American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth
  • DCDS Home
  • This Issue
  • Next Article
    Age-dependent equations with non-linear diffusion
April  2010, 26(2): 713-736. doi: 10.3934/dcds.2010.26.713

Boundary dynamics of a two-dimensional diffusive free boundary problem

Micah Webster 1, and  Patrick Guidotti 2,

1. 

Mathematics and Computer Science Department, Goucher College, 1021 Dulaney Valley Road, Baltimore, MD, 21204, United States
2. 

Department of Mathematics, University of California, Irvine, CA 92697-3875

Received  February 2009 Revised  July 2009 Published  October 2009

Numerous models of industrial processes such as diffusion in glassy polymers or solidification phenomena, lead to general one-phase free boundary value problems with phase onset. In this paper we develop a framework viable to prove global existence and stability of planar solutions to one such multi-dimensional model whose application is in controlled-release pharmaceuticals. We utilize a boundary integral reformulation to allow for the use of maximal regularity. To this effect, we view the operators as pseudo-differential and exploit knowledge of the relevant symbols. Within this framework, we give a local existence and continuous dependence result necessary to prove planar solutions are locally exponentially stable with respect to two-dimensional perturbations.
Keywords: Free Boundary Problem, Well-Posedness, Non-Regular Pseudodifferential Operators., Maximal Regularity.
Mathematics Subject Classification: Primary: 35A07, 35C15, 35M1.
Citation: Micah Webster, Patrick Guidotti. Boundary dynamics of a two-dimensional diffusive free boundary problem. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 713-736. doi: 10.3934/dcds.2010.26.713
[1]

Wenming Hu, Huicheng Yin. Well-posedness of low regularity solutions to the second order strictly hyperbolic equations with non-Lipschitzian coefficients. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1891-1919. doi: 10.3934/cpaa.2019088
[2]

Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi. Local well-posedness in low regularity of the MKDV equation with periodic boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1635-1654. doi: 10.3934/dcds.2010.28.1635
[3]

Lars Diening, Michael Růžička. An existence result for non-Newtonian fluids in non-regular domains. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 255-268. doi: 10.3934/dcdss.2010.3.255
[4]

Yuri Trakhinin. On well-posedness of the plasma-vacuum interface problem: the case of non-elliptic interface symbol. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1371-1399. doi: 10.3934/cpaa.2016.15.1371
[5]

Fujun Zhou, Shangbin Cui. Well-posedness and stability of a multidimensional moving boundary problem modeling the growth of tumor cord. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 929-943. doi: 10.3934/dcds.2008.21.929
[6]

Joachim Escher, Anca-Voichita Matioc. Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 573-596. doi: 10.3934/dcdsb.2011.15.573
[7]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307
[8]

Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321
[9]

Hyungjin Huh, Bora Moon. Low regularity well-posedness for Gross-Neveu equations. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1903-1913. doi: 10.3934/cpaa.2015.14.1903
[10]

Barbara Kaltenbacher, Irena Lasiecka. Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions. Conference Publications, 2011, 2011 (Special) : 763-773. doi: 10.3934/proc.2011.2011.763
[11]

George Avalos, Pelin G. Geredeli, Justin T. Webster. Semigroup well-posedness of a linearized, compressible fluid with an elastic boundary. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1267-1295. doi: 10.3934/dcdsb.2018151
[12]

Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61
[13]

Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387
[14]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic & Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032
[15]

Yoshihiro Shibata. Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface. Evolution Equations & Control Theory, 2018, 7 (1) : 117-152. doi: 10.3934/eect.2018007
[16]

Daniel Coutand, Steve Shkoller. A simple proof of well-posedness for the free-surface incompressible Euler equations. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 429-449. doi: 10.3934/dcdss.2010.3.429
[17]

Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315
[18]

Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831
[19]

Magdalena Czubak, Nina Pikula. Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1669-1683. doi: 10.3934/cpaa.2014.13.1669
[20]

Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences