Self-Similar Blowup Solutions to an Aggregation Equation in Rn

We present numerical simulations of radially symmetric finite time blowup for the aggregation equation ut = ∇· (u∇K ∗ u), where the kernel K(x )= |x|. The dynamics of the blowup exhibits self-similar behavior in which zero mass concentrates at the core at the blowup time. Computations are performed in R n for n between 2 and 10 using a method based on characteristics. In all cases studied, the self-similarity exhibits second-kind (anomalous) scaling.

[1]  Hatem Zaag,et al.  Stability of the blow-up profile for equations of the type $u_t=\Delta u+|u|^{p-1}u$ , 1997 .

[2]  Liu,et al.  Anomalous dimensions and the renormalization group in a nonlinear diffusion process. , 1990, Physical review letters.

[3]  L. Peletier,et al.  On the Barenblatt equation of elasto-plastic filtration , 1991 .

[4]  Darryl D. Holm,et al.  Formation of clumps and patches in self-aggregation of finite-size particles , 2005, nlin/0506020.

[5]  J. Eggers,et al.  The role of self-similarity in singularities of PDE's , 2008, 0812.1339.

[6]  Jose L. Rodrigo,et al.  Refined blowup criteria and nonsymmetric blowup of an aggregation equation , 2009 .

[7]  G. I. Barenblatt,et al.  Self-similar solutions of the second kind in nonlinear filtration: PMM vol. 33, n≗5, 1969, pp. 861–870 , 1969 .

[8]  Li Dong,et al.  FINITE-TIME SINGULARITIES OF AN AGGREGATION EQUATION IN R WITH FRACTIONAL DISSIPATION , 2008 .

[9]  J. Dold.  ANALYSIS OF THE EARLY STAGE OF THERMAL RUNAWAY. , 1985 .

[10]  T. Laurent,et al.  The behavior of solutions of multidimensional aggregation equations with mildly singular interaction kernels , 2009 .

[11]  Thomas P. Witelski,et al.  Blowup and dissipation in a critical-case unstable thin film equation , 2004, European Journal of Applied Mathematics.

[12]  A. Mogilner,et al.  A non-local model for a swarm , 1999 .

[13]  Martin Burger,et al.  On an aggregation model with long and short range interactions , 2007 .

[14]  G. Toscani,et al.  Long-Time Asymptotics of Kinetic Models of Granular Flows , 2004 .

[15]  A. P. Mikhailov,et al.  Blow-Up in Quasilinear Parabolic Equations , 1995 .

[16]  Andrea L. Bertozzi,et al.  Finite-Time Blow-up of Solutions of an Aggregation Equation in Rn , 2007 .

[17]  Andrea L. Bertozzi,et al.  Axisymmetric Surface Diffusion: Dynamics and Stability of Self-Similar Pinchoff , 1998 .

[18]  C. Villani,et al.  Homogeneous Cooling States¶are not always¶Good Approximations to Granular Flows , 2002 .

[19]  M. Sharif,et al.  Spherically Symmetric Gravitational Collapse , 2008, 0812.0646.

[20]  G. I. Barenblatt.  Scaling: Self-similarity and intermediate asymptotics , 1996 .

[21]  Chen,et al.  Numerical renormalization-group calculations for similarity solutions and traveling waves. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[22]  J. Vázquez,et al.  Anomalous exponents in nonlinear diffusion , 1993 .

[23]  Robert D. Russell,et al.  A study of moving mesh PDE methods for numerical simulation of blowup in reaction diffusion equations , 2008, J. Comput. Phys..

[24]  D. Aronson,et al.  A selfsimilar solution to the focusing problem for the porous medium equation , 1993, European Journal of Applied Mathematics.

[25]  Dong Li,et al.  Finite-Time Singularities of an Aggregation Equation in $${\mathbb {R}^n}$$ with Fractional Dissipation , 2009 .

[26]  D. Aronson,et al.  Renormalization study of two-dimensional convergent solutions of the porous medium equation , 1999, patt-sol/9908006.

[27]  Josephus Hulshof,et al.  Self-similar solutions of the second kind for the modified porous medium equation , 1994, European Journal of Applied Mathematics.

[28]  Martin Burger,et al.  Large time behavior of nonlocal aggregation models with nonlinear diffusion , 2008, Networks Heterog. Media.

[29]  J. Eggers,et al.  The role of self-similarity in singularities of partial differential equations , 2008 .

[30]  Thomas Laurent,et al.  Local and Global Existence for an Aggregation Equation , 2007 .

[31]  Gadi Fibich,et al.  Self-similar optical wave collapse: observation of the Townes profile. , 2003, Physical review letters.

[32]  Robert V. Kohn,et al.  Refined asymptotics for the blowup of ut –Δu = up , 1992 .

[33]  Robert D. Russell,et al.  Moving Mesh Methods for Problems with Blow-Up , 1996, SIAM J. Sci. Comput..

[34]  Andrea L. Bertozzi,et al.  Blow-up in multidimensional aggregation equations with mildly singular interaction kernels , 2009 .

[35]  R. Russell,et al.  Precise computations of chemotactic collapse using moving mesh methods , 2005 .

[36]  Leo P. Kadanoff,et al.  Diffusion, attraction and collapse , 1999 .

[37]  M. Bodnar,et al.  An integro-differential equation arising as a limit of individual cell-based models , 2006 .

[38]  Darryl D. Holm,et al.  Aggregation of finite-size particles with variable mobility. , 2005, Physical review letters.

[39]  E. Garmire,et al.  Self-trapping of optical beams , 1966 .

[40]  J. F. Williams,et al.  Self-Similar Blow-Up in Higher-Order Semilinear Parabolic Equations , 2004, SIAM J. Appl. Math..

[41]  J. A. Carrillo,et al.  Numerical Simulation of Diffusive and Aggregation Phenomena in Nonlinear Continuity Equations by Evolving Diffeomorphisms , 2009, SIAM J. Sci. Comput..

[42]  R. Glassey.  On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations , 1977 .

[43]  Andrea L. Bertozzi,et al.  Swarming Patterns in a Two-Dimensional Kinematic Model for Biological Groups , 2004, SIAM J. Appl. Math..

[44]  Y. Giga,et al.  Asymptotically self‐similar blow‐up of semilinear heat equations , 1985 .

[45]  L. Peletier.  The anomalous exponent of the Barenblatt equation , 1994, European Journal of Applied Mathematics.

[46]  H. Bhadeshia.  Diffusion , 1995, Theory of Transformations in Steels.

[47]  Chris Budd,et al.  Stability and spectra of blow–up in problems with quasi–linear gradient diffusivity , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[48]  N. Gavish,et al.  New singular solutions of the nonlinear Schrödinger equation , 2005 .

[49]  T. Laurent,et al.  Lp theory for the multidimensional aggregation equation , 2011 .

[50]  L. Segel,et al.  Initiation of slime mold aggregation viewed as an instability. , 1970, Journal of theoretical biology.

[51]  A. Bertozzi,et al.  A Nonlocal Continuum Model for Biological Aggregation , 2005, Bulletin of mathematical biology.

[52]  Papanicolaou,et al.  Focusing singularity of the cubic Schrödinger equation. , 1986, Physical review. A, General physics.

[53]  Andrea L. Bertozzi,et al.  Finite-time blow-up of L∞-weak solutions of an aggregation equation , 2010 .