
Previous Article
Low Mach number asymptotics for reacting compressible fluid flows
 DCDS Home
 This Issue

Next Article
Looking for critical nonlinearity in the onedimensional quasilinear SmoluchowskiPoisson system
The surface diffusion flow on rough phase spaces
1.  Institute for Applied Mathematics, Leibniz University of Hanover, D30167 Hanover 
2.  Institute of Applied Mathematics and Mechanics, University of Warsaw, 02097 Warszawa, Poland 
Combining tools from harmonic analysis, such as Besov spaces, multiplier results with abstract results from the theory of maximal regularity we present an analytic framework in which we can investigate weak solutions to the original evolution equation. This approach allows us to prove wellposedness on a large (Besov) space of initial data which is in general larger than $C^2$ (and which is in the distributional sense almost optimal). Our second main result shows that the set of all compact embedded equilibria, i.e. the set of all spheres, is an invariant manifold in this phase space which attracts all solutions which are close enough (which respect to the norm of the phase space) to this manifold. As a consequence we are able to construct nonconvex initial data which generate global solutions, converging finally to a sphere.
[1] 
Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reactiondiffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems  A, 2019, 0 (0) : 00. doi: 10.3934/dcds.2020033 
[2] 
Irena Lasiecka, Mathias Wilke. Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system. Discrete & Continuous Dynamical Systems  A, 2013, 33 (11&12) : 51895202. doi: 10.3934/dcds.2013.33.5189 
[3] 
Pascal Auscher, Sylvie Monniaux, Pierre Portal. The maximal regularity operator on tent spaces. Communications on Pure & Applied Analysis, 2012, 11 (6) : 22132219. doi: 10.3934/cpaa.2012.11.2213 
[4] 
Tôn Việt Tạ. Nonautonomous stochastic evolution equations in Banach spaces of martingale type 2: Strict solutions and maximal regularity. Discrete & Continuous Dynamical Systems  A, 2017, 37 (8) : 45074542. doi: 10.3934/dcds.2017193 
[5] 
Minghua Yang, Jinyi Sun. Gevrey regularity and existence of NavierStokesNernstPlanckPoisson system in critical Besov spaces. Communications on Pure & Applied Analysis, 2017, 16 (5) : 16171639. doi: 10.3934/cpaa.2017078 
[6] 
Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete & Continuous Dynamical Systems  A, 2001, 7 (4) : 763780. doi: 10.3934/dcds.2001.7.763 
[7] 
Shouming Zhou. The Cauchy problem for a generalized $b$equation with higherorder nonlinearities in critical Besov spaces and weighted $L^p$ spaces. Discrete & Continuous Dynamical Systems  A, 2014, 34 (11) : 49674986. doi: 10.3934/dcds.2014.34.4967 
[8] 
Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible NavierStokes. Kinetic & Related Models, 2016, 9 (1) : 75103. doi: 10.3934/krm.2016.9.75 
[9] 
Peter Weidemaier. Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$norm. Electronic Research Announcements, 2002, 8: 4751. 
[10] 
Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems  B, 2016, 21 (5) : 14211434. doi: 10.3934/dcdsb.2016003 
[11] 
Antonios Zagaris, Christophe Vandekerckhove, C. William Gear, Tasso J. Kaper, Ioannis G. Kevrekidis. Stability and stabilization of the constrained runs schemes for equationfree projection to a slow manifold. Discrete & Continuous Dynamical Systems  A, 2012, 32 (8) : 27592803. doi: 10.3934/dcds.2012.32.2759 
[12] 
Ariadna Farrés, Àngel Jorba. On the high order approximation of the centre manifold for ODEs. Discrete & Continuous Dynamical Systems  B, 2010, 14 (3) : 9771000. doi: 10.3934/dcdsb.2010.14.977 
[13] 
Jingzhi Yan. Existence of torsionlow maximal isotopies for area preserving surface homeomorphisms. Discrete & Continuous Dynamical Systems  A, 2018, 38 (9) : 45714602. doi: 10.3934/dcds.2018200 
[14] 
Hiroshi Matsuzawa. A free boundary problem for the FisherKPP equation with a given moving boundary. Communications on Pure & Applied Analysis, 2018, 17 (5) : 18211852. doi: 10.3934/cpaa.2018087 
[15] 
Jan Haškovec, Dietmar Oelz. A free boundary problem for aggregation by short range sensing and differentiated diffusion. Discrete & Continuous Dynamical Systems  B, 2015, 20 (5) : 14611480. doi: 10.3934/dcdsb.2015.20.1461 
[16] 
HongMing Yin. A free boundary problem arising from a stressdriven diffusion in polymers. Discrete & Continuous Dynamical Systems  A, 1996, 2 (2) : 191202. doi: 10.3934/dcds.1996.2.191 
[17] 
JiaFeng Cao, WanTong Li, Meng Zhao. On a free boundary problem for a nonlocal reactiondiffusion model. Discrete & Continuous Dynamical Systems  B, 2018, 23 (10) : 41174139. doi: 10.3934/dcdsb.2018128 
[18] 
Harunori Monobe, Hirokazu Ninomiya. Multiple existence of traveling waves of a free boundary problem describing cell motility. Discrete & Continuous Dynamical Systems  B, 2014, 19 (3) : 789799. doi: 10.3934/dcdsb.2014.19.789 
[19] 
Esther S. Daus, JosipaPina Milišić, Nicola Zamponi. Global existence for a twophase flow model with crossdiffusion. Discrete & Continuous Dynamical Systems  B, 2020, 25 (3) : 957979. doi: 10.3934/dcdsb.2019198 
[20] 
Gabriella Pinzari. Global Kolmogorov tori in the planetary $\boldsymbol N$body problem. Announcement of result. Electronic Research Announcements, 2015, 22: 5575. doi: 10.3934/era.2015.22.55 
2018 Impact Factor: 1.143
Tools
Metrics
Other articles
by authors
[Back to Top]