Existence of global weak solutions to the kinetic Hookean dumbbell model for incompressible dilute polymeric fluids
John W. Barrett, Endre S\"uli
TL;DR
This paper proves the existence of global weak solutions for the kinetic Hookean dumbbell model in 2D and weak subsolutions in 3D, linking microscopic polymer models to macroscopic fluid equations.
Contribution
It establishes the existence of large-data global weak solutions for the Hookean dumbbell model in two dimensions and introduces the concept of weak subsolutions with defect measures in three dimensions.
Findings
Existence of large-data global weak solutions in 2D.
Existence of weak subsolutions with defect measures in 3D.
Rigorous connection between the Hookean dumbbell model and the Oldroyd-B model.
Abstract
We explore the existence of global weak solutions to the Hookean dumbbell model, a system of nonlinear partial differential equations that arises from the kinetic theory of dilute polymers, involving the unsteady incompressible Navier--Stokes equations in a bounded domain in two or three space dimensions, coupled to a Fokker--Planck-type parabolic equation. We prove the existence of large-data global weak solutions in the case of two space dimensions. Indirectly, our proof also rigorously demonstrates that, in two space dimensions at least, the Oldroyd-B model is the macroscopic closure of the Hookean dumbbell model. In three space dimensions, we prove the existence of large-data global weak subsolutions to the model, which are weak solutions with a defect measure, where the defect measure appearing in the Navier--Stokes momentum equation is the divergence of a symmetric positive…
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Taxonomy
TopicsNavier-Stokes equation solutions · Fluid Dynamics and Turbulent Flows · Gas Dynamics and Kinetic Theory
EXISTENCE OF GLOBAL WEAK SOLUTIONS TO
THE KINETIC HOOKEAN DUMBBELL MODEL FOR
INCOMPRESSIBLE DILUTE POLYMERIC FLUIDS
*John W. Barrett
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*Endre Süli
Mathematical Institute, University of Oxford, Oxford OX1 3LB, UK
( )
Abstract
We explore the existence of global weak solutions to the Hookean dumbbell model, a system of nonlinear partial differential equations that arises from the kinetic theory of dilute polymers, involving the unsteady incompressible Navier–Stokes equations in a bounded domain in two or three space dimensions, coupled to a Fokker–Planck-type parabolic equation. We prove the existence of large-data global weak solutions in the case of two space dimensions. Indirectly, our proof also rigorously demonstrates that, in two space dimensions at least, the Oldroyd-B model is the macroscopic closure of the Hookean dumbbell model. In three space dimensions, we prove the existence of large-data global weak subsolutions to the model, which are weak solutions with a defect measure, where the defect measure appearing in the Navier–Stokes momentum equation is the divergence of a symmetric positive semidefinite matrix-valued Radon measure.
Keywords: Kinetic polymer models, Hookean dumbbell model, Navier–Stokes–Fokker–Planck system, dilute polymer, Oldroyd-B model
1 Introduction
The aim of this paper is to explore the existence of global weak solutions to the Hookean dumbbell model, — a system of nonlinear partial differential equations involving the coupling of the time-dependent incompressible Navier–Stokes equations to a parabolic Fokker–Planck type equation, — which arises from the kinetic theory of dilute polymeric fluids. In this model the solvent is assumed to be an isothermal, viscous, incompressible, Newtonian fluid in a bounded open Lipschitz domain , or . We will admit both and for the vast majority of the paper, even though our main result concerning the existence of large-data global weak solutions is, ultimately, restricted to the case of . In the model the equation for conservation of linear momentum in the Navier–Stokes system involves, as a source term, an elastic extra-stress tensor
(i.e., the polymeric part of the Cauchy stress tensor), to be defined below in terms of the solution of a coupled Fokker–Planck type equation.
Given , we seek a nondimensional velocity field {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}\,:\,({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},t)\in\overline{\Omega}\times[0,T]\mapsto{\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},t)\in{\mathbb{R}}^{d} (which, for simplicity, we shall require to satisfy a no-slip boundary condition on ) and a nondimensional pressure p\,:\,({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},t)\in\Omega\times(0,T]\mapsto p({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},t)\in{\mathbb{R}}, such that
[TABLE]
In these equations is the reciprocal of the Reynolds number and is the nondimensional density of body forces.
The simplest kinetic model for a dilute polymeric fluid is the dumbbell model, where long polymer chains suspended in the viscous incompressible Newtonian solvent are assumed not to interact with each other, and each chain is idealized as a pair of massless beads, connected with an elastic spring. The associated elastic extra-stress tensor
is defined by the Kramers expression in terms of , the probability density function of the (random) conformation vector
of the spring (cf. (1.11) below), and the domain of admissible conformation vectors
is either the whole of or a bounded open -dimensional ball centred at the origin {\vtop{\hbox{0}\hbox{\scriptscriptstyle\sim}}}{}\in\mathbb{R}^{d}. The evolution of from a given nonnegative initial datum, , is governed by a second-order parabolic partial differential equation, the Fokker–Planck equation, whose transport coefficients depend on the velocity field
.
In [6] we were concerned with models where is a bounded open ball in , , resulting in, what are known as, finitely extensible nonlinear (FENE) models. Here, as in [7], we shall be concerned with the technically more subtle case when , i.e., the spring between the beads is allowed to have an arbitrarily large extension. In fact, in both [6] and [7] we considered the more general case of polymer models involving linear chains of beads coupled with springs, where . Much of the analysis below carries across to , but as our final result is restricted to we shall confine ourselves to this case from the start for simplicity. Although springs with arbitrarily large extension are physically unrealistic, thanks to their simplicity models of this kind are, nevertheless, frequently used in practice. The elastic spring-force {\vtop{\hbox{F}\hbox{\scriptscriptstyle\sim}}}{}\,:\,D=\mathbb{R}^{d}\rightarrow\mathbb{R}^{d} of the spring is then defined by
[TABLE]
where , and is monotonic nondecreasing and unbounded on . We note that, among all such potentials , only the Hookean potential , with associated spring force {{\vtop{\hbox{F}\hbox{\scriptscriptstyle\sim}}}{}}({{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}})={{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}}, {{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}}\in{D}=\mathbb{R}^{d}, yields, formally at least, closure to a macroscopic model, the Oldroyd-B model (cf. [24]), which we shall state below.
In our paper [7], we further assumed that there exist constants , , such that the (normalized) Maxwellian , is defined on by
[TABLE]
and the associated spring potential satisfies, for ,
[TABLE]
Therefore, in [7] we were unable to cover the case of the Hookean model, which corresponds to the choice in (1.4a–c). This was due to the failure of a compactness argument, used in passing to a limit on the extra-stress tensor in the existence proof there, which required that the mapping had superlinear growth at infinity; see Section 6 below for further details. We note that apart from this step, all the other results in [7] remain valid with in (1.4a–c). Thus in [7] we could deal with a spring potential of the form
[TABLE]
for any and , which approximates the Hookean potential . Hence our use of the terminology Hookean-type model throughout the paper [7] (instead of Hookean model, which would have corresponded to taking in the above).
In this paper, we overcome the difficulties encountered with the compactness result used on the extra-stress term in [7] and so we are able to address the Hookean model. This is achieved by relating the Hookean model to its macroscopic closure, the Oldroyd-B model, and using an existence result, and establishing regularity results, for the latter. We shall assume henceforth, in place of (1.4a–c), that
[TABLE]
By recalling (1.3), we observe that the Maxwellian satisfies
[TABLE]
It follows from (1.3) and (1.8) that, for any ,
[TABLE]
The governing equations of the Hookean dumbbell model are (1.1a–d), where the extra-stress tensor
, dependent on the probability density function \psi:({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{},t)\in\Omega\times D\times[0,T]\mapsto\psi({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{},t)\in{\mathbb{R}}_{\geq 0}, is defined by the Kramers expression:
[TABLE]
where the dimensionless constant is a constant multiple of the product of the Boltzmann constant and the absolute temperature ,
is the unit tensor,
[TABLE]
and the density of polymer chains located at
at time is given by
[TABLE]
The probability density function is a solution, in , of the Fokker–Planck equation
[TABLE]
where, for {\vtop{\hbox{v}\hbox{\scriptscriptstyle\sim}}}{}={\vtop{\hbox{v}\hbox{\scriptscriptstyle\sim}}}{}({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},t)\in\mathbb{R}^{d}, ({\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\approx}}}{}_{x}\,{\vtop{\hbox{v}\hbox{\scriptscriptstyle\sim}}}{})({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},t)\in{\mathbb{R}}^{d\times d} and \{{\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\approx}}}{}_{x}\,{\vtop{\hbox{v}\hbox{\scriptscriptstyle\sim}}}{}\}_{ij}:=\textstyle\frac{\partial v_{i}}{\partial x_{j}}. In (1.13), is the dimensionless centre-of-mass diffusion coefficient and the dimensionless parameter is the Deborah number; we refer the reader to [6, 7, 5, 8] for further details.
Finally, we impose the following decay/boundary and initial conditions on :
[TABLE]
where
is the unit outward normal vector to .
Here is a nonnegative function defined on , with \int_{D}\psi_{0}({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}){\,\rm d}{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}=1 for a.e. {\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{}\in\Omega. The boundary conditions for on and the decay conditions for on as |{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}|\rightarrow\infty have been chosen so as to ensure that
[TABLE]
Definition 1.1
The set of equations and hypotheses (1.1a–d), (1.3), (1.8) and (1.11)–(1.14a–c)* will be referred to henceforth as model , or as the Hookean dumbbell model (with centre-of-mass diffusion). *
Next, for any {\vtop{\hbox{a}\hbox{\scriptscriptstyle\sim}}}{}\in{\mathbb{R}}^{d}, we note the following results:
[TABLE]
Multiplying (1.13) by {\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}\,{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}^{\rm T}, integrating over , performing integration by parts (assuming that and {\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\sim}}}{}_{q}\psi decay to zero, sufficiently fast as |{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}|\rightarrow\infty) and noting (1.16), and, similarly, integrating (1.13) over and noting (1.14a), yields formally that {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(\psi)({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},t)\in{\mathbb{R}}^{d\times d} and \rho(\psi)({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},t)\in{\mathbb{R}} satisfy
[TABLE]
subject to the boundary and initial conditions
[TABLE]
As it is assumed that \int_{D}\psi_{0}({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{})\,\,{\rm d}{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}=1 for a.e. {\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{}\in\Omega, it follows that is the unique solution of (1.17b,c).
Definition 1.2
The collection of equations (1.1a–d), (1.11) and (1.17a–c) will be referred to throughout the paper as model , or as the Oldroyd-B model (with stress-diffusion).
A remark is in order concerning the evolution equation (1.17a) for the extra stress tensor
.
Remark 1.1
*By suppressing in our notation the dependence of
and on and setting , equation (1.17a) becomes*
[TABLE]
This is precisely the classical Oldroyd-B evolution equation for the elastic extra-stress, with our factor usually replaced by , which is easily derived from equation (59) in Oldroyd’s paper [24]. Indeed, by interpreting the convective derivative in equation (59) in [24] as the upper convected derivative
[TABLE]
*and by additively splitting the (total) Cauchy stress tensor into a part
, to be defined below, related to the distortion (deformation at constant volume), and an isotropic tensor that is a scalar multiple of
, as in equation (52) in [24], equation (59) in [24] states that*
[TABLE]
*is the symmetric velocity gradient, is the sum of the solvent viscosity and the polymeric viscosity ; and , the relaxation time, and , the retardation time, are two constants with the dimension of time. By splitting the tensor
additively into its solvent part {\vtop{\hbox{T}\hbox{\scriptscriptstyle\approx}}}{}_{s} and polymeric part {\vtop{\hbox{T}\hbox{\scriptscriptstyle\approx}}}{}_{p} as {\vtop{\hbox{T}\hbox{\scriptscriptstyle\approx}}}{}={\vtop{\hbox{T}\hbox{\scriptscriptstyle\approx}}}{}_{s}+{\vtop{\hbox{T}\hbox{\scriptscriptstyle\approx}}}{}_{p}, where {\vtop{\hbox{T}\hbox{\scriptscriptstyle\approx}}}{}_{s}:=2\mu_{s}\,{\vtop{\hbox{D}\hbox{\scriptscriptstyle\approx}}}{}, and setting , it follows that*
[TABLE]
As \overset{\tiny\nabla}{{\vtop{\hbox{I}\hbox{\scriptscriptstyle\approx}}}{}}=-2{\vtop{\hbox{D}\hbox{\scriptscriptstyle\approx}}}{}, we then have that {\vtop{\hbox{T}\hbox{\scriptscriptstyle\approx}}}{}_{p}+\lambda\,\overset{\tiny\nabla}{{\vtop{\hbox{T}\hbox{\scriptscriptstyle\approx}}}{}_{p}}=-\mu_{p}\,\overset{\tiny\nabla}{{\vtop{\hbox{I}\hbox{\scriptscriptstyle\approx}}}{}}. Thus, by defining {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}:=\frac{\rho\lambda}{\mu_{p}}\,{\vtop{\hbox{T}\hbox{\scriptscriptstyle\approx}}}{}_{p}+\rho\,{\vtop{\hbox{I}\hbox{\scriptscriptstyle\approx}}}{}, where is the solution of \frac{\partial\rho}{\partial t}+({\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}\cdot{\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\sim}}}{}_{x})\rho=0 (i.e. (1.17b) with ) subject to a given initial condition \rho({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},0)=\rho_{0}({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{}), we deduce that
[TABLE]
*which is precisely (1.18) upon replacing Oldroyd’s by our ; in particular, the choice of \rho_{0}({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{})\equiv 1 yields \rho({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},t)\equiv 1 for all ({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},t)\in\Omega\times(0,T], in agreement with the statement in the sentence preceding Definition 1.2. In summary then, {\vtop{\hbox{T}\hbox{\scriptscriptstyle\approx}}}{}:={\vtop{\hbox{T}\hbox{\scriptscriptstyle\approx}}}{}_{s}+{\vtop{\hbox{T}\hbox{\scriptscriptstyle\approx}}}{}_{p}, with {\vtop{\hbox{T}\hbox{\scriptscriptstyle\approx}}}{}_{s}:=2\mu_{s}\,{\vtop{\hbox{D}\hbox{\scriptscriptstyle\approx}}}{} and {\vtop{\hbox{T}\hbox{\scriptscriptstyle\approx}}}{}_{p}:=\frac{\mu_{p}}{\rho\lambda}({\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}-\rho{\vtop{\hbox{I}\hbox{\scriptscriptstyle\approx}}}{}), where
and are solutions to the partial differential equations appearing in the previous sentence, with being an admissible special case. For an alternative, thermodynamically consistent, derivation of the Oldroyd-B model we refer to [22].*
We continue with a brief literature survey. In our paper [7] we proved the existence and equilibration of large-data global weak solutions to general noncorotational Hookean-type bead-spring chain models with stress-diffusion in both two and three space dimensions, under the assumption that the spring potentials appearing in the model exhibit superlinear growth at infinity. We were, however, unable to cover the classical Hookean dumbbell model, where the spring potential has linear growth at infinity. Our objective here is to close this gap, in the case of two space dimensions at least. The relevance of the Hookean dumbbell model is that it has a formal macroscopic closure: the Oldroyd-B model. Lions and Masmoudi [21] proved global existence of large-data weak solutions to a corotational Oldroyd-B model (i.e. one where the gradient of the velocity field in the stress evolution equation is replaced by the skew-symmetric part of the velocity gradient) without stress-diffusion, in both two and three space dimensions. Returning to the general noncorotational case, Hu and Lin [20] proved the global existence of weak solutions to incompressible viscoelastic flows, including the Oldroyd-B model, without stress-diffusion, in two spatial dimensions, under the assumption that the initial deformation gradient is close to the identity matrix in {\vtop{\hbox{L}\hbox{\scriptscriptstyle\approx}}}{}^{2}(\Omega)\cap{\vtop{\hbox{L}\hbox{\scriptscriptstyle\approx}}}{}^{\infty}(\Omega) and the initial velocity is small in {\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}^{2}(\Omega) and bounded in {\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}^{p}(\Omega) for some . In [3], Barrett & Boyaval proved the existence of large-data global weak solutions to the Oldroyd-B model, in the presence of stress-diffusion, again in two spatial dimensions. Constantin and Kliegl [11] subsequently showed the global regularity of solutions to the Oldroyd-B model with stress-diffusion in two space dimensions. Motivated by [3] and maximal regularity results for the unsteady Stokes and Navier–Stokes systems (cf. [19], [26], [14], for example, and references therein), we revisit the classical Hookean dumbbell model and prove, in the general noncorotational case, in two space dimensions, the existence of large-data global weak solutions. Indirectly, our argument also rigorously proves that, in two space dimensions at least, the Oldroyd-B model with stress-diffusion is the macroscopic closure of the Hookean dumbbell model with centre-of-mass diffusion. In the case of three dimensions the question of existence of large-data global weak solutions to both the general noncorotational Hookean dumbbell model and the noncorotational Oldroyd-B model with stress-diffusion remains open, although we will show here the existence of large-data global weak subsolutions to the general noncorotational Hookean dumbbell model for .
The paper is structured as follows. In the next section we introduce our notation and useful results, such as compactness theorems. Henceforth, we shall frequently write
[TABLE]
In Section 3, we recall from [3] a global-in-time existence result for the Oldroyd-B model when . We then establish some regularity results for this solution, ({\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\rm OB},{\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}), and a uniqueness result for the stress equation. In Section 4, we prove the existence of a global-in-time weak solution, , to the Fokker–Planck equation, for a given velocity field {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\star}, via regularization and time discretization. In addition, we show that {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\widehat{\psi}_{\star}) solves the corresponding Oldroyd-B stress equation with given velocity field {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\star}. In Section 5 we combine the results of the previous two sections to establish the the existence of a global-in-time weak solution, ({\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\rm OB},\widehat{\psi}_{\rm OB}), to the Hookean dumbbell model when . Moreover, we show that ({\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\rm OB},{\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\widehat{\psi}_{\rm OB})) solves the Oldroyd-B model. Finally in Section 6 we explain why we had to resort to the reasoning upon which our existence proof is based, and why a more direct argument is only capable of showing, for both and , the existence of large-data global weak subsolutions (in a sense to be made precise in Section 6).
2 Preliminaries
Let , or 3, be a bounded open set with a Lipschitz-continuous boundary , and the set of elongation vectors {\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}\in D\equiv{\mathbb{R}}^{d}. Let
[TABLE]
where the divergence operator {\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\sim}}}{}_{x}\cdot is to be understood in the sense of vector-valued distributions on . Let [{\vtop{\hbox{H}\hbox{\scriptscriptstyle\sim}}}{}^{1}_{0}(\Omega)]^{\prime} denote the dual of {\vtop{\hbox{H}\hbox{\scriptscriptstyle\sim}}}{}^{1}_{0}(\Omega). We recall the following well-known Gagliardo–Nirenberg inequality. Let if , and if and . Then, there is a constant , such that, for all :
[TABLE]
Let , , denote the Maxwellian-weighted space over with norm
[TABLE]
Similarly, we introduce , the Maxwellian-weighted space over . On defining
[TABLE]
we then set
[TABLE]
Similarly, we introduce , the Maxwellian-weighted space over . It is shown in Appendix A of [7] that
[TABLE]
In addition, we note that the embeddings
[TABLE]
are compact; see Appendix D and Appendix F of [7], respectively.
Next, we note that (1.12a) and (1.10) yield, for and any , that
[TABLE]
In addition, the following simple lemma will be useful (cf. Lemma 5.1 in [5] for the proof).
Lemma 2.1
Suppose that a sequence converges in to , and is bounded in , i.e., there exists such that for all . Then, for all , and the sequence converges to in for all .
We shall use the symbol to denote the absolute value when the argument is a real number, the Euclidean norm when the argument is a vector, and the Frobenius norm when the argument is a square matrix. For a square matrix {\vtop{\hbox{B}\hbox{\scriptscriptstyle\approx}}}{}\in\mathbb{R}^{d\times d}, we recall that the symbol \mathfrak{tr}({\vtop{\hbox{B}\hbox{\scriptscriptstyle\approx}}}{}) will signify the trace of
.
We state a simple integration-by-parts formula (cf. Lemma 3.1 in [7] and note that ).
Lemma 2.2
Suppose that and let {\vtop{\hbox{B}\hbox{\scriptscriptstyle\approx}}}{}\in\mathbb{R}^{d\times d} be a square matrix such that \mathfrak{tr}({\vtop{\hbox{B}\hbox{\scriptscriptstyle\approx}}}{})=0; then,
[TABLE]
Let be defined by
[TABLE]
Clearly, , and is a nonnegative, strictly convex function. We note the following result.
Lemma 2.3
For all such that we have that
[TABLE]
*where . *
- Proof
See the proof of Lemma 4.1 in [7], where it is proved with \frac{1}{2}\,|{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}|^{2} and replaced by (\frac{1}{2}\,|{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}|^{2})^{\vartheta} and , respectively, and , recall (1.3), based on , satisfying (1.4a–c); but the proof given there is valid for and hence for based on (1.8).
We recall the Aubin–Lions–Simon compactness theorem, see, e.g., Temam [27] and Simon [25]. Let , and be Banach spaces, where , , are reflexive, with a compact embedding and a continuous embedding . Then, for , , the embedding
[TABLE]
is compact.
We shall also require the following generalization of the Aubin–Lions–Simon compactness theorem due to Dubinskiĭ [13]; see also [9]. Before stating the result, we recall the concept of a seminormed set (in the sense of Dubinskiĭ). A subset of a linear space over is said to be a seminormed set if , for any and , and there exists a functional (namely the seminorm of ), denoted by , such that:
- (i)
; and if, and only if, ; 2. (ii)
, .
A subset of a seminormed set is said to be bounded if there exists a positive constant such that for all . A seminormed set contained in a normed linear space with norm is said to be continuously embedded in , and we write , if there exists a such that for all . The embedding of a seminormed set into a normed linear space is said to be compact if from any bounded infinite set of elements of one can extract a subsequence that converges in .
Theorem 2.1** (Dubinskiĭ [13])**
Suppose that and are Banach spaces, , and is a seminormed subset of such that the embedding is compact. Then, for , , the embedding
[TABLE]
is compact.
3 The Oldroyd-B model
We start with noting the following existence result for problem (Q), Oldroyd-B, with .
Theorem 3.1
Let and . In addition, let {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{0}\in{\vtop{\hbox{H}\hbox{\scriptscriptstyle\sim}}}{}, {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{0}\in{\vtop{\hbox{L}\hbox{\scriptscriptstyle\approx}}}{}^{2}(\Omega) with {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{0}={\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{0}^{\rm T}\geq 0 a.e. in and {\vtop{\hbox{f}\hbox{\scriptscriptstyle\sim}}}{}\in{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}^{2}(0,T;[{\vtop{\hbox{H}\hbox{\scriptscriptstyle\sim}}}{}^{1}_{0}(\Omega)]^{\prime}). Then there exist
[TABLE]
with {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}={\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}^{\rm T}\geq 0 a.e. in such that
[TABLE]
- Proof
Existence of a solution to (Q) with is proved in [3] via a finite element approximation for a polygonal domain under the stronger assumption {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{0}\in{\vtop{\hbox{L}\hbox{\scriptscriptstyle\approx}}}{}^{\infty}(\Omega) with {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{0}={\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{0}^{\rm T}>0 a.e. in . The restriction on was purely for ease of exposition for the finite element approximation. Existence of a solution to a compressible version of (Q) is proved in [4] under the stronger assumptions , for , and {\vtop{\hbox{f}\hbox{\scriptscriptstyle\sim}}}{}\in L^{\infty}(\Omega\times(0,T)), with {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{0}\in{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}^{2}(\Omega). The proof there is easily adapted to the far simpler incompressible case for and {\vtop{\hbox{f}\hbox{\scriptscriptstyle\sim}}}{}\in{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}^{2}(0,T;[{\vtop{\hbox{H}\hbox{\scriptscriptstyle\sim}}}{}^{1}_{0}(\Omega)]^{\prime}), with {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{0}\in{\vtop{\hbox{H}\hbox{\scriptscriptstyle\sim}}}{}. For example, in the existence proof for the compressible version of (Q) discussed in [4] the assumption is only needed because one requires high regularity for a parabolic Neumann problem; see Lemma 3.2 below, which, however, is not needed in the existence proof for (Q) in the incompressible case.
Remark 3.1
Since the test functions in
are divergence-free, the pressure has been eliminated in (3.2); it can be recovered in a very weak sense following the same procedure as for the incompressible Navier–Stokes equations discussed on p. 208 in [27]; i.e., one obtains that .**
We now deduce additional regularity for this Oldroyd-B model.
Lemma 3.1
Let and suppose that the hypotheses of Theorem 3.1 hold; then,
[TABLE]
- Proof
The first two results in (3.3) follow directly from (3.1) and (2.2). The remaining results in (3.3) follow directly from the first two results in (3.3) and (3.1) using Hölder’s inequality.
In order to improve the regularity results (3.1) and (3.3), we consider the parabolic initial-boundary-value problem:
[TABLE]
where . We require the following definitions to state a regularity result for (3.4a–c). First we introduce fractional-order Sobolev spaces. For any , and , we define
[TABLE]
where
[TABLE]
and . We then define , for , to be the completion of \{\zeta\in C^{\infty}(\overline{\Omega}):{\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\sim}}}{}_{x}\zeta\,\cdot\,{\vtop{\hbox{n}\hbox{\scriptscriptstyle\sim}}}{}=0\ \mbox{on }\partial\Omega\} in the norm of . We now recall the following regularity result for (3.4a–c); see e.g. Lemma 7.37 in [23].
Lemma 3.2
Let with , for . In addition, let and , for . Then, there exists a unique function
[TABLE]
solving (3.4a–c). Here (3.4b) is satisfied in the sense of the normal trace, which is well defined since . Moreover, we have that
[TABLE]
We now apply Lemma 3.2 to the stress equation (3.2).
Lemma 3.3
Let , , for , and {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{0}\in{\vtop{\hbox{W}\hbox{\scriptscriptstyle\approx}}}{}_{n}^{\frac{1}{2},\frac{4}{3}}(\Omega) with {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{0}={\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{0}^{\rm T}\geq 0 a.e. in . Then we have that
[TABLE]
- Proof
Applying Lemma 3.2 with to each component of (3.2) and noting (3.3) yields the first, and hence the second, result in (3.5). The final result in (3.5) follows from the second result and Sobolev embedding as .
To improve the regularity results (3.1), (3.3) and (3.5) further, we now consider the Stokes initial-boundary-value problem for :
[TABLE]
and the Navier–Stokes initial-boundary-value problem, where (3.6a) is replaced by
[TABLE]
In order to state a regularity result for (3.6a–c), we require the following definitions. The first is a generalisation of
, let {\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}_{\rm div}^{s}(\Omega) be the completion of \{{\vtop{\hbox{w}\hbox{\scriptscriptstyle\sim}}}{}\in{\vtop{\hbox{C}\hbox{\scriptscriptstyle\sim}}}{}^{\infty}_{0}(\Omega):{\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\sim}}}{}_{x}\cdot{\vtop{\hbox{w}\hbox{\scriptscriptstyle\sim}}}{}=0\mbox{ in }\Omega\} in {\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}^{s}(\Omega) for . So {\vtop{\hbox{H}\hbox{\scriptscriptstyle\sim}}}{}\equiv{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}_{\rm div}^{2}(\Omega). Next, we introduce, for and ,
[TABLE]
where is the Stokes operator with domain D(A_{s})={\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}_{\rm div}^{s}(\Omega)\cap{\vtop{\hbox{W}\hbox{\scriptscriptstyle\sim}}}{}^{1,s}_{0}(\Omega)\cap{\vtop{\hbox{W}\hbox{\scriptscriptstyle\sim}}}{}^{2,s}(\Omega) and P_{s}:{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}^{s}(\Omega)\rightarrow{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}_{\rm div}^{s}(\Omega) is the Helmholtz projection, see [19] for details and Remark 3.2 below. We now recall the following regularity result for (3.6a–c), see Theorem 2.8 in [19].
Lemma 3.4
*Let with , for . In addition, let {\vtop{\hbox{v}\hbox{\scriptscriptstyle\sim}}}{}_{0}\in{\vtop{\hbox{D}\hbox{\scriptscriptstyle\sim}}}{}^{1-\frac{1}{r},r}_{s}(\Omega) and {\vtop{\hbox{g}\hbox{\scriptscriptstyle\sim}}}{}\in L^{r}(0,T;{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}_{\rm div}^{s}(\Omega)), for . Then, there exist unique functions
and {\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\sim}}}{}_{x}\pi satisfying*
[TABLE]
and solving (3.6a–c). Moreover, we have that
[TABLE]
In addition, we recall the following regularity result for (3.7), (3.6b,c), see Theorem 3.10 on p.213 in [27].
Lemma 3.5
Let , , {\vtop{\hbox{v}\hbox{\scriptscriptstyle\sim}}}{}_{0}\in{\vtop{\hbox{V}\hbox{\scriptscriptstyle\sim}}}{} and {\vtop{\hbox{g}\hbox{\scriptscriptstyle\sim}}}{}\in L^{2}(0,T;{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}_{\rm div}^{2}(\Omega)). Then, there exists a function
[TABLE]
solving (3.7), (3.6b,c). Moreover, we have that
[TABLE]
Remark 3.2
We note that Lemmas 3.4 and 3.5 can be applied to {\vtop{\hbox{g}\hbox{\scriptscriptstyle\sim}}}{}\in L^{r}(0,T;{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}^{s}(\Omega)) (with in the case of Lemma 3.4, and in the case of Lemma 3.5), by using the Helmholtz decomposition of
and adjusting the pressure, since any such
can be uniquely decomposed as
[TABLE]
On applying Lemmas 3.4 and 3.5 to the flow equation (3.2), we have the following result.
Lemma 3.6
Let the assumptions of Lemma 3.3 hold. In addition let {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{0}\in{\vtop{\hbox{V}\hbox{\scriptscriptstyle\sim}}}{}\cap{\vtop{\hbox{D}\hbox{\scriptscriptstyle\sim}}}{}_{\mathfrak{s}}^{1-\frac{1}{\mathfrak{r}},\mathfrak{r}}(\Omega) and {\vtop{\hbox{f}\hbox{\scriptscriptstyle\sim}}}{}\in L^{2}(0,T;{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}^{2}(\Omega)) \cap L^{\mathfrak{r}}(0,T;{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}^{\mathfrak{s}}(\Omega)), for and . Then, we have that
[TABLE]
In addition, (3.9) yields, for any and any , that
[TABLE]
Moreover, we have that
[TABLE]
- Proof
Applying Lemma 3.5 to the flow equation (3.2), on noting (3.1) and Remark 3.2, yields the first result in (3.9). The second result in (3.9) follows immediately from the first. The third result in (3.9) follows from the second and (2.2). As , it follows from the first result in (3.9) and Sobolev embedding that the first result in (3.10) holds. The second result in (3.10) follows from this and the last result in (3.9).
Applying Lemma 3.4 with and to the flow equation (3.2), on noting (3.5), (3.10) and Remark 3.2, yields the first result in (3.11). The second result in (3.11) follows from the first and Sobolev embedding as .
Theorem 3.2
Let , , for , {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{0}\in{\vtop{\hbox{V}\hbox{\scriptscriptstyle\sim}}}{}\cap{\vtop{\hbox{D}\hbox{\scriptscriptstyle\sim}}}{}_{\mathfrak{s}}^{1-\frac{1}{\mathfrak{r}},\mathfrak{r}}(\Omega) and {\vtop{\hbox{f}\hbox{\scriptscriptstyle\sim}}}{}\in L^{2}(0,T;{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}^{2}(\Omega))\cap L^{\mathfrak{r}}(0,T;{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}^{\mathfrak{s}}(\Omega)), for and , and {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{0}={\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}^{\rm T}_{0}\in{\vtop{\hbox{W}\hbox{\scriptscriptstyle\approx}}}{}^{1,2}_{n}(\Omega). Then,
[TABLE]
solves (3.2,b).
Moreover, for given {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\rm OB} satisfying (3.12a), the solution {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB} to (3.2) is unique.
- Proof
The result (3.12a) for {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\rm OB} follows immediately from (3.9) and (3.11). It follows from (3.3), (3.9), (3.10) and (3.1) that ({\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\approx}}}{}_{x}\,{\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\rm OB})\,{\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}\in L^{2}(0,T;{\vtop{\hbox{L}\hbox{\scriptscriptstyle\approx}}}{}^{2}(\Omega)) and ({\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\rm OB}\cdot{\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\sim}}}{}_{x}){\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}\in L^{2}(0,T;{\vtop{\hbox{L}\hbox{\scriptscriptstyle\approx}}}{}^{\mathfrak{z}}(\Omega)) for any . Applying Lemma 3.2 with and to each component of (3.2) and noting the above yields that {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}\in C([0,T];{\vtop{\hbox{W}\hbox{\scriptscriptstyle\approx}}}{}^{1,\mathfrak{z}}(\Omega))\cap L^{2}(0,T;{\vtop{\hbox{W}\hbox{\scriptscriptstyle\approx}}}{}^{2,\mathfrak{z}}(\Omega))\cap H^{1}(0,T;{\vtop{\hbox{L}\hbox{\scriptscriptstyle\approx}}}{}^{\mathfrak{z}}(\Omega)). From this and Sobolev embedding, as , we obtain that {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}\in L^{2}(0,T,{\vtop{\hbox{W}\hbox{\scriptscriptstyle\approx}}}{}^{1,\mathfrak{y}}(\Omega)) for any . Hence, combining this with (3.10) we now have that ({\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\rm OB}\cdot{\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\sim}}}{}_{x}){\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}\in L^{2}(0,T;{\vtop{\hbox{L}\hbox{\scriptscriptstyle\approx}}}{}^{2}(\Omega)). Applying Lemma 3.2 again with now to each component of (3.2) yields the desired result (3.12b).
If there existed another solution {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}^{\prime}\in L^{\infty}(0,T;{\vtop{\hbox{L}\hbox{\scriptscriptstyle\approx}}}{}^{2}(\Omega))\cap L^{2}(0,T;{\vtop{\hbox{H}\hbox{\scriptscriptstyle\approx}}}{}^{1}(\Omega)) to (3.2) for given {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\rm OB} satisfying (3.12a), then, as in Lemma 7.1 and the above, one could establish that {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}^{\prime} satisfies the same regularity as {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB} in (3.12b). Hence the difference {\vtop{\hbox{z}\hbox{\scriptscriptstyle\approx}}}{}={\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}-{\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}^{\prime} satisfies
[TABLE]
and {\vtop{\hbox{z}\hbox{\scriptscriptstyle\approx}}}{}(\cdot,0)={\vtop{\hbox{0}\hbox{\scriptscriptstyle\approx}}}{}. Choosing {\vtop{\hbox{\xi}\hbox{\scriptscriptstyle\approx}}}{}=\chi_{[0,t]}\,{\vtop{\hbox{z}\hbox{\scriptscriptstyle\approx}}}{}, where denotes the characteristic function of the interval , in the above yields for all that
[TABLE]
Applying a Grönwall inequality yields that {\vtop{\hbox{z}\hbox{\scriptscriptstyle\approx}}}{}\equiv{\vtop{\hbox{0}\hbox{\scriptscriptstyle\approx}}}{}, and hence the required uniqueness result.
Remark 3.3
As is noted in [19], Solonnikov proved in [26] that, for , {\vtop{\hbox{D}\hbox{\scriptscriptstyle\sim}}}{}_{\mathfrak{s}}^{1-\frac{1}{\mathfrak{r}},\mathfrak{r}}(\Omega) is the completion of in {\vtop{\hbox{W}\hbox{\scriptscriptstyle\sim}}}{}^{2-\frac{2}{\mathfrak{r}},\mathfrak{s}}(\Omega) when and . For the characterization of {\vtop{\hbox{D}\hbox{\scriptscriptstyle\sim}}}{}_{\mathfrak{s}}^{1-\frac{1}{\mathfrak{r}},\mathfrak{r}}(\Omega) in terms of Besov spaces of divergence-free vector functions, we refer to the Appendix in [14], where and , and Theorem 3.4 in Amann [2], where and . Specifically, {\vtop{\hbox{D}\hbox{\scriptscriptstyle\sim}}}{}_{\mathfrak{s}}^{1-\frac{1}{\mathfrak{r}},\mathfrak{r}}(\Omega)={\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{2-\frac{2}{\mathfrak{r}}}_{\mathfrak{s},\mathfrak{r},0}(\Omega)\cap{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}_{\rm div}^{\mathfrak{s}}(\Omega), where {\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{2-\frac{2}{\mathfrak{r}}}_{\mathfrak{s},\mathfrak{r},0}(\Omega):=\{{\vtop{\hbox{w}\hbox{\scriptscriptstyle\sim}}}{}\in{\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{2-\frac{2}{\mathfrak{r}}}_{\mathfrak{s},\mathfrak{r}}(\Omega)\,:\,{\vtop{\hbox{w}\hbox{\scriptscriptstyle\sim}}}{}|_{\partial\Omega}={\vtop{\hbox{0}\hbox{\scriptscriptstyle\sim}}}{}\} for ; {\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{2-\frac{2}{\mathfrak{r}}}_{\mathfrak{s},\mathfrak{r},0}(\Omega):=\{{\vtop{\hbox{w}\hbox{\scriptscriptstyle\sim}}}{}\in{\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{2-\frac{2}{\mathfrak{r}}}_{\mathfrak{s},\mathfrak{r}}(\mathbb{R}^{d})\,:\,\mbox{supp }({\vtop{\hbox{w}\hbox{\scriptscriptstyle\sim}}}{})\subset\overline{\Omega}\} for ; and {\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{2-\frac{2}{\mathfrak{r}}}_{\mathfrak{s},\mathfrak{r},0}(\Omega):={\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{2-\frac{2}{\mathfrak{r}}}_{\mathfrak{s},\mathfrak{r}}(\Omega) for , with . We note, for example, that {\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{r}_{s,s}(\Omega)={\vtop{\hbox{W}\hbox{\scriptscriptstyle\sim}}}{}^{r,s}(\Omega), for and fractional , and {\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{r}_{2,2}(\Omega)={\vtop{\hbox{H}\hbox{\scriptscriptstyle\sim}}}{}^{r}(\Omega), for (cf. Triebel [28], Sec. 4.4.1, Remark 2 and Sec. 4.6.1, Theorem (b)). Consequently, for we have that {\vtop{\hbox{D}\hbox{\scriptscriptstyle\sim}}}{}_{2}^{\frac{1}{2},2}(\Omega)={\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{1}_{2,2,0}(\Omega)\cap{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}_{\rm div}^{2}(\Omega)=\{{\vtop{\hbox{w}\hbox{\scriptscriptstyle\sim}}}{}\in{\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{1}_{2,2}(\Omega)\,:\,{\vtop{\hbox{w}\hbox{\scriptscriptstyle\sim}}}{}|_{\partial\Omega}={\vtop{\hbox{0}\hbox{\scriptscriptstyle\sim}}}{}\}\cap{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}_{\rm div}^{2}(\Omega)={\vtop{\hbox{H}\hbox{\scriptscriptstyle\sim}}}{}^{1}_{0}(\Omega)\cap{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}_{\rm div}^{2}(\Omega)={\vtop{\hbox{V}\hbox{\scriptscriptstyle\sim}}}{}. As {\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{1}_{2,2}(\Omega)\hookrightarrow{\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{1}_{2,\infty}(\Omega)\hookrightarrow{\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{1-\varepsilon}_{2,1}(\Omega)\hookrightarrow{\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{1-\varepsilon}_{2,\mathfrak{r}}(\Omega)\hookrightarrow{\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{2-\frac{2}{\mathfrak{r}}}_{\mathfrak{s},\mathfrak{r}}(\Omega) for all such that and , (cf. [28], Sec. 4.6.1, Theorem (a) and (c)), it follows that {\vtop{\hbox{V}\hbox{\scriptscriptstyle\sim}}}{}={\vtop{\hbox{D}\hbox{\scriptscriptstyle\sim}}}{}_{2}^{\frac{1}{2},2}(\Omega)={\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{1}_{2,2,0}(\Omega)\cap{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}_{\rm div}^{2}(\Omega)\subset{\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{2-\frac{2}{\mathfrak{r}}}_{\mathfrak{s},\mathfrak{r},0}(\Omega)\cap{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}_{\rm div}^{2}(\Omega)={\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{2-\frac{2}{\mathfrak{r}}}_{\mathfrak{s},\mathfrak{r},0}(\Omega)\cap({\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}^{\mathfrak{s}}(\Omega)\cap{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}_{\rm div}^{2}(\Omega))={\vtop{\hbox{B}\hbox{\scriptscriptstyle\sim}}}{}^{2-\frac{2}{\mathfrak{r}}}_{\mathfrak{s},\mathfrak{r},0}(\Omega)\cap{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}_{\rm div}^{\mathfrak{s}}(\Omega) (cf. Theorem III.2.3 in Galdi [18] for the final equality), and therefore {\vtop{\hbox{V}\hbox{\scriptscriptstyle\sim}}}{}\subset{\vtop{\hbox{D}\hbox{\scriptscriptstyle\sim}}}{}_{\mathfrak{s}}^{1-\frac{1}{\mathfrak{r}},\mathfrak{r}}(\Omega) for all such . Thus, in particular, {\vtop{\hbox{V}\hbox{\scriptscriptstyle\sim}}}{}\subset{\vtop{\hbox{D}\hbox{\scriptscriptstyle\sim}}}{}_{\mathfrak{s}}^{1-\frac{1}{\mathfrak{r}},\mathfrak{r}}(\Omega) for , , and ; hence, for also {\vtop{\hbox{V}\hbox{\scriptscriptstyle\sim}}}{}\cap{\vtop{\hbox{D}\hbox{\scriptscriptstyle\sim}}}{}_{\mathfrak{s}}^{1-\frac{1}{\mathfrak{r}},\mathfrak{r}}(\Omega)={\vtop{\hbox{V}\hbox{\scriptscriptstyle\sim}}}{} for and . **
4 The Fokker–Planck equation
On setting {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}={\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\star}\in L^{\infty}(0,T;{\vtop{\hbox{V}\hbox{\scriptscriptstyle\sim}}}{})\cap L^{1}(0,T;{\vtop{\hbox{W}\hbox{\scriptscriptstyle\approx}}}{}^{1,\infty}(\Omega)) in (1.13), we now want to prove the existence of a weak solution, , to this Fokker–Planck equation. We restate this as the following problem.
: Find \widehat{\psi}_{\star}:({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{},t)\mapsto\widehat{\psi}_{\star}({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{},t) such that
[TABLE]
subject to the following decay/boundary and initial conditions:
[TABLE]
We shall assume that
[TABLE]
In addition, we shall assume in this section that
[TABLE]
4.1 A discrete-in-time regularized problem,
Similarly to [6] and [7], in order to prove existence of a weak solution to (FP), we consider a discrete-in-time approximation, (FP), of a regularization of (FP) based on the parameter , where the drag term, i.e. the term involving {\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\approx}}}{}_{x}\,{\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\star}, in (4.1) and the corresponding term in (4.2a) are modified using the cut-off function defined as
[TABLE]
The weak formulation of the regularization of (FP) leads to the following problem involving the cut-off function .
: Find such that
[TABLE]
We now formulate our discrete-in-time approximation of (FPL). We set, for ,
[TABLE]
It follows from (4.4) and (4.9) that
[TABLE]
Next, we shall assign a certain ‘smoothed’ initial datum,
[TABLE]
to the given initial datum such that
[TABLE]
For , let
[TABLE]
In the Appendix of [8] it is proved for FENE-type potentials and satisfying (4.3), with replaced by the weaker assumption , recall (2.9), that , satisfying (4.11), is such that ,
[TABLE]
The proof given in [8] for FENE-type potentials carries across immediately to potentials satisfying (1.8). In addition, with the stronger assumption (4.3) on , it is easy to show that
[TABLE]
Moreover, it follows from (2.7b), (4.14a) and (4.3), for any that
[TABLE]
Our discrete-in-time approximation of (FPL) is then defined as follows.
: Let . Then, for , given , find such that
[TABLE]
We note that if in (4.16) is replaced by then the resulting integral is not well-defined.
Lemma 4.1
Let the assumptions (4.3) and (4.4) hold; then, there exists a solution to (FP).
- Proof
It is convenient to rewrite (4.16) as
[TABLE]
where, for all ,
[TABLE]
On noting (4.9) and that , it is easily deduced that is a continuous nonsymmetric coercive bilinear functional on , and is a continuous linear functional on for all .
In order to prove existence of a solution to (4.16), i.e., (4.17), we consider a regularized system for a given : Find such that
[TABLE]
where . In order to prove the existence of a solution to (4.19), we consider a fixed-point argument. Given , let be such that
[TABLE]
The Lax–Milgram theorem yields the existence of a unique solution to (4.20) for each . Thus the nonlinear map is well-defined. On recalling (2.6b), we have that is compact. Next, we show that is continuous. Let be such that strongly in as . It follows immediately that strongly in as . As , independent of , it follows from (2.6b) that there exists a subsequence and a function such that weakly in and strongly in , as ; see the argument on p. 1233 in [6] for details. We deduce from the above, the definition of and the density result (2.5) that
[TABLE]
Noting again (2.5) yields that (4.21) holds for all , and hence . Therefore the whole sequence strongly in , as , and so is continuous.
Finally, to show that has a fixed point, i.e. there exists a solution to (4.19), using Schauder’s fixed point theorem we need to show that there exists a such that for every and satisfying ; that is,
[TABLE]
In order to prove this, we introduce the following convex regularization of defined, for any and , by
[TABLE]
We note that
[TABLE]
Choosing in (4.22), noting (4.27d), (4.9) and that \widehat{\psi}\,{\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\sim}}}{}_{x}[{\cal F}_{\delta}^{L}]^{\prime}(\widehat{\psi})={\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\sim}}}{}_{x}{\cal G}_{\delta}^{L}(\widehat{\psi}), where , yields that
[TABLE]
It is easy to show that is nonnegative for all , with . In addition, for any , if or , and also if . Thus we deduce that for all , and . Hence, on applying the above bound and (4.27c) to (4.28) yields that with dependent only on , , , {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\star}^{n} and . Therefore has a fixed point. Thus we have proved the existence of a solution to (4.19).
Choosing in (4.19) yields, similarly to (4.28), that
[TABLE]
where is independent of as . We obtain from (4.29) and (4.27c) that . Similarly to the continuity argument for the mapping above, it follows from (2.6b) that there exists a subsequence and a function such that weakly in and strongly in , as . The fact that follows from the first term on the left-hand side in (4.29) and the bound (4.27c). Hence, we have that strongly in , as . Therefore, we can pass to the limit in (4.19) for to obtain (4.17) for . The desired result (4.17) for all then follows from the density result (2.5). Finally, to conclude that , we need to show the integral constraint, \rho(M\,\widehat{\psi}_{\star,L}^{n})({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{})\in[0,1] for a.e. {\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{}\in\Omega, on recalling (4.12) and (1.12b). This follows from a maximum principle, see p. 1234 in [6] for details.
Next, we note the following result.
Lemma 4.2
Under the assumptions of Lemma 4.1 the solution to (FP) is such that
[TABLE]
*for , and satisfy *
[TABLE]
- Proof
On noting (2.4), (2.3) and (1.10), we have that , for any , and \widehat{\varphi}\in{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}\,{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}^{\rm T}:{\vtop{\hbox{\zeta}\hbox{\scriptscriptstyle\approx}}}{}\in\widehat{X}, for any {\vtop{\hbox{\zeta}\hbox{\scriptscriptstyle\approx}}}{}\in{\vtop{\hbox{C}\hbox{\scriptscriptstyle\approx}}}{}^{\infty}(\overline{\Omega}). The first choice of in (4.16) immediately yields (4.30b) for any .
The second choice of in (4.16) yields, on noting (1.12a) and (1.16), that, for ,
[TABLE]
Noting (2.5), we can approximate , for fixed and , by a sequence such that
[TABLE]
Then we have for any {\vtop{\hbox{\zeta}\hbox{\scriptscriptstyle\approx}}}{}\in{\vtop{\hbox{C}\hbox{\scriptscriptstyle\approx}}}{}^{\infty}(\overline{\Omega}) that
[TABLE]
It follows from (1.16), (1.9) and (1.12a,b) that
[TABLE]
Next we note that (1.12a,b), (2.3) and (1.10) yield
[TABLE]
Hence, it follows from (4.31)–(4.35) that (4.30a) holds for any {\vtop{\hbox{\zeta}\hbox{\scriptscriptstyle\approx}}}{}\in{\vtop{\hbox{C}\hbox{\scriptscriptstyle\approx}}}{}^{\infty}(\overline{\Omega}).
Finally, similarly to (2.7), we have, for , that
[TABLE]
Hence , recall (4.14a) for , yields that {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\widehat{\psi}_{\star,L}^{n}),\,{\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\beta^{L}(\widehat{\psi}_{\star,L}^{n}))\in{\vtop{\hbox{H}\hbox{\scriptscriptstyle\approx}}}{}^{1}(\Omega), , for . Combining these, the fact that {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\star}^{n}\in{\vtop{\hbox{W}\hbox{\scriptscriptstyle\sim}}}{}^{1,\infty}(\Omega), , and that is dense in yield that (4.30a,b) hold.
4.2 Uniform bounds on the solution of
We note the following result.
Lemma 4.3
Let the assumptions of Lemma 4.1 hold. Then, we have, for any , that
[TABLE]
- Proof
We first prove (4.37) for any . Similarly to the proof of Lemma 4.2, we can choose, on noting (1.10), \widehat{\varphi}=|{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}|^{r}, for any , in (4.16). This yields, on noting (1.16), that, for ,
[TABLE]
We then approximate , for fixed and , by a sequence satisfying (4.32). Hence
[TABLE]
It follows from (1.9) and (1.8) that
[TABLE]
Next we note from (2.3) and (1.10) that
[TABLE]
Hence, on setting, for any ,
[TABLE]
combining (4.38)–(4.41) and noting (4.32), and (1.12b) yields, for and any , that
[TABLE]
Choosing in (4.43), and summing for , yields, on noting , (1.10) and (4.7), that, for
[TABLE]
Therefore, applying a discrete Grönwall inequality to (4.44) yields, on noting (4.15) and (4.10b), the desired result (4.37) for . The result (4.37) for follows immediately from (4.37) for and , the latter holding as , .
We now introduce the following definitions, in line with (4.9):
[TABLE]
We shall adopt as a collective symbol for , . We note that
[TABLE]
where and .
Using the above notation, (4.16) summed for can be restated in the following form.
: satisfy
[TABLE]
and the initial condition . We emphasize that (4.2) is an equivalent restatement for which existence of a solution has been established under assumptions (4.3) and (4.4) on the data (cf. Lemma 4.1).
Similarly, we rewrite (4.30a,b), , using the notation (4.9) and (4.45a,b) to obtain:
(S): {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\widehat{\psi}_{\star,L}^{\Delta t,+})(t),\,{\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\beta^{L}(\widehat{\psi}_{\star,L}^{\Delta t,+}))(t),\,\rho(M\,\widehat{\psi}_{\star,L}^{\Delta t,+})(t)\in{\vtop{\hbox{H}\hbox{\scriptscriptstyle\approx}}}{}^{1}(\Omega) satisfy
[TABLE]
and the initial conditions {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\widehat{\psi}_{\star,L}^{\Delta t}(\cdot,\cdot,0))={\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\widehat{\psi}^{0}(\cdot,\cdot)), .
On noting (4.45a,b), (1.12b), (4.37) and that , , we have, for all , that
[TABLE]
Moreover, we have the following result.
Lemma 4.4
Under the assumptions of Lemma 4.1 we have that
[TABLE]
*In addition, we have that *
[TABLE]
- Proof
Similarly to (4.26), we introduce the following convex regularization of defined, for any , by
[TABLE]
We have the following analogues of (4.27c,b) for all :
[TABLE]
For any , choosing , , in (4.50), noting (4.55), (4.9) and that (\widehat{\psi}_{\star,L}^{\Delta t,+}+\alpha)\,{\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\sim}}}{}_{x}[{\cal F}^{L}]^{\prime}(\widehat{\psi}_{\star,L}^{\Delta t,+}+\alpha)={\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\sim}}}{}_{x}{\cal G}^{L}(\widehat{\psi}_{\star,L}^{\Delta t,+}+\alpha), where for , yields, similarly to (4.29), that
[TABLE]
As {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\star}^{\Delta t,+}\in L^{\infty}(0,T;{\vtop{\hbox{V}\hbox{\scriptscriptstyle\sim}}}{}), it follows from (2.8) that
[TABLE]
As, for all ,
[TABLE]
we have, on noting (4.49a), (4.10a,b) and (1.10) that
[TABLE]
Combining (4.56) and (4.58) yields, on noting (4.55), that, for and any ,
[TABLE]
Passing to the limit in (4.59), noting that , as , (4.55), (4.13a) and (4.3) yield the desired result (4.50) with in the first and third terms replaced by . It follows from (4.14a) and (4.3) that (4.50) holds with in the first and third terms replaced by . It is then a simple matter to derive the desired result (4.50) on recalling (4.45a,b), the convexity of and that |{\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\sim}}}{}_{x}\sqrt{\widehat{\psi}_{\star,L}^{\Delta t}}|^{2}\leq 2(|{\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\sim}}}{}_{x}\sqrt{\widehat{\psi}_{\star,L}^{\Delta t,+}}|^{2}+|{\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\sim}}}{}_{x}\sqrt{\widehat{\psi}_{\star,L}^{\Delta t,-}}|^{2}), see p. 44 in [5] for details of the latter result.
Finally, to obtain the bound (4.51) from (4.2), we have, on noting (4.50), (4.49a,b) and (4.10a), that
[TABLE]
The remaining two terms in (4.2) are bounded similarly.
Lemma 4.5
Let the assumptions of Lemma 4.1 hold. In addition, if then we have that
[TABLE]
- Proof
Choosing {\vtop{\hbox{\zeta}\hbox{\scriptscriptstyle\approx}}}{}=\chi_{[0,t_{n}]}\,{\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\widehat{\psi}_{\star,L}^{\Delta t,+}), , in (4.30a) yields, on noting (2.7), (4.14a), (4.3), (4.10a,b), (4.49a,b), (4.7) and (1.10) that
[TABLE]
Therefore, applying a discrete Grönwall inequality to (4.62) yields, on noting (2.7), (4.14a), (4.3) and (4.10b) that the bounds in (4.61a) hold.
Similarly to the above, choosing , , in (4.30b) yields the bounds (4.61b).
Remark 4.1
We note that Lemma 4.5 is valid for , as well as , as it exploits the L^{\infty}(0,T;{\vtop{\hbox{V}\hbox{\scriptscriptstyle\sim}}}{})\cap L^{1}(0,T;{\vtop{\hbox{W}\hbox{\scriptscriptstyle\sim}}}{}^{1,\infty}(\Omega)) regularity of {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\star}, recall (4.4). If {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\star} were only in L^{2}(0,T;{\vtop{\hbox{V}\hbox{\scriptscriptstyle\sim}}}{}), then one has to use the Gagliardo-Nirenberg inequality, (2.2), in the proof of Lemma 4.5, and this will lead to a restriction to . This is the same reason why the existence proof for the Oldroyd-B model, via the convergence of a finite element approximation, is restricted to in [3], see Theorem 7.1 there. **
4.3 Passage to the limit
We are now ready to pass to the limit , in (FP), (4.2), and (S), (4.48a,b). In view of the assumption on and in Lemma 4.5 we shall choose as .
Theorem 4.1
*Let the assumptions (4.3) and (4.4) hold on the data, and let as . Then, there exists a subsequence of (not indicated), and a function such that *
[TABLE]
with
[TABLE]
and finite relative entropy and Fisher information, with
[TABLE]
such that, as (and thereby ),
[TABLE]
*In addition, for , the function satisfies *
[TABLE]
- Proof
We shall apply Dubinskiĭ’s theorem, Theorem 2.1, to the sequence . On noting (4.50), we select and
[TABLE]
and, for , we define
[TABLE]
Note that is a seminormed subset of the Banach space . As is compactly embedded in , recall (2.6b), one can deduce that the embedding is compact by applying the argument on p. 1251 in [6]. On noting (4.51) and as Sobolev embedding yields that , for , we choose , where is the dual of , equipped with the norm For such , it follows from Sobolev embedding, for any , that
[TABLE]
Hence, we have that . Thus, our choices of , and satisfy the conditions of Theorem 2.1. Applying this theorem with and , implies that the embedding
[TABLE]
is compact. Using this compact embedding, together with the bounds (4.50) and (4.51), in conjunction with (4.49a) and Sobolev embedding, we deduce (upon extraction of a subsequence) strong convergence of in to an element , as .
Thanks to the bound on the second term in (4.50), (4.45a,b) and (1.10), we have that
[TABLE]
On recalling that , together with (4.69) and the strong convergence of to in , we deduce, as , strong convergence of in to the same element . This completes the proof of (4.66d) for with and .
From the first bound in (4.51) we have that are bounded in . By Lemma 2.1, the strong convergence of these to in , shown above, then implies strong convergence in to the same limit for all values of . That completes the proof of (4.66d) for with and any .
Strong convergence in for , implies convergence almost everywhere on of a subsequence. Hence it follows from (4.49a) that a.e. on . Applying Fubini’s theorem, one can deduce from the above and (4.49b) that
[TABLE]
Since is nonnegative, one can deduce from Fatou’s lemma and (4.50) that, for a.e. ,
[TABLE]
As the expression on the left-hand side of (4.71) is nonnegative, we deduce the first result in (4.65). Similarly, one can deduce from (4.49a) that, for any and a.e. ,
[TABLE]
Hence (4.63a) holds.
Since for any , we have, for any and , on noting (4.72) that
[TABLE]
Hence, for any , the desired result (4.66d) for with any follows from (4.66d) for with ; thus, by Hölder’s inequality, it is also true for . Therefore, we have completed the proof of (4.66d) for . The proof of (4.66d) for follows, on noting (4.7), that, for any and ,
[TABLE]
The first term converges to zero using Lebesgue’s dominated convergence and the convergence of to a.e. on as . Hence, (4.66d) for follows from (4.66d) for .
It follows from for any and (4.66d) with that
[TABLE]
as . The weak convergence results (4.66a–c) are then easily deduced, see p. 1268 in [6] for details. In addition, (4.63b) and the second result in (4.65) hold.
We now pass to the limit (and ) in (FP), (4.2). We shall take at first . Integration by parts with respect to on the first term in (4.2) gives
[TABLE]
Using (4.66d) and (4.14b), we immediately have that, as (and ), the first term on the right-hand side of (4.76) converges to the first term on the left-hand side of (4.1) and the second term on the right-hand side of (4.76) converges to -\int_{\Omega\times D}\widehat{\psi}_{0}({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{})\,\widehat{\varphi}({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{},0)\,{\rm d}{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}\,{\rm d}{\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{}, resulting in the first term on the right-hand side of (4.1). On rewriting {\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\sim}}}{}_{x}\widehat{\psi}_{\star,L}^{\Delta t,+}=2\,\sqrt{\widehat{\psi}_{\star,L}^{\Delta t,+}}\,{\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\sim}}}{}_{x}\sqrt{\widehat{\psi}_{\star,L}^{\Delta t,+}}, and similarly {\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\sim}}}{}_{q}\widehat{\psi}_{\star,L}^{\Delta t,+}, it is a simple matter to pass to the limit (and ) in the remaining terms of (4.2) using (4.63a), (4.66a,b,d), (4.75) and (4.10a,b) to obtain (4.1) for all .
Finally, we note that for any , is dense in , and so is a dense linear subspace of the linear space of functions vanishing at . It follows from this, (4.63a,b), (4.65) and (4.4) that (4.1) holds for all , for any , vanishing at .
Lemma 4.6
Let the assumptions of Theorem 4.1 hold. Then we have, on possibly extracting a further subsequence of , that, as (and thereby ),
[TABLE]
and
[TABLE]
for any . In addition, it follows that
[TABLE]
satisfy
[TABLE]
- Proof
The desired result (4.77a) follows immediately from (1.12a) and (4.66d) as
[TABLE]
The desired result (4.78a) follows immediately from (4.66d) as
[TABLE]
The desired results (4.77b,c) and (4.78b,c) follow immediately from (4.61a,b), (4.77a) and (4.78a). Hence, (4.79) holds.
We now pass to the limit (and ) in (S), (4.48a,b). We shall take at first {\vtop{\hbox{\xi}\hbox{\scriptscriptstyle\approx}}}{}\in C^{1}([0,T];{\vtop{\hbox{C}\hbox{\scriptscriptstyle\approx}}}{}^{\infty}(\overline{\Omega})) with {\vtop{\hbox{\xi}\hbox{\scriptscriptstyle\approx}}}{}(\cdot,\cdot,T)={\vtop{\hbox{0}\hbox{\scriptscriptstyle\approx}}}{} and with . Integration by parts with respect to on the first term in (4.48a) gives, for all {\vtop{\hbox{\xi}\hbox{\scriptscriptstyle\approx}}}{}\in C^{1}([0,T];{\vtop{\hbox{C}\hbox{\scriptscriptstyle\approx}}}{}^{\infty}(\overline{\Omega})) with {\vtop{\hbox{\xi}\hbox{\scriptscriptstyle\approx}}}{}(\cdot,\cdot,T)={\vtop{\hbox{0}\hbox{\scriptscriptstyle\approx}}}{},
[TABLE]
Using (4.77b), (2.7) and (4.14b), we immediately have that, as (and ), the first term on the right-hand side of (4.76) converges to the first term on the left-hand side of (4.80) and the second term on the right-hand side of (4.83) converges to
[TABLE]
resulting in the first term on the right-hand side of (4.80). It is a simple matter to pass to the limit (and ) in the remaining terms of (4.48a) using (4.77a–c), (4.78b) and (4.10a,b) to obtain (4.80) for all {\vtop{\hbox{\xi}\hbox{\scriptscriptstyle\approx}}}{}\in C^{1}([0,T];{\vtop{\hbox{C}\hbox{\scriptscriptstyle\approx}}}{}^{\infty}(\overline{\Omega})) such that {\vtop{\hbox{\xi}\hbox{\scriptscriptstyle\approx}}}{}(\cdot,T)={\vtop{\hbox{0}\hbox{\scriptscriptstyle\approx}}}{}. Similarly to the above, we can pass to the limit (and ) in (4.48b) using (4.78a–c) and (4.10a,b) to obtain (4.80b) for all such that .
Finally, we note that is a dense linear subspace of the linear space . It follows from this, (4.79) and (4.4) that (4.80) holds for all {\vtop{\hbox{\xi}\hbox{\scriptscriptstyle\approx}}}{}\in W^{1,1}(0,T;{\vtop{\hbox{H}\hbox{\scriptscriptstyle\approx}}}{}^{1}(\Omega)) vanishing at and (4.80b) holds for all vanishing at .
Remark 4.2
As we assume that [\rho(M\,\widehat{\psi}_{0})]({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{})=\int_{D}M({\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{})\,\widehat{\psi}_{0}({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{})\,{\rm d}{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}=1\mbox{ for a.e. }{\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{}\in\Omega, recall (4.3), it follows that is the unique solution to the linear problem (4.80b). Similarly, , is the unique solution of (4.30b).**
5 The Hookean dumbbell model
Putting together the results in the previous two sections we have the following result.
Theorem 5.1
Let and , for . Let
[TABLE]
for and . Let satisfy (4.3) with {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{0}:={\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\widehat{\psi}_{0})\in W^{1,2}_{n}(\Omega). It follows that there exist {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\rm OB} and {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}={\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}^{\rm T} satisfying (3.12a,b) and solving the Oldroyd-B system (3.2,b).
In addition, there exists satisfying
[TABLE]
and solving, for ,
[TABLE]
Moreover, we have that {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}={\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\widehat{\psi}_{\rm OB}). Hence, ({\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\rm OB},\widehat{\psi}_{\rm OB}) solve the Hookean dumbbell model, (3.2) and (5.1); and ({\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\rm OB},{\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}={\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\widehat{\psi}_{\rm OB})) solve the Oldroyd-B model, (3.2,b).
- Proof
Theorem 3.2 immediately yields the existence of {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\rm OB} and {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}={\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}^{\rm T} satisfying (3.12a,b) and solving the Oldroyd-B system (3.2,b). By applying Theorem 4.1 with {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\star}={\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\rm OB} yields the existence of satisfying (5.2a–c) and solving (5.1). Finally, by applying Lemma 4.6 with {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\star}={\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\rm OB} yields that {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\widehat{\psi}_{\rm OB})\in L^{\infty}(0,T;{\vtop{\hbox{L}\hbox{\scriptscriptstyle\approx}}}{}^{2}(\Omega))\cap L^{2}(0,T;{\vtop{\hbox{H}\hbox{\scriptscriptstyle\approx}}}{}^{1}(\Omega)) satisfies
[TABLE]
where we have noted from Remark 4.2 that . Comparing (Proof) and (3.2), and recalling the uniqueness result of Theorem 3.2 yields that {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}={\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\widehat{\psi}_{\rm OB}).
Hence, ({\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\rm OB},\widehat{\psi}_{\rm OB}) solve the Hookean dumbbell model, (3.2) and (5.1); and ({\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{\rm OB},{\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}_{\rm OB}={\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\widehat{\psi}_{\rm OB})) solve the Oldroyd-B model, (3.2,b).
6 Concluding remarks: subsolutions and stress-defect measure
We close the paper with a discussion aimed at explaining why an alternative, apparently more direct, approach to proving the existence of large-data global weak solutions to problem (P), along the lines of our paper [7], fails in the case of the Hookean model, and why we had to resort in this paper to a different line of reasoning than in [7]. We shall describe this direct approach for both and space dimensions, and will also indicate the improvements that can be achieved in the case of . As will be explained below, however, these improvements are still insufficient to complete the proof of existence of global weak solutions to the model with that approach, even in the case of . On the other hand, for both and , this direct approach still establishes the existence of global weak subsolutions, in a sense to be made precise below.
A direct existence proof for problem (P) would follow the approach in [7] based on a discrete-in-time regularization of (P), similar to (FP), (4.2):
: Find ({\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{L}^{\Delta t,+}(t),\widehat{\psi}_{L}^{\Delta t,+}(t))\in{\vtop{\hbox{V}\hbox{\scriptscriptstyle\sim}}}{}\times(\widehat{X}\cap\widehat{Z}_{2}) s.t.
[TABLE]
subject to the initial conditions {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{L}^{\Delta t}(\cdot,0)={\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}^{0}\in{\vtop{\hbox{H}\hbox{\scriptscriptstyle\sim}}}{} and . Following the line of reasoning in [7] one can establish, for , or 3, with , the existence of a solution to (P) satisfying the uniform bounds
[TABLE]
In addition, one can establish that a.e. in , a.e. in , and
[TABLE]
where {\vtop{\hbox{V}\hbox{\scriptscriptstyle\sim}}}{}_{\mu} is the completion of \{{\vtop{\hbox{w}\hbox{\scriptscriptstyle\sim}}}{}\in{\vtop{\hbox{C}\hbox{\scriptscriptstyle\sim}}}{}^{\infty}_{0}(\Omega):{\vtop{\hbox{\nabla}\hbox{\scriptscriptstyle\sim}}}{}_{x}\cdot{\vtop{\hbox{w}\hbox{\scriptscriptstyle\sim}}}{}=0\mbox{ in }\Omega\} in {\vtop{\hbox{H}\hbox{\scriptscriptstyle\sim}}}{}_{0}^{1}(\Omega)\cap{\vtop{\hbox{W}\hbox{\scriptscriptstyle\sim}}}{}^{\mu,2}(\Omega). It follows from the fourth bound in (6.2), (2.10), (1.3) and (1.8) that
[TABLE]
Motivated by the form of the denominator of the prefactor of the multiple integral in the fifth term on the left-hand side of (6.2) we let as , so as to drive the multiple integral appearing in that term to [math] in the limit. Then, one can establish the existence of a subsequence of \{({\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{L}^{\Delta t},\widehat{\psi}_{L}^{\Delta t})\}_{L>1} (not indicated), and a pair of functions ({\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{},\widehat{\psi}) such that
[TABLE]
and
[TABLE]
with a.e. on , \int_{D}M({\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{})\,\widehat{\psi}({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{},t)\,{\rm d}{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}=1 for a.e. ,
[TABLE]
whereby (1+|{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}|^{2})\,\widehat{\psi}\in L^{\infty}(0,T;L^{1}_{M}(\Omega\times D)); such that, as (and thereby ),
[TABLE]
where if and if ; and
[TABLE]
The result (6.9d) for follows using Dubinskiĭ’s theorem, Theorem 2.1, and Lemma 2.1, as in the proof of Theorem 4.1. The result (6.9d) for is proved similarly to (4.73). We have for any and any that
[TABLE]
where we have noted (6.4) and that (1+|{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}|^{2})\,\widehat{\psi} belongs to . Therefore, it follows from (6.10) that, for any and any ,
[TABLE]
We deduce from (6.11) that, for any and for any ,
[TABLE]
Hence, the desired result (4.66d) follows immediately from (6.12).
Unfortunately, although (1.12a) and (6.4) imply that
[TABLE]
where is independent of and , and therefore there exists a symmetric positive semidefinite {\vtop{\hbox{z}\hbox{\scriptscriptstyle\approx}}}{}\in L^{\infty}(0,T;[{\vtop{\hbox{C}\hbox{\scriptscriptstyle\approx}}}{}(\overline{\Omega})]^{\prime}) such that
[TABLE]
it does not follow that {\vtop{\hbox{z}\hbox{\scriptscriptstyle\approx}}}{}={\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\widehat{\psi}). The reason why we cannot identify
with {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\widehat{\psi}) is because (6.9d) only holds for (but not for ), and the boundedness of the sequence \{|{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}|^{2}\,\widehat{\psi}_{L}^{\Delta t,+}\}_{L\geq 1} in , which is the strongest uniform bound that we have on |{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}|^{2}\,\widehat{\psi}_{L}^{\Delta t,+}, does not suffice to deduce that
[TABLE]
If this were true, it would of course automatically follow by the uniqueness of the weak* limit that {\vtop{\hbox{z}\hbox{\scriptscriptstyle\approx}}}{}={\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\widehat{\psi}), thus completing the proof of the existence of global weak solutions to the Hookean model in both and dimensions.
Nevertheless, motivated by notion of weak solution with a defect measure in the work of Feireisl (cf. in particular Sec. 4.3.2 in [15], the discussion on p.21 in [16], and [17], as well as pp. 7, 8 in the work of DiPerna & Lions [12] and Definition 1.3 in the paper of Alexandre & Villani [1]), we will show that this direct approach implies the existence of a subsolution to the Navier–Stokes–Fokker–Planck system, which is a weak solution with a defect measure, in a sense to be made precise below, for both and . We begin by noting that, since in , we have that
[TABLE]
where {\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}\mapsto\chi_{R}({\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}) is the characteristic function of the ball of radius in centred at the origin, and {\vtop{\hbox{\xi}\hbox{\scriptscriptstyle\approx}}}{}\in L^{1}(0,T;{\vtop{\hbox{C}\hbox{\scriptscriptstyle\approx}}}{}(\overline{\Omega})) is a symmetric positive semidefinite matrix-valued test function. By fixing and passing to the limit (with as ) using (6.9d) with , it then follows that
[TABLE]
where denotes the duality pairing between [{\vtop{\hbox{C}\hbox{\scriptscriptstyle\approx}}}{}(\overline{\Omega})]^{\prime} and {\vtop{\hbox{C}\hbox{\scriptscriptstyle\approx}}}{}(\overline{\Omega}).
In order to pass to the limit in the last inequality, we note that the sequence of real-valued nonnegative functions \{\chi_{R}\,M\,{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}\,{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}^{\rm T}\,\widehat{\psi}:{\vtop{\hbox{\xi}\hbox{\scriptscriptstyle\approx}}}{}\}_{R>0}, where |{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}|^{2}\,\widehat{\psi}\in L^{\infty}(0,T;L^{1}_{M}(\Omega\times D)), is nondecreasing and converges to M\,{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}\,{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}^{\rm T}\,\widehat{\psi}:{\vtop{\hbox{\xi}\hbox{\scriptscriptstyle\approx}}}{} a.e. on as . Hence, by Lebesgue’s monotone convergence theorem, we can pass to the limit in the last inequality, resulting in
[TABLE]
for all symmetric positive semidefinite matrix-valued test functions {\vtop{\hbox{\xi}\hbox{\scriptscriptstyle\approx}}}{}\in L^{1}(0,T;{\vtop{\hbox{C}\hbox{\scriptscriptstyle\approx}}}{}(\overline{\Omega})). In other words,
[TABLE]
where the last inclusion is a direct consequence of the fact that (1+|{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}|^{2})\,\widehat{\psi}\in L^{\infty}(0,T;L^{1}_{M}(\Omega\times D)).
One can now pass to the limit (and thereby ) in all other terms of (6.1,b) using (6.8a–d) and (6.9a–d) similarly to (4.2) as in the proof of Theorem 4.1.
Thus we have shown the existence of ({\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{},{\vtop{\hbox{z}\hbox{\scriptscriptstyle\approx}}}{},\widehat{\psi}), such that
and satisfy (6.5)–(6.7), with a.e. on , \int_{D}M({\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{})\,\widehat{\psi}({\vtop{\hbox{x}\hbox{\scriptscriptstyle\sim}}}{},{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{},t){\,\rm d}q=1 for a.e. , and
[TABLE]
satisfying, for ,
[TABLE]
The triple ({\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{},{\vtop{\hbox{z}\hbox{\scriptscriptstyle\approx}}}{},\widehat{\psi}) can be therefore viewed as a large-data global weak subsolution to the Navier–Stokes–Fokker–Planck system, which becomes a large-data global weak solution if the stress-defect measure
[TABLE]
which, as was proved above, is a symmetric positive semidefinite Radon measure, is in fact equal to
[math]
in L^{\infty}(0,T;[{\vtop{\hbox{C}\hbox{\scriptscriptstyle\approx}}}{}(\overline{\Omega})]^{\prime}).
Although with this direct approach one cannot prove the analogue of (4.37), as one only has a uniform bound on {\vtop{\hbox{u}\hbox{\scriptscriptstyle\sim}}}{}_{L}^{\Delta t(,\pm)} in L^{\infty}(0,T;{\vtop{\hbox{L}\hbox{\scriptscriptstyle\sim}}}{}^{2}(\Omega))\cap L^{2}(0,T;{\vtop{\hbox{H}\hbox{\scriptscriptstyle\sim}}}{}^{1}(\Omega)) and not in L^{1}(0,T;{\vtop{\hbox{W}\hbox{\scriptscriptstyle\sim}}}{}^{1,\infty}(\Omega)), one can still establish for , in the case at least, the analogue of (4.61a,b); that is,
[TABLE]
As was noted in Remark 4.1, in the analogue of (4.62) one bounds
[TABLE]
where we have used (2.2) as . In addition, we have that
[TABLE]
The key difference between (4.66a–d) and (6.9a–d) is (6.9d).
As (6.9d) is not valid for , unlike (4.66d), we still cannot identify the limit
in (6.16) with {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\widehat{\psi}) using (4.81). This is the only step that fails in this direct existence proof for (P), and therefore the existence of a nonzero stress-defect cannot be ruled out even in the case of by using this direct approach.
The failure of identifying the limit
in (6.16) with {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\widehat{\psi}) is why for the existence proof in [7], which covered both and , we required that the mapping had superlinear growth at infinity, recall (1.4a–c). If satisfies (1.4a–c), then the analogue of Lemma 2.10, Lemma 4.1 in [7], and the fourth bound in (6.2) yield that
[TABLE]
the analogue of (6.4). Setting \omega({\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{})=M({\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{})\,(1+|{\vtop{\hbox{q}\hbox{\scriptscriptstyle\sim}}}{}|)^{2\vartheta} and defining like , but with replaced by , one can show that is compact if , see Appendix F in [7]. One can then establish the strong convergence result (6.9d) for any , and hence one can then identify
in (6.16) with {\vtop{\hbox{\sigma}\hbox{\scriptscriptstyle\approx}}}{}(M\,\widehat{\psi}), and thus prove the existence of large-data global weak solutions to (P) with satisfying (1.4a–c). Finally, we note that if , e.g. , one can demonstrate that the above embedding is not compact by considering a counterexample; e.g. see Remark 3.17 in [10].
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] R. Alexandre and C. Villani , On the Boltzmann equation for long-range interactions , Comm. Pure Appl. Math., 55 (2002), pp. 30–70.
- 2[2] H. Amann , On the strong solvability of the Navier–Stokes equations , J. Math. Fluid Mech., 2 (2000), pp. 16–98.
- 3[3] J. W. Barrett and S. Boyaval , Existence and approximation of a (regularized) Oldroyd-B model , Math. Models Methods Appl. Sci., 21 (2011), pp. 1783–1837.
- 4[4] J. W. Barrett, Y. Lu, and E. Süli , Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model , Comm. Math. Sci., 15 (2017), pp. 1265–1323.
- 5[5] J. W. Barrett and E. Süli , Existence and equilibration of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers , Available from: ar Xiv:1004.1432, http://arxiv.org/abs/1004.1432 , (2010).
- 6[6] , Existence and equilibration of global weak solutions to kinetic models for dilute polymers I: Finitely extensible nonlinear bead-spring chains , Math. Models Methods Appl. Sci., 21 (2011), pp. 1211–1289.
- 7[7] , Existence and equilibration of global weak solutions to kinetic models for dilute polymers II: Hookean-type bead-spring chains , Math. Models Methods Appl. Sci., 22 (2012), p. 1150024 (84 pages).
- 8[8] , Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers , M 2AN Math. Model. Numer. Anal., 46 (2012), pp. 949–978.