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  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 & 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.  Google Scholar

[2]

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

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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].  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

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.  Google Scholar

[2]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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].  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

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