Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion

Transcription

1 Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion Christian Stinner Christina Surulescu # Michael Winkler Technische Universität Kaiserslautern, Felix-Klein-Zentrum für Mathematik, Paul-Ehrlich-Str. 3, Kaiserslautern, Germany Institut für Mathematik, Universität Paderborn, 3398 Paderborn, Germany Abstract We prove the global existence, along with some basic boundedness properties, of weak solutions to a PDE-ODE system modeling the multiscale invasion of tumor cells through the surrounding tissue matrix. The model has been proposed in [] accounts on the macroscopic level for the evolution of cell tissue densities, along with the concentration of a chemoattractant, while on the subcellular level it involves the binding of integrins to soluble insoluble components of the peritumoral region. The connection between the two scales is realized with the aid of a contractivity function characterizing the ability of the tumor cells to adapt their motility behavior to their subcellular dynamics. The resulting system, consisting of three partial three ordinary differential equations including a temporal delay, in particular involves chemotactic haptotactic cross-diffusion. In order to overcome technical obstacles stemming from the corresponding highest-order interaction terms, we base our analysis on a certain functional, inter alia involving the cell tissue densities in the diffusion haptotaxis terms respectively, which is shown to enjoy a quasi-dissipative property. This will be used as a starting point for the derivation of a series of integral estimates finally allowing for the construction of a generalized solution as the limit of solutions to suitably regularized problems. Key words: chemotaxis; haptotaxis; asymptotic behavior; multiscale model; delay. AMS Classification: 35Q9, 9C7, 35K57, 35B4. stinner@mathematik.uni-kl.de # surulescu@mathematik.uni-kl.de michael.winkler@math.uni-paderborn.de

2 Introduction The model. The migration of cancer cells from the original tumor toward blood vessels the subsequent transport to distal sites eventually leading to metastasis establishment is influenced by a plethora of biochemical processes [8]. These take place at several scales, ranging from the level of subcellular events up to the macroscopic behavior of tissue cell populations. Thereby, the time scales spread from seconds (or less) for the processes inside cells or on their surface, up to months for the doubling times of tumors or for the time it takes to develop clinically detectable secondary tumors. Chemotaxis haptotaxis are two of the main mechanisms conditioning cancer cell motility. The former terms the directed cell motion in response to concentration gradients of some chemoattractant/chemorepellent. The latter refers to changes in the orientation speed of the cell according to interactions (mainly adhesion) with the fibers of the extracellular matrix (ECM): The cells will migrate from a region of low concentration of relevant adhesive molecules towards a region with a higher concentration. Mathematical models set up on this macrolevel involving at least one of these two mechanisms have been proposed analyzed in literature, see e.g., [, 34] the references therein. The macroscopic features mentioned above are, however, influenced by the internal state of cells, hence by microscopic processes, e.g., receptor binding to chemoattractant or adhesion molecules initiating intracellular signaling pathways, which in turn lead to restructuring of the cytoskeleton, polarization, production of matrix degrading enzymes (MDEs), proliferation, etc. [7, 3, 4]. Including this microlevel dynamics into the mathematical model leads to a multiscale setting coupling a system of partial differential equations (PDEs) for the quantities on the macroscale with ordinary differential equations (ODEs) modeling the subcellular events. These continuum micro-macro models explicitly accounting for subcellular events are rather new, especially in the context of cancer cell migration. Other (mainly individual based) multiscale models connecting the macrolevel with the subcellular /or cellular scale have been proposed e.g., by Macklin et al. [9] for vascular tumor growth also involving mechanic effects or by Ramis-Conde et al. [7, 8], where an individual force-based approach is considered, focusing on the influence of intra- intercellular dynamics on the cell movement. Hybrid models using cellular automata or agent-based approaches (see e.g., [9, 39, 44]) provide a natural framework for coupling individual events with macrolevel features, thereby allowing for a high level of detail. However, when the number of cells greatly increases to biologically realistic values this approach can become very expensive or even intractable from a computational point of view. In contrast, continuum models describe the evolution of population-averaged phenomena are more efficient from a numerical point of view. In this framework Webb et al. [4] studied a population based micro-macro model coupling the tumor cell ECM densities with concentrations of MMP (matrix metalloproteinases) protons, the latter both in their intracellular extracellular forms. Mallet & Pettet [] proposed a model for the integrin-mediated haptotactic migration, which assigns particular attention to the role played by cell surface receptors in the adhesion to ECM. All previously mentioned settings focused on the modeling illustration of the involved dynamics by way of numerical simulations, paying less attention to the mathematical well posedness. The latter is a nontrivial issue, due to the fact that in the multiscale models different types of equations are coupled in a highly nonlinear way the regularity assumptions which can be made are rather modest, according to biologically motivated requirements. Also, the models are sensitive to changes

3 in the initial data in the parameter ranges, which in turn are difficult to assess in the absence of adequate quantitative information. Hence, the solutions to the corresponding systems can exhibit oscillatory behavior even blow-up. Such problems are often encountered in chemotaxis systems (see e.g., [3, 5]). Global existence of solutions to systems modeling tumor invasion has been investigated among others by Liţcanu & Morales-Rodrigo [7] Tao [3] (accounting for haptotaxis only) by Tao et al. [9, 3, 3, 34] (chemotaxis haptotaxis jointly influencing the motion). The analyzed models were set on the macroscale described the evolution of tumor cells in interaction with the surrounding tissue MDEs, hence coupling reaction-diffusion ordinary differential equations in two or three space dimensions. In the context of multiscale cancer cell migration the well posedness issue was addressed e.g., in [5, 8] (micro-meso-macro models) in [, 3] (micro-macro models), respectively. The present work is concerned with this problem, too. Here the subcellular (microlevel) dynamics is represented by the binding of integrins to the respective ligs this is seen as the onset of all subsequent intracellular signaling its effects on the cell responses. The coupling with the macrolevel is realized by way of a contractivity function capturing the effects of subcellular dynamics on the cell motility carrying them over into the diffusion haptotaxis coefficients of the corresponding terms in the equation describing the evolution of tumor cells. This function satisfies itself a differential equation involving the concentrations of bound integrins as inputs. As the conversion of biochemical signals initiated by integrin binding into motile behavior needs a long sequence of events to happen, we allow for a time lag in the contractivity equation, which hence takes the form of a delay differential equation (DDE). We recall here the micro-macro setting in [] refer to that work for the modeling details. Upon non-dimensionaliziation, the model proposed there becomes ( ) ( ) ( ) t c = κ +cv c κv +v c v c +cl l + µ c c( c η v), x, t >, t v = µ v v( v) λcv, x, t >, t l = l l + cv, x, t >, (.) t y = k ( y y )v k y, x, t >, t y = k ( y y )l k y, x, t >, t κ = κ + My (,t τ) +y (,t τ), x, t >, the dependent variables respectively representing the following: c(x, t): cancer cell density; v(x, t): density of tissue fibers in the ECM; l(x, t): concentration of chemoattractant (i.e., proteolytic rests resulting from degradation of ECM fibers by matrix degrading enzymes produced by the tumor cells); y (x, t): concentration of integrins bound to ECM fibers; y (x, t): concentration of integrins bound to proteolytic residuals; κ(x, t): contractivity function; µ c : proliferation rate of tumor cells; 3

4 µ v : production/re-establishment rate of ECM fibers; δ: degradation rate of ECM fibers due to the action of MDEs; η, η : interaction rates allowing to take crowding into account; k, k, k, k : reaction rates in the binding of integrins to insoluble (ECM fibers) soluble (proteolytic rests) ligs in the peritumoral environment. Moreover, we abbreviate λ := δ + µ v η. Mathematical challenges. The goal of the present work is to establish a result on global solvability of (.) in an adequate framework. From a mathematical point of view, this amounts to appropriately overcome several difficulties inherent to (.), in particular, the strong couplings therein. Indeed, as a subsystem the model (.) contains a Keller-Segel-type chemotaxis system, the original form of which, that is, { t c = c (c l), x, t >, (.) t l = l l + c, x, t >, is known to exhibit phenomena of blow-up in finite time when either n 3 (see [43]) or n = the total mass of cells is supercritical [, 4]. As compared to (.), both the diffusivity D(c, v) := κ +cv the chemotactic sensitivity χ(c, l) := c +cl in (.) depend nonlinearly on c, v, l, thus leading to an even higher complexity of interaction. In [4] it has been shown that in the neighboring case when D = D(c) = c +c χ = χ(c) = +c, blow-up in the correspondingly adapted variant of (.) does as well occur at least when n = 3. Anyhow, the logistic-type cell kinetic term in the first equation in (.), is known to have a certain blow-up preventing effect on chemotaxis systems: When accordingly introduced in (.), for instance, proliferation terms of the form µ c c( c) are known to enforce boundedness of solutions when n (see [5]), also when n 3 µ c is suitably large [4]; however, this apparently does not exclude the possibility of explosions when n = 3 µ c is small. A second subsystem is obtained upon focusing on the cross-diffusive interaction with v, resulting in a variant of the prototypical haptotaxis system { t c = c (c v), x, t >, (.3) t v = f(c, v), x, t >, with given f. Here the lack of diffusibility of v is reflected in the absence of any regularizing effect on v during evolution. Correspondingly, the mathematical literature on (.3) is comparatively thin, the main contributions on haptotaxis systems concentrating on particular choices of f [6, 38] or on modified variants involving additional dampening effects such as logistic growth inhibition [, 7]. More recently, certain combined chemotaxis-haptotaxis models with linear cell diffusion stard cross-diffusive terms as in (.) (.3), have been investigated, some results on global existence also on asymptotic solution behavior could be gained for various special choices of f in the respective ODE t v = f(c, v, l) [9, 33, 35, 36, ]. Main results. In the sequel, we shall consider the PDE-ODE system (.) in a bounded domain 4

5 R n, n 3, with smooth boundary, under no-flux boundary conditions (ν denotes the outer normal to ): κ + cv νc κv + v c νv = ν l =, x, t >, (.4) the initial conditions where we assume that satisfy c(x, ) = c (x), v(x, ) = v (x), l(x, ) = l (x), κ(x, ) = κ (x), x, y (x, t) = y (x, t), y (x, t) = y (x, t), x, t [ τ, ], (.5) c C ( ), v W, () C ( ), l W, () C ( ), κ W,4 (), y C ([ τ, ]; W,4 ()) y C ([ τ, ]; W,4 ()) (.6) c, v, l >, κ > in as well as y, y y + y in ( τ, ). (.7) The parameters µ c, η, µ v, λ := δ + µ v η, k, k, k, k, M τ are assumed to be positive throughout. In this mathematical setting, the main outcome of this paper then establishes the global existence of solutions in an appropriate generalized sense, along with some basic global qualitative properties. Besides providing accessibility to our mathematical approach described below, such a resort to weak solutions may be adequate also from a biological point of view: As cell migration is a process featured by very complex phenomena influenced by a strongly heterogeneous rapidly modifying environment, the cell fiber densities though existing globally in time should be allowed to be less regular the conditions for their existence to be less restrictive than, for instance, required in frameworks of classical solutions. In its precise formulation, our main result reads as follows. Theorem. Let n 3 R n be a bounded domain with smooth boundary, suppose that c, v, l, y, y κ comply with (.6) (.7). Then (.), (.4), (.5) possesses at least one global weak solution in the sense specified in Definition 5. below. Moreover, this solution has the following global boundedness properties: There exists C > such that t+ c(, t) C c C for all t >, v C in (, ), l(, t) W, () C y C t in (, ), t+ y C in (, ) κ C in (, ). t l(, s) W, () ds C for all t >, (.8) 5

6 Main ideas underlying our approach. A cornerstone of our analysis will consist of identifying a certain quasi-dissipative property of the functional E := c ln c + κ v λ + v. Indeed, we shall see in Lemma 4. below that a certain regularized variant thereof satisfies an inequality of the form d E + D F (t)e + H(t τ), t >, (.9) dt with some nonnegative F L loc [, )), a function H L loc ([, )) allowing for the estimate H(t τ) CE(t τ) for all t > τ with some C >, with the dissipation rate given by D := κ c + κc v + cv c ( + v) + µ c c ln( + c) (cf. (4.3)). In particular, this will imply an integral estimate of the form E(t) C(T ) + C(T ) t E(σ)dσ for all t > with suitably large C > (see (4.36)), whereupon the Grønwall lemma will provide bounds for sup t (,T ) E(t) T D(t)dt for any fixed T >. These in turn will form the fundament for our derivation of appropriate compactness properties of solutions to the regularized problems (.) below (see Section 5.), thereby finally allow for the construction of weak solutions as limits of global smooth solutions to these approximate problems in Section 5.. Finally, in Section 6 we will comment on some possible model extensions. A family of approximate problems In order to construct a solution of (.), (.4), (.5), let us first consider the regularized problems ( ) ( ) ( ) t c = c + κ +c v c κv +v c v c +c l l +µ c c ( c η v ) c θ, x, t >, t v = v + µ v v ( v ) λc v, x, t >, t l = l l + c v, x, t >, t y = k ( y y )v k y, x, t >, t y = k ( y y )l k y, x, t >, t κ = κ + My (,t τ) +y (,t τ), x, t >, ν c = ν v = ν l =, x, t >, c (x, ) = c (x), v (x, ) = v (x), l (x, ) = l (x), κ (x, ) = κ (x), x, y (x, t) = y (x, t), y (x, t) = y (x, t), x, t [ τ, ] 6 (.)

7 for (, ), where θ > max{, n is a fixed parameter (c ) (,), (v ) (,), (l ) (,), (κ ) (,), (y ) (,) (y ) (,) are families of functions which are all positive satisfy c C 3 ( ), v C 3 ( ), l C 3 ( ), κ C 3 ( ), y C 3 ( [ τ, ]) y C 3 ( [ τ, ]) (.) ν c = ν v = ν l = on y + y < in [ τ, ] (.3) for all (, ) as well as c c in C ( ), v v in W, () C ( ), l l in W, () C ( ), κ κ in W,4 (), y y in C ([ τ, ]; W,4 ()) y y in C ([ τ, ]; W,4 ()) (.4) as. We shall see in Lemma 3. below that each of these problems possesses a global classical solution. 3 Global existence basic estimates for the regularized problems 3. Local existence Let us first assert the local existence of smooth solutions to (.). Lemma 3. For any (, ) there exist T (, ] a collection of positive functions c, v, l, y, y κ, all belonging to C, ( [, T )), which solve (.) in the classical sense in (, T ) have the additional property that t T if T < then lim sup { c (, t) C +β ( ) + v (, t) C +β ( ) + l (, t) C +β ( ) + y (, t) C +β ( ) + y (, t) C +β ( ) + κ (, t) C +β ( ) = for all β (, ). (3.) Proof. We fix (, ) β (, ) define A := c C +β ( ) + v C +β ( ) + l C +β ( ) + y C +β,+ β ( [ τ,]) + y C +β,+ β ( [ τ,]) + κ C +β ( ). Furthermore, we define c t (x) to be the right-h side of the first equation of (.) evaluated at (x, t) = (x, ). This means that B := c C β ( ) + c t C ( ) C (A) < (3.) 7

8 is satisfied with a constant C (A) depending on A. Next, we fix T (, ] set X := {c C β, β ( [, T ]) : c, c Cβ, β B +. ( [,T ]) Given a fixed c X, by [6, Theorems V.7.4 IV.5.3], (.), (.3), v > the (strong) parabolic maximum principle, there exists a solution v to the second equation of (.) with the homogeneous Neumann boundary condition initial data v such that < v max{, v C ( ) in [, T ] v C +β,+ β C (A) (3.3) ( [,T ]) with some β (, β] some constant C (A) > which in view of (3.) may depend on A. Then, using similar arguments we obtain the existence of a positive solution l to the third equation of (.) with the homogeneous Neumann boundary condition initial data l satisfying l C +β,+ β ( [,T ]) C 3 (A) (3.4) with some constant C 3 (A) > some β (, β ]. Next, as the fourth fifth equations of (.) form a linear system of ODEs for (y, y ), by using (3.3), (3.4) (.) we deduce that there exists a solution to this system with initial data (y, y ) which fulfills y i C +β,+ β ( [, T ]) as well as y i C β, β + y i ( [ τ,t τ]) x j + y i C β, β ( [ τ,t τ]) x j x k C C β, β 4 (A) (3.5) ( [ τ,t τ]) for all i {, j, k {,..., n with some C 4 (A) >. Since moreover Y with Y := {(y, y ) (, ) : y + y < is a positive invariant set for this system of ODEs, (.3), the positivity of the initial data applications of the comparison principle to y, y y + y, respectively, imply that (y, y ) cannot reach the boundary of Y in finite time therefore remains inside the positive cone in R. Next, in view of (.), (3.5) the positivity of κ, there is a solution κ to the linear ODE formed by the sixth equation of (.) with initial data κ which is positive by the comparison principle satisfies κ C +β,+ β C 5 (A) (3.6) ( [,T ]) with some constant C 5 (A) depending on A. Taking now these functions v, l κ using [6, Theorems V.7.4 IV.5.3], (.), (.3), (3.3), (3.4), (3.6) the positivity of c, we obtain a solution c C +β 3,+ β 3 ( [, T ]) to the first equation of (.) with the homogeneous Neumann boundary condition initial data c for some β 3 (, β ] which is positive by the strong maximum principle fulfills c C +β 3,+ β 3 C 6 (A) (3.7) ( [,T ]) with some constant C 6 (A) depending on A. In particular, we have t c C β 3, β 3 ( [,T ]) C 6 (A) (3.8) 8

9 the regularity of c implies that c t (x) = t c (x, ) for x with the definition of c t given in the beginning of this proof. Therefore, (3.) (3.8) yield the existence of T (, ] just depending on A such that c C β, β ( [,T ]) B + for T := min{t, T, where we have used the fact that f C β ([,T ]) f C ([,T ]) holds for f C ([, T ]) due to T. Hence, setting now T := T, we have c X so that the map F : X X, F (c ) := c for c X, is well-defined compact in view of (3.7). Since X is a closed, bounded convex subset of C β, β ( [, T ]), F has a fixed point c by Schauder s fixed point theorem. Starting with this c, the above reasoning gives the existence of a classical solution to (.) in [, T ] such that all components are positive belong to C, ( [, T ]). Finally, since T just depends on A, the solution to (.) can be extended up to some T > T if all components are uniformly bounded in C +β ( ) for t (, T ) (because in case of T > τ > then y y are uniformly bounded in C +β,+ β ( [, T 3 τ]) for some T 3 > T which in view of (3.5) is sufficient to extend the solution up to some T (T, T 3 ]). Hence, (3.) is proved as β (, ) was arbitrary. 3. Some elementary -independent estimates Some basic but important properties of c v are summarized in the next two lemmata. Lemma 3. The solution of (.) satisfies { c (, t) m := max sup (,) c, for all t (, T ) (3.9) t+ as well as Proof. t t+ t c µ c + µ c m for all t (, T ) (3.) c θ (µ c + )m for all t (, T ). (3.) A spatial integration of the first equation in (.) yields d c = µ c c ( c η v ) c θ dt µ c c µ c c c θ for all t (, T ), (3.) so that since c ( c ) by the Cauchy-Schwarz inequality, an ODE comparison shows (3.9). Thereafter, (3.) (3.) easily result upon an integration of (3.) over (t, t + ). Lemma 3.3 For each (, ), we have { v (x, t) K v := max v L (), sup (,) for all x t (, T ). 9

10 Proof. This is a straightforward consequence of the parabolic maximum principle applied to the second equation in (.). Our next goal is to derive some first regularity properties of l. For the argument in the respective Lemma 3.5 below we prepare the following auxiliary statement. Lemma 3.4 Let T >, suppose that z is a nonnegative absolutely continuous function on [, T ) satisfying z (t) + az(t) f(t) for a.e. t (, T ) (3.3) with some a > a nonnegative function f L loc ([, T )) for which there exists b > such that Then Proof. z(t) z(t ) + t+ t f(s)ds b for all t [, T ). z(t) max {z() + b, ba + b for all t (, T ). (3.4) In view of the nonnegativity of z f, an integration of (3.3) shows that t t f(s) ds z(t ) + b for all t [, T ) all t [t, t + ] [, T ). (3.5) Next, setting γ := max{z(), b a + b i := max{i N : i < T, we claim that z(i) γ for all i {,..., i. (3.6) Obviously, z() γ is fulfilled. Assuming that z(i) γ for some i {,..., i, we either have z(t) b a for all t [i, i + ], so that an integration of (3.3) yields z(i + ) z(i) a b + b = z(i) γ, a or z( t) < b a for some t [i, i + ], which means z(i + ) z( t) + b γ by (3.5). Hence, (3.6) holds by induction. Finally, a combination of (3.6) (3.5) with t = i for i {,..., i proves (3.4). We can now collect a list of estimates for l. Lemma 3.5 There exists C > such that for all (, ), l (, t) W, () C for all t (, T ) (3.7) t+ l (, t) W, () ds C for all t (, T ) (3.8) as well as t+ Moreover, t t ( t l ) C for all t (, T ). (3.9) { l (x, t) inf inf l (x) e t for all x t (, T ). (3.) (,) x

11 Proof. that We first integrate the third equation in (.) use Lemma 3. Lemma 3.3 to see d l = l + c v l + mk v for all t (, T ), dt which by an ODE comparison implies that { l (, t) C := max sup (,) l, mk v for all t (, T ). (3.) We next test the PDE for l by l use Young s inequality to find, again relying on Lemma 3.3, that d l + l + l = c v l dt l + K v c for all t (, T ). (3.) In particular, this shows that z (t) := l (, t), t [, T ), satisfies z (t) + z (t) f (t) := Kv c (, t) for all t (, T ), where from Lemma 3. we know that t+ t Hence, Lemma 3.4 applies to warrant that { l (, t) max f (s)ds C := K v µc + µ c m for all t [, T ). (3.3) sup (,) l + C, 5 C for all t (, T ), (3.4) which combined with (3.) (.4) yields (3.7). Now (3.8) easily follows from this, again, (3.3) (3.4) upon a time integration in (3.). The inequality (3.9) is then immediate from (3.8) (3.). Finally, (3.) results upon an application of the parabolic comparison principle. For the last three components in (.) we can derive some pointwise bounds by elementary ODE comparison arguments in the following two lemmata. Lemma 3.6 Given any (, ), we have the pointwise estimate y (x, t) + y (x, t) K y := for all x t ( τ, T ). (3.5) Proof. It is clear by (.3) that the claimed inequality holds for all (x, t) ( τ, ]. As for positive t, we add the respective equations in (.) to obtain { t (y + y ) = (y + y ) {k v + k l k y k y { (y + y ) {k v + k l in (, T ), which immediately implies (3.5) by an ODE comparison.

12 Lemma 3.7 Let K y be as given by Lemma 3.6. Then for all (, ), the function κ has the properties { κ (x, t) K κ := max κ L (), MK y for all x t (, T ) (3.6) sup (,) as well as { κ (x, t) inf inf κ (x) e t for all x t (, T ) (3.7) (,) x t κ (x, t) K κ for all x t (, T ). (3.8) Proof. Since y i (x, t τ) K y for all (x, t) (, T ) i {, by Lemma 3.6, κ satisfies κ t κ κ + MK y MK y in (, T ). By integration, this first implies (3.6) (3.7) then proves (3.8) by using (3.6). 3.3 Global existence in the approximate problems We next aim at proving that all problems (.) are in fact globally solvable. To this end, in view of (3.) it is sufficient to derive, given any fixed (, ), bounds for all solution components with respect to the norm in C +β ( ) with some β >. We begin by performing a stard testing procedure to the first equation in (.). Lemma 3.8 Let (, ) p >. Then there exists C(, p) > such that d c p (p ) + c p c C(, p) c p p dt v + C(, p) c p l + µ c for all t (, T ). Proof. dropping nonnegative terms that d c p + (p ) p dt We multiply the first equation in (.) by c p c p (3.9) to see upon integrating by parts c p c κ v (p ) c p c v + v +(p ) c p c l + c l +µ c c p for all t (, T ). (3.3) Here, recalling Lemma 3.7 Lemma 3.3 using Young s inequality we see that κ v (p ) c p (p ) c v c p c + p K + v 4 κkv c p v, similarly we obtain (p ) c p (p ) c l + c l 4 c p c + p c p l

13 for all t (, T ). Hence, (3.9) results from (3.3). The next lemma will enable us to estimate the first two integrals on the right of (3.9) suitably. Its proof is the only place where we need our assumption that the exponent in the artificial absorptive term in the first equation of (.) satisfies θ > max{n,. Lemma 3.9 Let T >. Then for each (, ) one can find C(, T ) > such that T holds for T = min{t, T. Proof. ( ) v (, t) L () + l (, t) L () dt C(, T ) (3.3) From Lemma 3. we know that T c θ (µ c + )m(t + ), (3.3) so that since v K v by Lemma 3.3, f := t v v µ v v ( v ) λc v is bounded in L θ ( (, T )). Therefore, known results on maximal Sobolev regularity in parabolic equations ([]) assert that T v (, t) θ W,θ () dt C (, T ) (3.33) with some C (, T ) >. Similarly, also g := t l l + l c v belongs to L θ ( (, T )), whence applying the same regularity result to the third equation in (.) yields C (, T ) > fulfilling T l (, t) θ W,θ () dt C (, T ). (3.34) As θ > W,θ () W, () thanks to the additional assumption θ > n, (3.33) (3.34) imply (3.3). We can thereby estimate c in L ((, T ); L p ()) for any p (, ) each finite T T. Lemma 3. Let p > T >. Then for all (, ) there exists C(, p, T ) > such that c p (, t) C(, p, T ) for all t (, T ). (3.35) Proof. By Lemma 3.8, we can find C (, p) > such that d ( ) c p C (, p) v (, t) L dt () + l (, t) L () c p + pµ c c p for all t (, T ), so that an integration yields ( ) { t ( ) c p (, t) c p exp C (, p) v (, s) L () + l (, s) L () ds + pµ c t for all t (, T ). In view of Lemma 3.9, this proves (3.35). Now the desired result on global existence in (.) can be derived in a straightforward manner. 3

14 Lemma 3. For each (, ), the solution of (.) from Lemma 3. is global in time; that is, we have T =. Proof. Assuming on the contrary that T is finite for some (, ), from Lemma 3. parabolic regularity theory ([6]) applied to the second third equation in (.) in combination with Lemma 3.3 Lemma 3.7 we would obtain that the coefficient functions κ (x, t) a (x, t, q) := q + + c (x, t)v (x, t) q κ (x, t)v (x, t) c (x, t) v (x, t) + v (x, t) c (x, t) + c (x, t)l (x, t) l (x, t), (x, t, q) (, T ) R n, in ( ) b (x, t) := µ c c (x, t) c (x, t) η v (x, t) c θ (x, t), (x, t) (, T ), ( ) t c = a (x, t, c ) + b (x, t), (x, t) (, T ), would satisfy a (x, t, q)q q ψ (x, t) for all (x, t, q) (, T ) R n a (x, t, q) ( + K κ ) q + ψ (x, t) for all (x, t, q) (, T ) R n as well as b (x, t) ψ (x, t) for all (x, t) (, T ) with certain functions ψ, ψ ψ all belonging to p> Lp ( (, T )). Therefore, parabolic Hölder estimates ([6, Theorem.3 Remark.4]) would yield β (, ) C () > such that c C β, β ( [,T ]) C (). According to stard parabolic Schauder estimates ([6]), this in turn would provide bounds for v l in C +β,+ β ( [, T ]), which would directly entail corresponding estimates for y, y κ in the latter space. Again by parabolic Schauder theory, this would now imply that also c lies in C +β,+ β ( [, T ]) thereby contradict the extensibility criterion (3.) in Lemma An entropy-type functional The goal of the present section is to establish the following estimate which has its origin in (.9). 4

15 Lemma 4. Let T >. Then there exists C(T ) > such that for any choice of (, ) the solution of (.) satisfies { κ v T c T κ c sup c ln c t (,T ) + v c + c v c T v T + κ c ( + v ) + c ln( + c ) T + c θ ln( + c ) C(T ). (4.) The proof thereof will be prepared by a series of integral estimates. The first of these, to be given in Lemma 4.3, requires itself as a preliminary the following elementary observation. Lemma 4. There exists C > such that ξ ln ξ ξ ln ξ ξ ln( + ξ) + C for all ξ > (4.) ξ θ ln ξ ξθ ln( + ξ) + C for all ξ > θ >. (4.3) Proof. To see (4.), we note that φ : [, ) R with φ() := φ(ξ) := ξ ln ξ ξ ln ξ + ξ ln( + ξ), ξ >, defines a continuous function which satisfies φ(ξ) as ξ, so that φ(ξ) < for all ξ > ξ with some ξ >. Thus, (4.) follows if we pick any C > max ξ [,ξ ] φ(ξ). A verification of (4.3) can be achieved in much the same manner. We can thereby give some basic information on the time evolution of the first integral on the left of (4.). Lemma 4.3 There exists C > such that for any (, ) we have d c c ln c + + κ c + µ c c ln( + c ) + c θ ln( + c ) dt c + c v c κ v c v + c ( + c v ) + v κ ( + c l ) l + C for all t >. (4.4) Proof. Since c is positive in [, ) according to the strong maximum principle, we can use the first equation in (.) to compute d c ln c = ln c t c + t c dt c κ c κ v = + c v + c l c + c v c + v + c l +µ c {c ln c c ln c µ c η c ln c v c θ ln c +µ c c ( c η v ) c θ for all t >, (4.5) 5

16 where by Young s inequality c l + c l κ c + + c v c c ( + c v ) κ ( + c l ) l. (4.6) Moreover, from Lemma 4. we obtain C > C > such that µ c {c ln c c ln c µ c c ln( + c ) + C (4.7) whereas clearly, with K v as in Lemma 3.3, the latter because ξ ln ξ e c θ ln c c θ ln( + c ) + C, (4.8) µ c c ( c η v ) c θ µ c (4.9) µ c η K v c ln c v µ c η, (4.) e for all ξ >. Combining (4.5)-(4.) directly leads to (4.4). Next, the integr in the second term on the left of (4.) even satisfies a favorable pointwise inequality. We note that all terms appearing below are meaningful since v is smooth in (, T ) by stard parabolic regularity theory ([6]). Lemma 4.4 Let K κ be as given by Lemma 3.7. Then for all (, ) we have the pointwise inequality t κ v + v λ κ κ v v + v ( + v ) v v κ v v c v λκ c + v ( + v ) +(µ v + )K κ v + v for all x t >. (4.) Proof. By straightforward differentiation we obtain t κ v + v = { κ v ( t v ) κ v t v + v ( + v ) + tκ v where using the second equation in (.) in its differentiated version, ( t v ) = v + µ v v µ v v v λc v λv c, + v for all x t >, (4.) 6

17 we compute κ v ( t v ) κ v t v + v ( + v ) { v v v = κ v v + µ v 4µ v λ c v + v + v + v + v λ v c v + v ( + v ) v v v v µ v ( + v ) + µ v v v ( + v ) +λ c v v ( + v ) κ = κ v v + v ( + v ) v v λ κ v c v + v + κ v {µ ( + v ) v ( + v ) 4µ v v ( + v ) λc ( + v ) µ v v + µ v v + λc v κ = κ v v + v ( + v ) v v λ κ v c v + v + κ v {µ ( + v ) v 3µ v v 3µ v v λc λc v κ κ v v + v ( + v ) v v λ κ v c v + v +µ v κ v ( + v ) λκ c v ( + v ) for all (x, t) (, ). Since Lemma 3.7 says that κ K κ t κ K κ in (, ), using that v we can estimate κ v µ v ( + v ) + tκ v ) v (µ v K κ + K κ in (, ), + v + v thereby conclude from (4.) that indeed (4.) is valid. During our spatial integration of the above inequality (4.), a crucial role will be played by the following result of a straightforward computation. Lemma 4.5 For each (, ) we have κ κ v v + v ( + v ) v v = κ ( + v ) D ln( + v ) + for all t >. + v κ (D v v ) ( + v ) v κ v (4.3) 7

18 Proof. Using that ν v = on, we may integrate by parts to see that κ n κ v v = j v iij v + v + v = as well as κ ( + v ) v v = = + i,j= n i,j= n i,j= n i,j= n i,j= n i,j= κ + v ( ij v ) + i κ + v j v ij v κ ( + v ) ( jv ) ii v n i,j= κ ( + v ) iv j v ij v i κ ( + v ) iv ( j v ) κ ( + v ) iv j v ij v n for all t >. Adding both identities yields κ κ v v + v ( + v ) v v n { ( ij v ) = κ iv j v ij v i,j= + v ( + v ) + ( iv ) ( j v ) ( + v ) 3 κ (D v v ) + + v ( + v ) v κ v n κ = + v ijv iv j v + v i,j= + v κ (D v v ) + i,j= κ ( + v ) 3 ( iv ) ( j v ) ( + v ) v κ v (4.4) for all t >. As on the other h we have the pointwise identity ( + v ) D ln( + v ) n { = ( + v ) j v i + v i,j= n ij v = ( + v ) iv j v + v i,j= ( + v ) n = + v ijv iv j v, + v i,j= 8

19 from (4.4) we immediately obtain (4.3). In estimating those integrals in (4.3) which involve κ, we shall make use of the following result, which is a special case of a more general integral inequality in [4, Lemma 3.3]. Lemma 4.6 For each nonnegative ψ C ( ), the inequality ψ 4 ( + ψ) 3 ( + n) ( + ψ) D ln( + ψ) is valid. Now it is evident that even with this lemma at h we will not be able to completely suppress integrals involving κ. In view of Lemma 4.4 Lemma 4.5, these will appear along with factors of order ; more precisely, it will turn out (cf. the proof of Lemma 4. below) that we need to control κ 4 appropriately. Lemma 4.7 One can find C > such that for any (, ) we have { d κ 4 C y (, t τ) 4 + y (, t τ) 4 dt Proof. We compute ( t κ ) = κ + for all t >. (4.5) M + y (, t τ) y (, t τ) My (, t τ) ( + y (, t τ)) y (, t τ), x, t >, thus infer using Young s inequality that d κ 4 = κ 4 + M 4 dt + y (, t τ) κ κ y (, t τ) y (, t τ) M ( + y (, t τ)) κ κ y (, t τ) 7 3 M 4 ( + y (, t τ)) 4 y (, t τ) M 4 y 4 (, t τ) ( + y (, t τ)) 8 y (, t τ) M 4 because y K y by Lemma 3.6. y (, t τ) M 4 K 4 y y (, t τ) 4 for all t >, The above makes it necessary to characterize the evolution of y 4 y 4 as well. Lemma 4.8 There exists C > such that whenever (, ), the solution of (.) satisfies { { d y 4 + y 4 C + l (, t) dt L 4 () + l (, t) L () { + y 4 + y 4 + v 4 (4.6) 9

20 for all t >. Proof. By (.), we have ( t y ) = k ( y y ) v k v y k v y k y in (, ), so that d y 4 + d 4 dt 4 dt ( t y ) = k ( y y ) l k l y k l y k y y 4 = y y ( t y ) + y y ( t y ) = k ( y y ) y y v k v y 4 k v y y y k y 4 +k ( y y ) y y l k l y y y k l y 4 k y 4 for all t >. (4.7) Here we use Young s inequality Lemma 3.6 to find C > such that k ( y y ) y y v k ( + K y ) y 3 v v 4 + C y 4, (4.8) 4 whereas by the same token along with Lemma 3.3 we obtain C > such that k v y y y y 4 + C y 4. (4.9) As for the first of the integrals in (4.7) involving l, we first apply the Hölder inequality then Young s inequality to see that ( k ( y y ) y y l k ( + K y ) l L 4 () y 4) 3 4 { k ( + K y ) l L 4 () + y, 4 (4.)

21 in the second we again use Young s inequality to estimate k l y y y k l y 4 k l y 4 k l L () In summary, (4.8)-(4.) inserted into (4.7) readily prove (4.6). We can now pass to the proof of the main result of this section. y 4. (4.) Proof of Lemma 4.. In light of Lemma 4.5, upon integration over the inequality (4.) from Lemma 4.4 becomes d κ v + κ ( + v ) D ln( + v ) dt + v λ κ v + v c v λ κ c v ( + v ) v +(µ v + )K κ + v κ (D v v ) + v + ( + v ) v κ v for all t >. (4.) In order to estimate the two rightmost integrals appropriately, let us first apply Lemma 3.7 to find C (T ) > such that κ (x, t) C (T ) for all x t (, T ). (4.3) Then Lemma 4.6 ensures that v 4 ( + v ) 3 ( + n) ( + v ) D ln( + v ) C (T ) κ ( + v ) D ln( + v ) for all t (, T ) (4.4) with C (T ) := (+ n) C (T ). Now using once again the pointwise identity we obtain ij v = ( + v ) ij ln( + v ) + iv j v + v, (x, t) (, ), i, j {,..., n, κ (D v v ) + + v ( + v ) v κ v ) = κ (D ln( + v ) v ( + v ) v κ v for all t >,

22 where by Young s inequality, (4.3), (4.4) Lemma 3.3 we find that ) κ (D ln( + v ) v κ ( + v ) D ln( + v ) κ + v + v κ κ ( + v ) D ln( + v ) + v 4 C (T ) ( + v ) 3 + C (T ) ( + v ) κ 4 κ κ ( + v ) D ln( + v ) + C (T )( + K v ) C (T ) for all t (, T ). Similarly, ( + v ) v v 4 κ v C (T ) ( + v ) 3 + 7C3 (T ) ( + v ) κ 4 3 κ ( + v ) D ln( + v ) + 7C3 (T )( + K v) 3 for all t (, T ). Since clearly, again by (4.3), v + v C (T ) κ v + v, from (4.) we therefore obtain that d κ v + κ ( + v ) D ln( + v ) dt + v κ v λ c v λ + v κ v +C 4 (T ) + C 4 (T ) + v κ c v ( + v ) κ 4 for all t (, T ) κ 4 κ 4 with some C 4 (T ) >. When multiplied by λ added to (4.4), this shows that there exists C 5 > such that d dt { c ln c + λ κ v + v + + µ c + 4λ C 4(T ) λ κ c + c v + c v κ c ( + v ) c c ln( + c ) + + c θ ln( + c ) c κ ( + v ) D ln( + v ) κ v + c ( + c v ) + v κ ( + c l ) l + C 5 κ 4 for all t (, T ). (4.5) + C 4(T ) λ

23 Here the second integral on the right can be estimated using Lemma 3.5, which provides C 6 (T ) (, K v ] fulfilling l (x, t) C 6 (T ) for all x t (, T ), so that by (4.3) Lemma 3.3 we obtain the pointwise bound c ( + c v ) κ ( + c l ) = c ( + K v c ) C (T )( + C 6 (T )c ) K v C 6 (T ) (C 6(T )c ) ( C6 (T ) K v + C 6 (T )c ) C (T ) ( + C 6 (T )c ) K v C (T )C6 (T ) for all x t (, T ). By means of Lemma 3.5, we can thus find C 7 (T ) > such that c ( + c v ) κ ( + c l ) l K v C (T )C6 (T ) l C 7 (T ) for all t (, T ). (4.6) Now in order to compensate the last term on the right of (4.5), we recall that by Lemma 4.7 we have { d κ 4 C 8 y (, t τ) 4 + y (, t τ) 4 for all t >, (4.7) dt then invoke Lemma 4.8 to find C 9 > fulfilling { { d y 4 + y 4 v 4 + C 9 + l (, t) dt L 4 () + l (, t) L () { + y 4 + y 4 (4.8) for all t >. Once more thanks to Lemma 3.3 (4.4), we see that here v 4 ( + K v ) 3 v 4 ( + v ) 3 ( + K v) 3 C (T ) κ ( + v ) D ln( + v ), (4.9) so that if we let C (T ) := 4λ(+K v) 3 C then from (4.5)-(4.9) we obtain (T ) { d c ln c + κ v ( + κ 4 + C (T ) y 4 + y 4) dt λ + v + κ c v + κ c + c v c ( + v ) + µ c c c ln( + c ) + + c θ ln( + c ) c C 4(T ) κ v { + C 8 y (, t τ) 4 + y (, t τ) 4 λ + v { { +C 9 C (T ) + l (, t) L 4 () + l (, t) L () + y 4 + y 4 +C 7 (T ) + C 5 + C 4(T ) λ κ 4 for all t (, T ). (4.3) 3

24 Now for t we introduce some nonnegative functions by letting E (t) := c ln c + κ v { + κ 4 + C (T ) y 4 + λ + v as well as f (t) := + l (, t) L 4 () + l (, t) L () D (t) := κ c v + κ c + c v c ( + v ) + µ c c ln( + c ) c + + c θ ln( + c ) c h (t) := y (, t) 4 + y (, t) 4, thereby see that with some adequately large C (T ) >, the inequality y 4 + e d dt E (t) + D (t) C (T )f (t)e (t) + C (T )h (t τ) (4.3) holds for all t (, T ). In particular, an ODE comparison shows that this implies E (t) E () e C (T ) t f(σ)dσ + C (T ) t e C (T ) t s f(σ)dσ h (s τ)ds for all t (, T ). (4.3) Here, C := sup (,) E () is finite by (.4), Lemma 3.5 says that with some C 3 (T ) > we have T f (σ)dσ C 3 (T ) (4.33) for all (, ), where we have used that our assumption n 3 ensures that W, () is continuously embedded into both W,4 () L (). Consequently, (4.3) yields E (t) C e C (T )C 3 (T ) + C (T )e C (T )C 3 (T ) By definition of h E, however, we can estimate t h (s τ)ds t h (s τ)ds for all t (, T ). (4.34) { (t τ)+ y 4 + y 4 + h (σ)dσ τ t C 4 + E (σ)dσ for all t (, T ) (4.35) C (T ) 4

25 with C 4 := sup (,) ( y 4 L (( τ,);l 4 ()) + y 4 L (( τ,);l 4 ())) being finite due to (.4). Accordingly, (4.34) implies that with some C 5 (T ) > we have E (t) C 5 (T ) + C 5 (T ) whence the Grønwall lemma says that t E (σ)dσ for all t (, T ), (4.36) E (t) C 5 (T ) e C 5(T ) t C 5 (T ) e C 5(T ) T for all t (, T ). With this information at h, we return to (4.3), use once more (4.33) (4.35), recall the definition of D to readily end up with (4.). 5 Global weak solutions in the original problem Let us now specify our solution concept. In view of our intended compactness arguments, it will turn out to be convenient to formally rewrite c = + c + c. Definition 5. Let T >. Then by a weak solution of (.), (.4), (.5) we mean a collection of nonnegative functions c L ( (, T )) with v L ( (, T )) L ((, T ); W, ()), l L ((, T ); W, ()), y L ( (, T )), y L ( (, T )) κ L ( (, T )), which satisfy for all ϕ C ( [, T )) + c L 4 3 ((, T ); W, 4 3 ()), T T κ T κv c t ϕ c ϕ(, ) = + c + c ϕ + c v ϕ + cv + v T c T + + cl l ϕ + µ c c( c η v)ϕ (5.) T T { v t ϕ v ϕ(, ) = µ v v( v) λcv ϕ (5.) T T T l t ϕ l ϕ(, ) = l ϕ + ( l + cv)ϕ (5.3) 5

26 as well as T T y t ϕ y ϕ(, ) = {k ( y y )v k y ϕ (5.4) T T y t ϕ y ϕ(, ) = {k ( y y )l k y ϕ (5.5) T T { κ t ϕ κ ϕ(, ) = κ + My (, t τ) ϕ. (5.6) + y (, t τ) A global weak solution is a vector function (c, v, l, y, y, κ) : (, ) R 6 which is a weak solution of (.) in (, T ) for all T >. 5. Further -independent estimates for (.) Now besides the bounds provided in Section 3. in Lemma 4., in our limit procedure we shall rely on some further estimates which basically derivate from the former. A particular goal will consist of establishing some strong compactness properties which allow to extract a.e. convergent subsequences. 5.. Estimates for + c Let us first derive from Lemma 4. a bound for + c in a reflexive space. Lemma 5. Let T >. Then there exists C(T ) > such that for any (, ) we have T + c (, t) 4 3 W, 4 3 () dt C(T ). (5.7) Proof. According to Lemma 4. Lemma 3.7 we can find positive constants C (T ), C (T ) C 3 (T ) such that as well as T κ c C (T ) + c v c T c C (T ) κ C 3 (T ) in (, T ) 6

27 for all (, ). Therefore using the Hölder inequality Lemma 3.3 we can estimate T c = 4 3 = 4 3 T T c 4 3 ( + c ) 3 { κ { T 4 3 { T c v c c κ c + c v c κ c + c v c { 3 + c v κ 3 { T { 4 3 C 3 ( + (T ) Kv ) 3 C3 (T ) C (T ), which in view of Lemma 3. immediately proves (5.7). 3 { ( + Kv ) C 3 (T ) c + c 3 ( + c v ) c κ ( + c ) In order to obtain a strong compactness property for + c from this, we need to find a suitable control for the respective time derivative. This will be possible only in a temporal L framework. Lemma 5. Let k N be such that k > n+. Then for all T > there exists C(T ) > such that for all (, ). T t + c (, t) (W k, ()) T c 3 3 dt C(T ) (5.8) Proof. Since t + c = tc in (, ), for each fixed t (, T ) arbitrary ψ C +c (), an integration by parts in (.) yields t + c ψ = { κ c c + κ v c v + + c v + v ψ +µ c c ( c η v ) + c = 3 c ψ + c + +µ c c ( c η v ) κ ( + c v ) + c 3 c ψ κ v c ( + v ) + c 3 c v ψ + c l + c l + c c ψ c ( + c l ) + c 3 c l ψ + 7 ψ + c c θ ψ + c κ ( + c v ) + c c ψ κ v c ( + v ) + c v ψ ψ + c c ( + c l ) + c l ψ c θ ψ + c. (5.9)

28 Here several straightforward estimations involving Young s inequality, Lemma 3.3 Lemma 3.7 show that ( 3 c ψ c ) ψ + c L () c ψ + c as well as κ ( + c v ) c ψ + c { { ( κ ( + c v ) 3 + c c ψ { { c c c + c 4 c + 4 c + c ψ L () ψ L () κ c ) ψ + c v c L () κ c + + c v c 4 κ c + K κ + c v c 4 κ c ( + c v )( + c ) ψ L () κ v c ( + v ) 3 + c c v ψ { κ c + κ vc 3 ( + c v ) + c v c 6 ( + v ) ( + c ) 3 v ψ L () { κ c + K v ( + K v ) v κ c + c v c 6 ( + v ) ψ L () κ v c ( + v ) v ψ + c { v κ c ( + v ) + 4 { v κ c ( + v ) + K κkv 4 κ vc + c ψ L () ψ L (). ψ L () Now with C (T ) (, ) C (T ) (, ), as provided by Lemma 3.7 Lemma 3.5, such that κ C (T ) l C (T ) in (, T ), we similarly obtain c ( + c l ) 3 + c c l ψ { l + c 6 ( + c l ) ( + c ) 3 c ψ L () { l + K v κ c + ψ 6C (T )C (T ) + c v c L () 8

29 c ( + c l ) l ψ + c { { l + 4 l + 4C (T ) c ( + c l ) ( + c ) ψ L (). ψ L () Since clearly µ ψ c c ( c η v ) µ c {m + c + + η K v m ψ L () c by Lemma 3. c θ ( ) ψ c θ + ψ L (), c from (5.9) we obtain altogether t + c (, t) ψ f (t) ψ W, () for all ψ C (), (5.) where { f (t) := C 3 (T ) + c κ c + + c + c v c c + c θ + v κ c ( + v ) + l for t (, T ) (, ), with some suitably large C 3 (T ) >. Now since k > n+ warrants that W k, () is continuously embedded into W, (), (5.) ensures that with some C 4 > we have t + c (, t) = sup (W k, ()) t + c (, t) ψ C 4f (t) ψ C ψ () k, W () for all t (, T ) each (, ). A combination of Lemma 3., Lemma 3.5, Lemma 4. yields the existence of a C 5 (T ) > fulfilling which proves (5.8). T f (t)dt C 5 (T ) for all (, ), Now the latter two lemmata lead in a stard way to the following. Lemma 5.3 For each T >, ( + c ) (,) is strongly precompact in L 4 3 ( (, T )). Proof. This is a direct consequence of Lemma 5., Lemma 5., a variant of the Aubin-Lions lemma ([37, Theorem.3 Remark. in Chapter III]), where we use the compactness of the embedding W, 4 3 () L 4 3 () the fact that (W k, ()) is a Hilbert space for any k N. 9

30 5.. Estimates for v, y, y κ For the control of the gradient of all the components related to ODEs in (.), Lemma 4. will be essential; indeed, as a first consequence we note the following. Lemma 5.4 Let T >. Then there exists C(T ) > such that v (, t) C(T ) (5.) for all (, ). sup t (,T ) Proof. From Lemma 4. we obtain C (T ) > such that κ v C (T ), + v sup t (,T ) whereas Lemma 3.7 provides C (T ) > fulfilling Therefore, by Lemma 3.3 we conclude that v + K v C (T ) for all t (, T ) any (, ). κ C (T ) in (, T ). This result entails corresponding bounds for y y : sup t (,T ) κ v ( + K v)c (T ) + v C (T ) Lemma 5.5 For all T > there exists C(T ) > such that { y (, t) + y (, t) C(T ) (5.) for all (, ). Proof. We first proceed in much the same manner as in Lemma 4.8 to derive the identities d y = y ( t y ) dt = k ( y y ) y v k v y k v y y k y (5.3) d dt y = k ( y y ) y l k l y y k l y k y (5.4) 3

31 for all t >, where using Young s inequality, Lemma 3.6 Lemma 3.3 we can estimate as well as k k ( y y ) y v k v y y ( y y ) y l Again by Young s inequality, k l y y k y + k ( + K y) 4 y + k K v 4 y y + k ( + K y) 4 v l. l y k 4 l (, t) L () y, whence adding (5.3) to (5.4) we see that with some C > we have { { { d y + y C + l (, t) dt L () y + y +C v + C l for all t >. (5.5) Now due to Lemmata we can find C (T ) > such that T ( v + l ) C (T ). Moreover, thanks to the Cauchy-Schwarz inequality, Lemma 3.5, the fact that W, () L () by our assumption n 3, T l (, t) L ()dt C 3 T ( T l (, t) W, () dt ) C4 (T ) for all (, ) with some C 3 > C 4 (T ) >. Hence, integrating (5.5) easily yields (5.). This result implies in turn an estimate for κ : Lemma 5.6 Given T >, we can find C(T ) > such that κ (, t) C(T ) (5.6) for all (, ). sup t (,T ) Proof. Writing z (x, t) := My (x,t τ) +y (x,t τ) for (x, t) (, ) (, ), from Lemma 5.5 Lemma 3.6 we easily infer the existence of C (T ) > such that z (, t) C (T ) for all t (, T ). 3

32 Thus by (.) Young s inequality, d κ = κ + κ z dt 4 (5.6) is immediate. z C (T ) 4 for all t (, T ), Now some bounds for the time derivatives of the approximations to the non-diffusible components in (.) are summarized in the following. Lemma 5.7 There exists C > such that for all (, ) we have T t v (, t) L () dt C( + T ) for all T >, (5.7) sup t y (, t) L () C, (5.8) t> T t y (, t) L () dt C( + T ) for all T > (5.9) sup t κ (, t) L () C. (5.) t> Proof. The statements (5.8) (5.) are obvious from Lemma 3.3, Lemma 3.6, Lemma 3.7. Likewise, (5.9) results upon recalling Lemma 3.5, because for some C > we have T l (, t) L () dt C T l (, t) W, () dt for all T > by the Sobolev embedding theorem, again, the fact that n 3. Finally, to derive (5.7) we multiply the second equation in (.) by t v to obtain t v ) ( + d { v µ v v + µ v v 3 = λ c v t v for all t >. dt 3 Since by Young s inequality Lemma 3.3, λ c v t v ( t v ) + λ K v using Lemma 3. we readily arrive at (5.7). c for all t >, Combining the above lemmata, we directly obtain strong precompactness of all the considered solution components in L loc ( [, )). Corollary 5.8 Let T >. Then (l ) (,), (v ) (,), (y ) (,), (y ) (,) (κ ) (,) are strongly precompact in L ( (, T )). Proof. This is an evident by-product of Lemma 3.5, Lemma 5.4, Lemma 5.5, Lemma 5.6 when combined with Lemma 5.7 the Aubin-Lions lemma ([37, Theorem. in Chapter III]). 3

33 5..3 Further results on strong precompactness Now thanks to the presence of the logarithmic factor in the second last integral on the left of (4.) we can apply the Dunford-Pettis theorem to derive the following further strong compactness results which will be essential in our passage to the limit. Lemma 5.9 Let T >. Then (c ) (,) is relatively compact with respect to the strong topology in L ( (, T )), (5.) moreover ( κ + c Proof. + c v ) (,) is relatively compact with respect to the strong topology in L 4 ( (, T )). (5.) First, Lemma 4. provides C (T ) > such that T c ln( + c ) C (T ) for all (, ). (5.3) Therefore, the Dunford-Pettis theorem ([, A6.4]) asserts that (c ) (,) is relatively compact in L ( (, T )) with respect to the weak topology therein. This means that given ( j ) j N (, ) we can find a subsequence ( ji ) i N such that c ji z in L ( (, T )) as i with some z L ( (, T )). Upon a further extraction if necessary, by (5.3) Lemma 5.3 we may assume that also c ji z in L ( (, T )) c ji c a.e. in (, T ) as i with a certain z L ( (, T )) some measurable c : (, T ) R, whence an application of Egorov s theorem ensures that z = c z = c. As thus c ji c in L ( (, T )) as i, using constant test functions here we infer that T c ji T c hence conclude from c ji c in L ( (, T )) that in fact c c in L ( (, T )) as i, which proves (5.). To verify (5.), we proceed similarly, noting first that (5.3) Lemma 3.7 imply that w := κ +c +c v satisfies T T ( ) w 4 ln( + w) Kκ 4 ( + c ) ln + Kκ( + c ) C (T ) for all (, ) with a certain C (T ) >. This entails that with respect to the corresponding weak topologies, (w ) (,) is relatively compact in L 4 ( (, T )) (w) 4 (,) is relatively compact in L ( (, T )), the latter again thanks to the Dunford-Pettis theorem. Now since by Lemma 5.3 Corollary 5.8 any ( j ) j N (, ) contains a subsequence along which c c, v v κ κ a.e. in (, T ), a similar argument as that leading to (5.) applies to yield (5.). 5. Proof of Theorem. As a final preparation, let us state the following elementary observation. Lemma 5. Let N, G R N be measurable, (u j ) j N L (G) (w j ) j N L (G) be such that as j we have u j u in L (G) w j w a.e. in G with some u L (G) w L (G), such that sup j N w j L (G) <. Then u j w j uw in L (G) as j. 33

34 Proof. We may take C > such that w j C in G for all j N, then estimate, using the elementary inequality (A + B) (A + B ) for A B, (u j w j uw) (u j u) wj + u (w j w) G G G C (u j u) + u (w j w) for all j N. G Here the first integral vanishes in the limit j by assumption, whereas in the second we apply the dominated convergence theorem along with the uniform majorization u (w j w) 4C u L (G) to infer that also G u (w j w) as j. On the basis of the above estimates, we can now construct a global weak solution of (.) as a limit of an appropriate sequence of solutions to (.). Proof of Theorem.. First, Lemma 5.9 Lemma 5. along with a stard extraction argument involving suitable diagonal sequences allows us to find ( j ) j N (, ) such that j as j c c in L loc ( [, )) a.e. in (, ) (5.4) as well as G + c + c in L 4 3 loc ( [, )) (5.5) as = j with some nonnegative c L loc ( [, )) fulfilling + c L 4 3 loc ( [, )). In view of Corollary 5.8, Lemma 5.4, Lemma 3.3, Lemma 3.5, Lemma 3.6, Lemma 3.7 we may next pass to a subsequence to obtain nonnegative functions v L ( (, )) L loc ([, ); W, ()), such that l L ((, ); W, ()), y L ( (, ), y L ( (, ) κ L ( (, ) (5.6) v v in L loc ( [, )) a.e. in (, ), (5.7) v v in L loc ( [, )), (5.8) l l in L loc ( [, )) a.e. in (, ), (5.9) l l in L loc ( [, )), (5.3) y y in L loc ( [, )) a.e. in (, ), (5.3) y y in L loc ( [, )) a.e. in (, ) (5.3) κ κ in L loc ( [, )) a.e. in (, ) (5.33) 34