Tesis (Ecuaciones Diferenciales y Análisis Numérico)https://hdl.handle.net/11441/108372024-03-19T06:55:05Z2024-03-19T06:55:05ZAnalysis and optimal control for chemotaxis-consumption modelshttps://hdl.handle.net/11441/1494372024-02-13T20:03:29Z2023-07-19T00:00:00ZAnalysis and optimal control for chemotaxis-consumption models
En esta tesis investigamos el siguiente modelo de quimiotaxis-consumo en dominios
acotados de RN (N = 1, 2, 3):
∂tu − Δu = −∇ · (u∇v), ∂tv − Δv = −usv,
donde s ≥ 1, dotado de condiciones de contorno aisladas y condiciones iniciales para
(u, v), con u y v representando la densidad de células y la concentración de la señal
química, respectivamente. Bajo hipótesis poco exigentes sobre la regularidad del dominio
y a través de la convergencia de las soluciones de un modelo truncado adecuado,
se establecen dos resultados principales: existencia de soluciones débiles uniformes en
el tiempo en dominios 3D, y unicidad y regularidad en dominios 2D (o 1D). Utilizando
la teoría desarrollada en este análisis teórico, proponemos y estudiamos un esquema
discreto en tiempo implícito tipo Backward Euler para dicho modelo combinado con
el uso de una variable auxiliar, probando existencia de solución, estimaciones a priori
uniformes en el tiempo y convergencia hacia una solución débil (u, v) del modelo
quimiotaxis-consumo. A continuación abordamos problemas de control óptimo sujetos
al siguiente modelo de quimiotaxis-consumo controlado de forma bilineal en un
dominio acotado Ω ⊂ R3 durante un intervalo de tiempo (0, T):
∂tu − Δu = −∇ · (u∇v), ∂tv − Δv = −usv + fv1Ωc ,
siendo f el control que actúa en un subdominio Ωc ⊂ Ω. En primer lugar, abordamos
un problema de control óptimo relacionado con las soluciones débiles del modelo de
quimiotaxis-consumo controlado. Demostramos la existencia de soluciones débiles que
satisfacen una desigualdad de energía, la existencia de control óptimo sujeto a controles
acotados y discutimos la relación entre el problema de control considerado y otros dos
relacionados que pueden ser de interés. A continuación estudiamos un problema de
control óptimo sujeto a soluciones fuertes del citado modelo de quimiotaxis-consumo
controlado. Demostramos un criterio de regularidad que nos permite obtener existencia
y unicidad de soluciones fuertes globales en el tiempo, mostramos la existencia de
una solución óptima global y, utilizando un teorema de multiplicadores de Lagrange,
establecemos condiciones de optimalidad de primer orden para cualquier solución óptima
local, probando existencia, unicidad y regularidad de los multiplicadores de Lagrange
asociados. Finalmente, en el capítulo de conclusiones, discutimos una serie de
posibles trabajos futuros relacionados con los resultados presentados en esta tesis.; In this thesis we investigate the following chemotaxis-consumption model in bounded
domains of RN (N = 1, 2, 3):
∂tu − Δu = −∇ · (u∇v), ∂tv − Δv = −usv,
where s ≥ 1, endowed with isolated boundary conditions and initial conditions for
(u, v), with u and v representing the cell density and chemical signal concentration,
respectively. Under mild regularity assumptions on the domain and through the convergence
of solutions of an adequate truncated model, two main results are established:
existence of uniform in time weak solutions in 3D domains, and uniqueness and regularity
in 2D (or 1D) domains. Using the theory developed in this theoretical analysis,
we propose and study a Backward Euler implicit time discrete scheme combined with
the use of an auxiliary variable for the aforementioned model, proving existence of
solution, uniform in time a priori estimates and convergence towards a weak solution
(u, v) of the chemotaxis-consumption model. In the sequel we approach optimal
control problems subject to the following bilinear controlled chemotaxis-consumption
model in a bounded domain Ω ⊂ R3 during a time interval (0, T):
∂tu − Δu = −∇ · (u∇v), ∂tv − Δv = −usv + fv1Ωc ,
with f being the control acting in a subdomain Ωc ⊂ Ω. First, we approach an optimal
control problem related to weak solutions of the controlled chemotaxis-consumption
model. We prove the existence of weak solutions satisfying an energy inequality, the
existence of optimal control subject to bounded controls and discuss the relation between
the considered control problem and two other related ones that might be of
interest. Next we study an optimal control problem subject to strong solutions of the
aforementioned controlled chemotaxis-consumption model. We prove a regularity criterion
that allows us to get existence and uniqueness of global-in-time strong solutions,
we show the existence of a global optimal solution and, using a Lagrange multipliers
theorem, we establish first order optimality conditions for any local optimal solution,
proving existence, uniqueness and regularity of the associated Lagrange multipliers.
Finally, in the conclusions chapter, we discuss a series of possible future works related
to the results presented in this thesis.
2023-07-19T00:00:00ZDynamics of stochastic systems with delay and applications to real modelshttps://hdl.handle.net/11441/1478952024-02-14T09:09:32Z2023-06-22T00:00:00ZDynamics of stochastic systems with delay and applications to real models
In this thesis we investigate the long time behavior of random dynamical systems associ-
ated to several kinds of stochastic equations with delays in terms of stability for stationary
solutions, weak pullback mean random attractors, random attractors and numerical attrac-
tors. The thesis consists of three parts, where the rst part covers Chapters 1-3, the last two
cover Chapters 4 and 5, respectively.
Chapters 1-3 are devoted to the random dynamics of 3D Lagrangian-averaged Navier-
Stokes equations with in nite delay in three cases.
In Chapter 1 we consider the stability analysis of such systems in the case of bounded
domains. We rst use Galerkin's approximations to prove the existence and uniqueness of
solutions when the non-delayed external force is locally integrable and the delay terms are
globally Lipschitz continuous with an additional assumption. We then prove the existence
of a unique stationary solution to the corresponding deterministic equation via the Lax-
Milgram and the Schauder theorems. The stability and asymptotic stability of stationary
solutions (equilibrium solutions) are also established. The local stability of stationary so-
lutions for general delay terms is carried out by using a direct method and then apply the
abstract results to two kinds of in nite delays. It is worth mentioning that all conditions
are general enough to include several kinds of delays, where we mainly consider unbounded
variable delays and in nite distributed delays. As we know, it is still an open and challenging
problem to obtain su cient conditions ensuring the exponential stability of solutions in case
of unbounded variable delay. Fortunately, we obtained the exponential stability of stationary
solutions in the case of in nite distributed delay. However, we are able to further investigate
the asymptotic stability of stationary solutions in the case of unbounded variable delay by
constructing suitable Lyapunov functionals. Besides, we proved the polynomial asymptotic
stability of stationary solutions for the particular case of proportional delay.
In Chapter 2, we further discuss mean dynamics and stability analysis of stochastic sys-
tems in the case of unbounded domains. We rst prove the well-posedness of systems with
in nite delay when the non-delayed external force is locally integrable, the delay term is
globally Lipschitz continuous and the nonlinear di usion term is locally Lipschitz continu-
ous, which leads to the existence of a mean random dynamical system. We then obtain that
such a dynamical system possesses a unique weak pullback mean random attractor, which
is a minimal, weakly compact and weakly pullback attracting set. Moreover, we prove the
existence and uniqueness of stationary solutions to the corresponding deterministic equa-
tion via the classical Galerkin method, the Lax-Milgram and the Brouwer xed theorems.
1
We discuss in the last part of Chapter 2 with those stability results concerning stationary
solutions discussed in Chapter 1.
The last case is concerned with the invariant measures for the autonomous version of
stochastic equations in Chapter 3 by using the method of generalized Banach limit. We rst
use Galerkin approximations, a priori estimates and the standard Gronwall lemma to show
the well-posedness for the corresponding random equation, whose solution operators generate
a random dynamical system. Next, the asymptotic compactness for the random dynamical
system is established via the Ascoli-Arzel a theorem. Besides, we derive the existence of
a global random attractor for the random dynamical system. Moreover, we prove that
the random dynamical system is bounded and continuous with respect to the initial values.
Eventually, we construct a family of invariant Borel probability measures, which is supported
by the global random attractor.
It is well-known that lattice dynamical systems have wide applications in physics, chem-
istry, biology and engineering such as pattern formation, image processing, propagation
of nerve pulses, electric circuits and so on. The theory of attractors for deterministic or s-
tochastic lattice systems has been widely developed. Therefore, we focus on the asymptotical
behavior of attractors for lattice dynamical systems in the last two chapters.
Two problems related to FitzHugh-Nagumo lattice systems are analyzed in Chapter
4. The rst one is concerned with the asymptotic behavior of random delay FitzHugh-
Nagumo lattice systems driven by nonlinear Wong-Zakai noise. We obtain a new result
ensuring that such a system approximates the corresponding deterministic system when the
correlation parameter of Wong-Zakai noise goes to in nity rather than to zero. We rst
prove the existence of tempered random attractors for the random delay lattice systems
with a nonlinear drift function and a nonlinear di usion term. The pullback asymptotic
compactness of solutions is proved thanks to the Ascoli-Arzel a theorem and uniform tail-
estimates. We then show the upper semi-continuity of attractors as the correlation parameter
tends to in nity. As for the second problem, we consider the corresponding deterministic
version of the previous model, and study the convergence of attractors when the delay
approaches zero. Namely, the upper semicontinuity of attractors for the delay system to the
nondelay one is proved.
Eventually, existence and connection of numerical attractors for discrete-time p-Laplace
lattice systems via the implicit Euler scheme are proved in Chapter 5. So far, it remains
open to obtain a numerical attractor for a non-autonomous (or stochastic) lattice system,
and thus we can at least investigate numerical attractors for the deterministic and non-
delayed version of p-Laplace lattice equations. The numerical attractors are shown to have
an optimized bound, which leads to the continuous convergence of the numerical attractors
2
when the graph of the nonlinearity closes to the vertical axis or when the external force
vanishes. A new type of Taylor expansions without Fr echet derivatives is established and
applied to show the discretization error of order two, which is crucial to prove that the
numerical attractors converge upper semi-continuously to the global attractor of the original
continuous-time system as the step size of the time goes to zero. It is also proved that
the truncated numerical attractors for nitely dimensional systems converge upper semi-
continuously to the numerical attractor and the lower semi-continuity holds in special cases.
The results of our investigation in this thesis are included in the following papers:
S. Yang, Y. Li, Q. Zhang and T. Caraballo, Stability analysis of stochastic 3D Lagrangian-
averaged Navier-Stokes equations with in nite delay, J. Dynam. Di erential Equations,
Published Online, (2023), Doi: 10.1007/s10884-022-10244-0.
S. Yang, T. Caraballo and Y. Li, Dynamics and stability analysis for stochastic 3D
Lagrangian-averaged Navier-Stokes equations with in nite delay on unbounded do-
mains, Appl. Math. Optim., (submitted).
S. Yang, T. Caraballo and Y. Li, Invariant measures for stochastic 3D Lagrangian-
averaged Navier-Stokes equations with in nite delay, Commun. Nonlinear Sci. Nu-
mer. Simul., 118 (2023), pp. 107004, 21.
S. Yang, Y. Li and T. Caraballo, Dynamical stability of random delayed FitzHugh-
Nagumo lattice systems driven by nonlinear Wong-Zakai noise, J. Math. Phys., 63
(2022), pp. 111512, 32.
Y. Li, S. Yang and Tom as Caraballo, Optimization and convergence of numerical at-
tractors for discrete-time quasi-linear lattice system, SIAM J. Numer. Anal., (to ap-
pear), 2023.; In this thesis we investigate the long time behavior of random dynamical systems associated to several
kinds of stochastic equations with delays in terms of stability for stationary solutions, weak pullback
mean random attractors, random attractors and numerical attractors. The thesis consists of three parts,
where the first part covers Chapters 1-3, the last two cover Chapters 4 and 5, respectively.
Chapters 1-3 are devoted to the random dynamics of 3D Lagrangian-averaged Navier-Stokes equations
with infinite delay in three cases.
In Chapter 1 we consider the stability analysis of such systems in the case of bounded domains.
We first use Galerkin’s approximations to prove the existence and uniqueness of solutions when the
non-delayed external force is locally integrable and the delay terms are globally Lipschitz continuous
with an additional assumption. We then prove the existence of a unique stationary solution to the
corresponding deterministic equation via the Lax-Milgram and the Schauder theorems. The stability
and asymptotic stability of stationary solutions (equilibrium solutions) are also established. The local
stability of stationary solutions for general delay terms is carried out by using a direct method and
then apply the abstract results to two kinds of infinite delays. It is worth mentioning that all conditions
are general enough to include several kinds of delays, where we mainly consider unbounded variable
delays and infinite distributed delays. As we know, it is still an open and challenging problem to obtain
su cient conditions ensuring the exponential stability of solutions in case of unbounded variable
delay. Fortunately, we obtained the exponential stability of stationary solutions in the case of infinite
distributed delay. However, we are able to further investigate the asymptotic stability of stationary
solutions in the case of unbounded variable delay by constructing suitable Lyapunov functionals.
Besides, we proved the polynomial asymptotic stability of stationary solutions for the particular case
of proportional delay.
In Chapter 2, we further discuss mean dynamics and stability analysis of stochastic systems in the
case of unbounded domains. We first prove the well-posedness of systems with infinite delay when the
non-delayed external force is locally integrable, the delay term is globally Lipschitz continuous and
the nonlinear di usion term is locally Lipschitz continuous, which leads to the existence of a mean
random dynamical system. We then obtain that such a dynamical system possesses a unique weak
pullback mean random attractor, which is a minimal, weakly compact and weakly pullback attracting
set. Moreover, we prove the existence and uniqueness of stationary solutions to the corresponding
deterministic equation via the classical Galerkin method, the Lax-Milgram and the Brouwer fixed
theorems. We discuss in the last part of Chapter 2 with those stability results concerning stationary
solutions discussed in Chapter 1.
The last case is concerned with the invariant measures for the autonomous version of stochastic
equations in Chapter 3 by using the method of generalized Banach limit. We first use Galerkin approximations,
a priori estimates and the standard Gronwall lemma to show the well-posedness for
the corresponding random equation, whose solution operators generate a random dynamical system.
Next, the asymptotic compactness for the random dynamical system is established via the Ascoli-
Arzelà theorem. Besides, we derive the existence of a global random attractor for the random dynamical
system. Moreover, we prove that the random dynamical system is bounded and continuous with
respect to the initial values. Eventually, we construct a family of invariant Borel probability measures,
which is supported by the global random attractor.
It is well-known that lattice dynamical systems have wide applications in physics, chemistry,
biology and engineering such as pattern formation, image processing, propagation of nerve pulses,
electric circuits and so on. The theory of attractors for deterministic or stochastic lattice systems has
been widely developed. Therefore, we focus on the asymptotical behavior of attractors for lattice
dynamical systems in the last two chapters.
Two problems related to FitzHugh-Nagumo lattice systems are analyzed in Chapter 4. The first
one is concerned with the asymptotic behavior of random delay FitzHugh-Nagumo lattice systems
driven by nonlinear Wong-Zakai noise. We obtain a new result ensuring that such a system approximates
the corresponding deterministic system when the correlation parameter of Wong-Zakai noise
goes to infinity rather than to zero. We first prove the existence of tempered random attractors for
the random delay lattice systems with a nonlinear drift function and a nonlinear di usion term. The
pullback asymptotic compactness of solutions is proved thanks to the Ascoli-Arzelà theorem and uniform
tail-estimates. We then show the upper semi-continuity of attractors as the correlation parameter
tends to infinity. As for the second problem, we consider the corresponding deterministic version of
the previous model, and study the convergence of attractors when the delay approaches zero. Namely,
the upper semicontinuity of attractors for the delay system to the nondelay one is proved.
Eventually, existence and connection of numerical attractors for discrete-time p-Laplace lattice
systems via the implicit Euler scheme are proved in Chapter 5. So far, it remains open to obtain
a numerical attractor for a non-autonomous (or stochastic) lattice system, and thus we can at least
investigate numerical attractors for the deterministic and non-delayed version of p-Laplace lattice
equations. The numerical attractors are shown to have an optimized bound, which leads to the continuous
convergence of the numerical attractors when the graph of the nonlinearity closes to the vertical
axis or when the external force vanishes. A new type of Taylor expansions without Fréchet derivatives
is established and applied to show the discretization error of order two, which is crucial to prove
that the numerical attractors converge upper semi-continuously to the global attractor of the original
continuous-time system as the step size of the time goes to zero. It is also proved that the truncated numerical
attractors for finitely dimensional systems converge upper semi-continuously to the numerical
attractor and the lower semi-continuity holds in special cases.
The results of our investigation in this thesis are included in the following papers:
S. Yang, Y. Li, Q. Zhang and T. Caraballo, Stability analysis of stochastic 3D Lagrangianaveraged
Navier-Stokes equations with infinite delay, J. Dynam. Di erential Equations, Published
Online, (2023), Doi: 10.1007/s10884-022-10244-0.
S. Yang, T. Caraballo and Y. Li, Dynamics and stability analysis for stochastic 3D Lagrangianix
averaged Navier-Stokes equations with infinite delay on unbounded domains, Appl. Math. Optim.,
(submitted).
S. Yang, T. Caraballo and Y. Li, Invariant measures for stochastic 3D Lagrangian-averaged
Navier-Stokes equations with infinite delay, Commun. Nonlinear Sci. Numer. Simul., 118 (2023),
pp. 107004, 21.
S. Yang, Y. Li and T. Caraballo, Dynamical stability of random delayed FitzHugh-Nagumo
lattice systems driven by nonlinear Wong-Zakai noise, J. Math. Phys., 63 (2022), pp. 111512,
32.
Y. Li, S. Yang and Tomás Caraballo, Optimization and convergence of numerical attractors for
discrete-time quasi-linear lattice system, SIAM J. Numer. Anal., (to appear), 2023.
2023-06-22T00:00:00ZReduced Basis Method applied to the Smagorinsky Turbulence Modelhttps://hdl.handle.net/11441/1355142024-02-17T16:22:34Z2022-06-03T00:00:00ZReduced Basis Method applied to the Smagorinsky Turbulence Model
This PhD dissertation addresses the numerical simulation of models that simulate the
behavior of turbulent flows through reduced-order techniques. The numerical simulation
of this kind of flows is complex and pricey, as well as necessary to the Eco-efficient
building optimized design, among many others applications in engineering and architecture.
The integration of reduced-order techniques is the key to reducing by orders of magnitude
the time and computational cost in the numerical simulation of these problems.
In this dissertation, we use the Smagorinsky model, a Large Eddy Simulation (LES)
model based upon the Navier-Stokes equations that allows for the resolution of turbulent
flows with coarser meshes. Even then, the computational cost is high, especially in 3D
cases.
With respect to the Reduced Order Model (ROM), there exist some techniques to
obtain the ROM. In this dissertation, we focus on the development of Reduced Basis (RB)
method. Upon an ancillary basis, we shall use Proper Orthogonal Decomposition (POD).
For ROM obtainment, it is necessary to compute the error between the approximated
model and the reduced model through the RB method, which could become very expensive
and complex. This is the reason to study a posteriori estimates to estimate the error
and which computation is faster.
The Smagorinsky model is non-linear since it is derived by the Navier-Stokes equations
and the estimator development for non-linear problems requires the use of adapted
mathematical techniques.
For steady non-linear models, we find estimates based on the Brezzi-Rappaz-Raviart
(BRR) theory of non-singular branches approximation of parametric non-linear problems.
It is a theory essentially based upon the Implicit function theorem. This estimator has
already been developed for steady flows, although it has only be applied to 2D flows.
Therefore, we shall start applying this estimator to the 3D case, obtaining large reductions
in time and computational cost.
We also apply this estimator to a realistic design problem for a cloister focused on
the thermal comfort optimization of the ground floor. The parameters to consider for
the problem are the height and width of the corridors around the cloister. We obtain an
optimal answer by analyzing 625 possibilities in 16 minutes. One of the main challenges addressed in this dissertation is to extend the a posteriori
estimator to unsteady problems. To start, it is necessary to analyze previous studies on
a priori estimations that involve velocity and pressure. Thus, we are able to develop
an a posteriori estimator under some hypothesis.
Furthermore, we develop an alternative based Kolmogórov’s theory of statistical
equilibrium, based on the existence of an energy cascade that spreads the energy from
large eddies to the smallest ones, dissipating the energy thanks to viscosity. This cascade
generates an inertial energy spectrum with a determined shape that we use as the estimator.
We have validated this estimator with an academic test, using a POD+Greedy strategy.
We obtain similar results using the estimator and the committed real error.; Esta tesis se enmarca en la resolución numérica de modelos que simulan el comportamiento
de flujos turbulentos mediante técnicas de orden reducido. La resolución numérica de
este tipo de flujos es compleja y costosa, a la par que necesaria para el diseño óptimo de
edificios eco-eficientes, entre otras muchas aplicaciones en ingeniería y arquitectura. La
incorporación de técnicas de Orden Reducido es clave para la reducción en órdenes de
magnitud el tiempo y coste computacional en la resolución numérica de estos problemas.
En esta tesis utilizaremos el modelo de Smagorinsky, un modelo de turbulencia de
tipo LES (Large Eddy Simulation) basado en las ecuaciones de Navier-Stokes que permite
la resolución de flujo turbulento sin la necesidad de mallas muy finas. Aún así, el coste
computacional es elevado, sobre todo en casos 3D.
Dentro de la modelización de orden reducido, existen varias técnicas que permiten
la obtención del modelo reducido. En esta tesis nos centramos en el desarrollo de
métodos de Bases Reducidas (BR). Usaremos de forma auxiliar la “Proper Orthogonal
Decomposition” (POD). Para la obtención del modelo reducido, es necesario el cálculo
del error entre el modelo aproximado y el modelo reducido mediante el método BR, lo
cuál puede llegar a ser muy costoso y complejo. Es por ello que se estudia la deducción
de estimadores a posteriori que permiten estimar este error y cuyo cálculo sea más rápido.
El modelo de Smagorinsky es no lineal ya que deriva de las ecuaciones de Navier-Stokes
y el desarrollo de estimadores para problemas no lineales requiere la utilización de
técnicas matemáticas adaptadas.
Para un modelo estacionario no lineal, encontramos estimadores basados en la teoría
Brezzi-Rappaz-Raviart (BRR) de aproximación de ramas no singulares de problemas no
lineales paramétricos. Es una teoría esencialmente basada sobre el teorema de la función
implícita. Este estimador ha sido ya elaborado para flujos estacionarios, aunque solo se
ha aplicado a flujos 2D. Por lo que, comenzamos aplicando este estimador a un caso 3D,
obteniendo grandes reducciones en cuanto a tiempo y coste computacional.
También aplicamos este estimador a un problema realista de diseño de claustros
orientado a la optimización del confort térmico en la planta baja. Los parámetros a
considerar para el problema son la altura y anchura del pasillo que rodea el claustro.
Conseguimos obtener una respuesta óptima analizando 625 posibilidades en 16 minutos. Uno de los principales desafíos que abordamos es extender la obtención de estimadores
a posteriori a problemas no estacionarios. Para empezar, es necesario realizar estudios
previos sobre estimaciones a priori que involucren a la velocidad y presión. Así, somos
capaces de desarrollar un estimador a posteriori asumiendo ciertas hipótesis.
Por otro lado, desarrollamos una alternativa basándonos en la teoría de turbulencia
en equilibrio estadístico de Kolmogórov, por la cuál sabemos que existe una cascada de
energía que se propaga desde los grandes torbellinos hacia los más pequeños disipando
la energía en ellos gracias a la viscosidad. Esta cascada genera un espectro de energía
inercial con una forma determinada que utilizamos como estimador. Validamos dicho
estimador con un test académico, utilizando una estrategia POD+Greedy, obteniendo
resultados similares utilizando el estimador y el error real cometido.
2022-06-03T00:00:00ZFinite-dimensionality of attractors for dynamicl systems wigh applications: deterministic and random settingshttps://hdl.handle.net/11441/1343922024-02-17T17:48:39Z2021-01-27T00:00:00ZFinite-dimensionality of attractors for dynamicl systems wigh applications: deterministic and random settings
In this work we obtain estimates on the fractal dimension of attractors in three different settings: global attractors associated to autonomous dynamical systems, uniform attractors associated to non-autonomous dynamical systems and random uniform attractors associated to non-autonomous random dynamical systems. Firstly we give a simple proof of a result due to Mañé (Springer LNM 898, 230242, 1981) that the global attractor A (as a subset of a Banach space) for a map S is finite-dimensional if DS(x) =C(x)+L(x), where C is compact and L is a contraction (and both are linear). In particular, we show that if S is compact and differentiable then A is finite-dimensional. Using a smoothing property for the differential DS we also prove that A has finite fractal dimension and we make a comparison of this method with Mañés approach. We give applications to an abstract semilinear parabolic equation and to 2D Navier-Stokes equations. Secondly we prove using a smoothing method that uniform attractors have finite fractal dimension on Banach spaces, with bounds in terms of the dimension of the symbol space and a Kolmogorov entropy number. We also show that the smoothing property is useful to prove the finite-dimensionality of uniform attractors in more regular Banach spaces. In addition, we prove that the finite-dimensionality of the hull of a time-dependent function is fully determined by the tails of the function. We give applications to non-autonomous 2D Navier- Stokes and reaction-diffusion equations. Thirdly we prove using a smoothing and a squeezing method that random uniform attractors have finite fractal dimension. Neither of the two methods implies the other. Estimates on the dimension are given in terms of the dimension of the symbol space plus a term arising from the smoothing/squeezing property; the smoothing is applied also to more regular spaces. In this setting we give applications to a stochastic reaction-diffusion equation with scalar additive noise. In addition, a random absorbing set which absorbs itself after a deterministic period of time is constructed.
2021-01-27T00:00:00Z