques from PDE-constrained optimization, and then passing to the limit in the regularization parameters. Moreover, a function-space-based numerical procedure is developed by following the theoretical limit argument used in the derivation of the dual stationarity conditions. The performance of the algorithm is demonstrated by several examples; including barycenter matching and trajectory tracking.
{"title":"Finite horizon model predictive control of electrowetting on dielectric with pinning","authors":"Harbir Antil, M. Hintermüller, R. Nochetto, T. Surowiec, D. Wegner","doi":"10.4171/IFB/375","DOIUrl":"https://doi.org/10.4171/IFB/375","url":null,"abstract":"A time-discrete spatially-continuous electrowetting on dielectric (EWOD) model with contact line pinning is considered as the state system in an optimal control framework. The pinning model is based on a complementarity condition. In addition to the physical variables describing velocity, pressure, and voltage, the solid-liquid-air interface, i.e., the contact line, arises as a geometric variable that evolves in time. Due to the complementarity condition, the resulting optimal control of a free boundary problem is thus a mathematical program with equilibrium constraints (MPEC) in function space. In order to cope with the geometric variable, a finite horizon model predictive control approach is proposed. Dual stationarity conditions are derived by applying a regularization procedure, exploiting techniques from PDE-constrained optimization, and then passing to the limit in the regularization parameters. Moreover, a function-space-based numerical procedure is developed by following the theoretical limit argument used in the derivation of the dual stationarity conditions. The performance of the algorithm is demonstrated by several examples; including barycenter matching and trajectory tracking.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2017-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90584398","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate a general, local version of the cane toads equation, which models the spread of a population structured by unbounded motility. We use the thin-front limit approach of Evans and Souganidis in [Indiana Univ. Math. J., 1989] to obtain a characterization of the propagation in terms of both the linearized equation and a geometric front equation. In particular, we reduce the task of understanding the precise location of the front for a large class of equations to analyzing a much smaller class of Hamilton-Jacobi equations. We are then able to give an explicit formula for the front location in physical space. One advantage of our approach is that we do not use the explicit trajectories along which the population spreads, which was a basis of previous work. Our result allows for large oscillations in the motility.
{"title":"Super-linear propagation for a general, local cane toads model","authors":"Christopher Henderson, B. Perthame, P. Souganidis","doi":"10.4171/IFB/409","DOIUrl":"https://doi.org/10.4171/IFB/409","url":null,"abstract":"We investigate a general, local version of the cane toads equation, which models the spread of a population structured by unbounded motility. We use the thin-front limit approach of Evans and Souganidis in [Indiana Univ. Math. J., 1989] to obtain a characterization of the propagation in terms of both the linearized equation and a geometric front equation. In particular, we reduce the task of understanding the precise location of the front for a large class of equations to analyzing a much smaller class of Hamilton-Jacobi equations. We are then able to give an explicit formula for the front location in physical space. One advantage of our approach is that we do not use the explicit trajectories along which the population spreads, which was a basis of previous work. Our result allows for large oscillations in the motility.","PeriodicalId":13863,"journal":{"name":"Interfaces and Free Boundaries","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2017-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77475685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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