March  2020, 9(1): 61-86. doi: 10.3934/eect.2020017

On quasilinear parabolic equations and continuous maximal regularity

1. 

Department of Mathematics & Computer Science, 221 Richmond Way, University of Richmond, VA 23173, USA

2. 

Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA

* Corresponding author: Jeremy LeCrone

Received  August 2018 Revised  August 2019 Published  March 2020 Early access  October 2019

Fund Project: This work was supported by a grant from the Simons Foundation (#426729, Gieri Simonett).

We consider a class of abstract quasilinear parabolic problems with lower–order terms exhibiting a prescribed singular structure. We prove well–posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global existence of solutions and we extend the generalized principle of linearized stability to settings with initial values in critical spaces. These general results are applied to the surface diffusion flow in various settings.

Citation: Jeremy LeCrone, Gieri Simonett. On quasilinear parabolic equations and continuous maximal regularity. Evolution Equations and Control Theory, 2020, 9 (1) : 61-86. doi: 10.3934/eect.2020017
References:
[1]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.

[2]

S. B. Angenent, Nonlinear analytic semiflows, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 91-107.  doi: 10.1017/S0308210500024598.

[3]

T. Asai, Quasilinear parabolic equation and its applications to fourth order equations with rough initial data, J. Math. Sci. Univ. Tokyo, 19 (2013), 507-532. 

[4]

A. J. BernoffA. L. Bertozzi and T. P. Witelski, Axisymmetric surface diffusion: Dynamics and stability of self-similar pinchoff, J. Statist. Phys., 93 (1998), 725-776.  doi: 10.1023/B:JOSS.0000033251.81126.af.

[5]

P. Clément and G. Simonett, Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations, J. Evol. Equ., 1 (2001), 39-67.  doi: 10.1007/PL00001364.

[6]

J. EscherU. F. Mayer and G. Simonett, The surface diffusion flow for immersed hypersurfaces, SIAM J. Math. Anal., 29 (1998), 1419-1433.  doi: 10.1137/S0036141097320675.

[7]

J. Escher and P. B. Mucha, The surface diffusion flow on rough phase spaces, Discrete Contin. Dyn. Syst., 26 (2010), 431-453.  doi: 10.3934/dcds.2010.26.431.

[8]

H. Koch and T. Lamm, Geometric flows with rough initial data, Asian J. Math., 16 (2012), 209-235.  doi: 10.4310/AJM.2012.v16.n2.a3.

[9]

J. LeCroneJ. Prüss and M. Wilke, On quasilinear parabolic evolution equations in weighted $L_p$-spaces Ⅱ, J. Evol. Equ., 14 (2014), 509-533.  doi: 10.1007/s00028-014-0226-6.

[10]

J. LeCrone and G. Simonett, Continuous maximal regularity and analytic semigroups, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 8th AIMS Conference. Suppl., 2 (2011), 963–970.

[11]

J. LeCrone and G. Simonett, On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow, SIAM J. Math. Anal., 45 (2013), 2834-2869.  doi: 10.1137/120883505.

[12]

J. LeCrone and G. Simonett, On the flow of non-axisymmetric perturbations of cylinders via surface diffusion, J. Differential Equations, 260 (2016), 5510-5531.  doi: 10.1016/j.jde.2015.12.008.

[13]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1995. [2013 reprint of the 1995 original] [MR1329547].

[14]

J. McCoyG. Wheeler and G. Williams, Lifespan theorem for constrained surface diffusion flows, Math. Z., 269 (2011), 147-178.  doi: 10.1007/s00209-010-0720-7.

[15]

J. Prüss and G. Simonett, On the manifold of closed hypersurfaces in $\Bbb{R}^n$, Discrete Contin. Dyn. Syst., 33 (2013), 5407-5428.  doi: 10.3934/dcds.2013.33.5407.

[16]

J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, 105. Birkhäuser/Springer, [Cham], 2016. doi: 10.1007/978-3-319-27698-4.

[17]

J. PrüssG. Simonett and M. Wilke, Critical spaces for quasilinear parabolic evolution equations and applications, J. Differential Equations, 264 (2018), 2028-2074.  doi: 10.1016/j.jde.2017.10.010.

[18]

J. PrüssG. Simonett and R. Zacher, On convergence of solutions to equilibria for quasilinear parabolic problems, J. Differential Equations, 246 (2009), 3902-3931.  doi: 10.1016/j.jde.2008.10.034.

[19]

J. Prüss, G. Simonett and R. Zacher, On normal stability for nonlinear parabolic equations, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 7th AIMS Conference, suppl., (2009), 612–621.

[20]

J. Prüss and M. Wilke, Addendum to the paper "On quasilinear parabolic evolution equations in weighted $L_p$-spaces Ⅱ" [MR3250797], J. Evol. Equ., 17 (2017), 1381-1388.  doi: 10.1007/s00028-017-0382-6.

[21]

Y. Shao, A family of parameter-dependent diffeomorphisms acting on function spaces over a Riemannian manifold and applications to geometric flows, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 45-85.  doi: 10.1007/s00030-014-0275-0.

[22]

Y. Shao and G. Simonett, Continuous maximal regularity on uniformly regular Riemannian manifolds, J. Evol. Equ., 14 (2014), 211-248.  doi: 10.1007/s00028-014-0218-6.

[23]

G. Wheeler, Lifespan theorem for simple constrained surface diffusion flows, J. Math. Anal. Appl., 375 (2011), 685-698.  doi: 10.1016/j.jmaa.2010.09.043.

[24]

G. Wheeler, Surface diffusion flow near spheres, Calc. Var. Partial Differential Equations, 44 (2012), 131-151.  doi: 10.1007/s00526-011-0429-4.

show all references

References:
[1]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.

[2]

S. B. Angenent, Nonlinear analytic semiflows, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 91-107.  doi: 10.1017/S0308210500024598.

[3]

T. Asai, Quasilinear parabolic equation and its applications to fourth order equations with rough initial data, J. Math. Sci. Univ. Tokyo, 19 (2013), 507-532. 

[4]

A. J. BernoffA. L. Bertozzi and T. P. Witelski, Axisymmetric surface diffusion: Dynamics and stability of self-similar pinchoff, J. Statist. Phys., 93 (1998), 725-776.  doi: 10.1023/B:JOSS.0000033251.81126.af.

[5]

P. Clément and G. Simonett, Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations, J. Evol. Equ., 1 (2001), 39-67.  doi: 10.1007/PL00001364.

[6]

J. EscherU. F. Mayer and G. Simonett, The surface diffusion flow for immersed hypersurfaces, SIAM J. Math. Anal., 29 (1998), 1419-1433.  doi: 10.1137/S0036141097320675.

[7]

J. Escher and P. B. Mucha, The surface diffusion flow on rough phase spaces, Discrete Contin. Dyn. Syst., 26 (2010), 431-453.  doi: 10.3934/dcds.2010.26.431.

[8]

H. Koch and T. Lamm, Geometric flows with rough initial data, Asian J. Math., 16 (2012), 209-235.  doi: 10.4310/AJM.2012.v16.n2.a3.

[9]

J. LeCroneJ. Prüss and M. Wilke, On quasilinear parabolic evolution equations in weighted $L_p$-spaces Ⅱ, J. Evol. Equ., 14 (2014), 509-533.  doi: 10.1007/s00028-014-0226-6.

[10]

J. LeCrone and G. Simonett, Continuous maximal regularity and analytic semigroups, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 8th AIMS Conference. Suppl., 2 (2011), 963–970.

[11]

J. LeCrone and G. Simonett, On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow, SIAM J. Math. Anal., 45 (2013), 2834-2869.  doi: 10.1137/120883505.

[12]

J. LeCrone and G. Simonett, On the flow of non-axisymmetric perturbations of cylinders via surface diffusion, J. Differential Equations, 260 (2016), 5510-5531.  doi: 10.1016/j.jde.2015.12.008.

[13]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1995. [2013 reprint of the 1995 original] [MR1329547].

[14]

J. McCoyG. Wheeler and G. Williams, Lifespan theorem for constrained surface diffusion flows, Math. Z., 269 (2011), 147-178.  doi: 10.1007/s00209-010-0720-7.

[15]

J. Prüss and G. Simonett, On the manifold of closed hypersurfaces in $\Bbb{R}^n$, Discrete Contin. Dyn. Syst., 33 (2013), 5407-5428.  doi: 10.3934/dcds.2013.33.5407.

[16]

J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics, 105. Birkhäuser/Springer, [Cham], 2016. doi: 10.1007/978-3-319-27698-4.

[17]

J. PrüssG. Simonett and M. Wilke, Critical spaces for quasilinear parabolic evolution equations and applications, J. Differential Equations, 264 (2018), 2028-2074.  doi: 10.1016/j.jde.2017.10.010.

[18]

J. PrüssG. Simonett and R. Zacher, On convergence of solutions to equilibria for quasilinear parabolic problems, J. Differential Equations, 246 (2009), 3902-3931.  doi: 10.1016/j.jde.2008.10.034.

[19]

J. Prüss, G. Simonett and R. Zacher, On normal stability for nonlinear parabolic equations, Discrete Contin. Dyn. Syst., Dynamical systems, differential equations and applications. 7th AIMS Conference, suppl., (2009), 612–621.

[20]

J. Prüss and M. Wilke, Addendum to the paper "On quasilinear parabolic evolution equations in weighted $L_p$-spaces Ⅱ" [MR3250797], J. Evol. Equ., 17 (2017), 1381-1388.  doi: 10.1007/s00028-017-0382-6.

[21]

Y. Shao, A family of parameter-dependent diffeomorphisms acting on function spaces over a Riemannian manifold and applications to geometric flows, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 45-85.  doi: 10.1007/s00030-014-0275-0.

[22]

Y. Shao and G. Simonett, Continuous maximal regularity on uniformly regular Riemannian manifolds, J. Evol. Equ., 14 (2014), 211-248.  doi: 10.1007/s00028-014-0218-6.

[23]

G. Wheeler, Lifespan theorem for simple constrained surface diffusion flows, J. Math. Anal. Appl., 375 (2011), 685-698.  doi: 10.1016/j.jmaa.2010.09.043.

[24]

G. Wheeler, Surface diffusion flow near spheres, Calc. Var. Partial Differential Equations, 44 (2012), 131-151.  doi: 10.1007/s00526-011-0429-4.

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