Continuous dependence results for quasilinear evolution equations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914355617988608 |
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| author | Cellarosi, Francesco Dutta, Anirban Mazzone, Giusy |
| author_facet | Cellarosi, Francesco Dutta, Anirban Mazzone, Giusy |
| contents | We study continuous dependence of solutions to quasilinear evolution equations of parabolic-type in the framework of maximal $L^p$-regularity. For equations of the form \[ \frac{dϕ}{dt} + A(t,ϕ)ϕ= f(t,ϕ), \] we establish continuous dependence of strong solutions on initial data, and suitable approximations of the nonlinear operators $A$ and $f$. An important step for proving the main result is the fact that the maximal regularity constant of the operator $A(t,ϕ)$, with $t$ and $ϕ$ fixed, admits a uniform bound over compact subsets of the relevant Banach spaces.
As an application, we consider a class of non-Newtonian fluid models with a Carreau-type viscosity and mixed boundary conditions. We show that, as the nonlinear contribution in the viscosity vanishes and the initial data converge, solutions of the non-Newtonian fluid model converge to those of the classical Navier--Stokes equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_23486 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Continuous dependence results for quasilinear evolution equations Cellarosi, Francesco Dutta, Anirban Mazzone, Giusy Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35K59, 35K15, 35K90, 35Q30, 35Q35, 76A05 We study continuous dependence of solutions to quasilinear evolution equations of parabolic-type in the framework of maximal $L^p$-regularity. For equations of the form \[ \frac{dϕ}{dt} + A(t,ϕ)ϕ= f(t,ϕ), \] we establish continuous dependence of strong solutions on initial data, and suitable approximations of the nonlinear operators $A$ and $f$. An important step for proving the main result is the fact that the maximal regularity constant of the operator $A(t,ϕ)$, with $t$ and $ϕ$ fixed, admits a uniform bound over compact subsets of the relevant Banach spaces. As an application, we consider a class of non-Newtonian fluid models with a Carreau-type viscosity and mixed boundary conditions. We show that, as the nonlinear contribution in the viscosity vanishes and the initial data converge, solutions of the non-Newtonian fluid model converge to those of the classical Navier--Stokes equations. |
| title | Continuous dependence results for quasilinear evolution equations |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis 35K59, 35K15, 35K90, 35Q30, 35Q35, 76A05 |
| url | https://arxiv.org/abs/2602.23486 |