Existence and nonuniqueness of segregated solutions to a class of cross-diffusion systems
aa r X i v : . [ m a t h . A P ] N ov Existence and nonuniqueness of segregated solutions to a class ofcross-diffusion systems ∗ Gonzalo Galiano † Sergey Shmarev † Juli´an Velasco † Abstract
We study the the Dirichlet problem for the cross-diffusion system ∂ t u i = div ( a i u i ∇ ( u + u )) + f i ( u , u ) , i = 1 , , a i = const > , in the cylinder Q = Ω × (0 , T ]. The functions f i are assumed to satisfy the conditions f (0 , r ) = 0, f ( s,
0) = 0, f (0 , r ), f ( s,
0) are locally Lipschitz-continuous. It is provedthat for suitable initial data u , v the system admits segregated solutions ( u , u ) such that u i ∈ L ∞ ( Q ), u + u ∈ C ( Q ), u + u > u · u = 0 everywhere in Q . We showthat the segregated solution is not unique and derive the equation of motion of the surfaceΓ which separates the parts of Q where u >
0, or u >
0. The equation of motion of Γis a modification of the Darcy law in filtration theory. Results of numerical simulation arepresented.
Keywords:
Nonlinear parabolic equation, cross-diffusion system, segregated solutions, Lagrangiancoordinates.
AMS:
In the context of Population Dynamics, Gurney and Nisbet [15] derived from microscopic con-siderations the density-dependent population flux J ( u ) = c ∇ u + au ∇ u, with positive constants a and c . In this expression the term c ∇ u reflects a random dispersal ofthe population, while the population pressure au ∇ u prevents overcrowding. The correspondingevolution equation has the form ∂ t u − div J ( u ) = u (cid:18) α − uβ (cid:19) , (1)where the right-hand side is the logistic growth term, α > β > ∗ Supported by the Spanish MCINN Project MTM2010-18427 † Dpt. of Mathematics, Universidad de Oviedo, c/ Calvo Sotelo, 33007-Oviedo, Spain ( [email protected],[email protected], [email protected] )
1. Galiano, S. Shmarev, J. VelascoVarious generalizations of this model were proposed, from different points of view, by Shige-sada et al. [20], Busenberg and Travis [5], or [16, 11], among others, and have given rise to theso-called cross-diffusion models . The authors of [5] assume that the individual population flow J i is proportional to the gradient of a potential function Ψ which depends only on the totalpopulation density U = u + u : J i ( u , u ) = a u i U ∇ Ψ( U ) . In this model the collective flow is still given in the form (1): J ( U ) = a ∇ Ψ( U ) with c = 0.Assuming the power law Ψ( s ) = s /
2, we arrive at the individual population flows given by J i ( u , u ) = au i ∇ U. This model was introduced by Gurtin and Pipkin [16] and mathematically analyzed by Bertschet al. [2, 4]. As remarked in [16], when considering a set of species with different characteristics,such as size, behavior with respect to overcrowding, etc., it is natural to assume that instead ofthe total population density u + u the individual flows J i depend on a general linear combinationof both population densities, possibly different for each population. This assumption leads tothe following expressions for the flows: J i ( u , u ) = u i ∇ ( a i u + a i u ) . (2)A more general evolution problem which included the flows of this type has been analyzed in[13]. A finite element fully discretized scheme was used to prove the existence of solutions underrather general assumptions on the data.The present article addrresses the singular case a ij = a i for i, j = 1 ,
2. Due to the lossof ellipticity of the diffusion matrix, this case is more complicated for the study. One of thepossible approaches consists in considering the contact-inhibition problem, see [6], assuming thatthe components of the solution are initially segregated:supp u ( x, ∪ supp u ( x,
0) = Ω , supp u ( x, ∩ supp u ( x,
0) = Γ , (3)where Ω ⊂ R n is the problem domain and Γ ∈ Ω is a given hypersurface. In the one-dimensionalcase Ω = ( − L, L ) and Γ = x c ∈ ( − L, L ). A segregated solution u = ( u , u ) of the cross-diffusion system ∂ t u = div (cid:16) u ∇ ( A · u ⊥ ) (cid:17) + f ( u ) , f = ( f , f ) , (4)with a 2 × A , is a solution which possesses the following property: u · u = 0 and u + u > a ij = 1 for i, j = 1 , n = 1 for the system u it = a i ( u i φ x ( u + u )) + f i ( u, v ) , i = 1 , , a i = const > , (5)2n a cross-diffusion systemin the rectangular domain ( − L, L ) × (0 , T ] under the zero-flux boundary conditions for u + u on the lateral boundaries. The proofs in [3, 4] rely on the observation that the introduction ofthe new thought function w := u + u transforms systems (4), (5) into systems composed of aparabolic equation for w and a transport equation for the function r := u /w with the velocityfield defined by ∇ w . Apart from the possibility to show the existence of segregated solutions,this method allowed the authors of [3] to derive the equation of motion of the curve x = ζ ( t )separating the parts of the problem domain where either u >
0, or u >
0. The question ofuniqueness of the segregated solutions for systems (4), (5) was left open.
Let Ω ⊂ R n be a bounded domain. We consider the problem of finding nonnegative functions( u, v ) satisfying the conditions u t = div ( a + u ∇ ( u + v )) + f + ( u, v ) in D = Ω × (0 , T ] ,v t = div ( a − v ∇ ( u + v )) + f − ( u, v ) in D, a ± = const > ,u + v = h on ∂ Ω × (0 , T ] ,u ( x,
0) = u ( x ), v ( x,
0) = v ( x ) in Ω . (6)It is assumed that the initial data are smooth and segregated: u ≥ v ≥ u · v = 0 in Ω ,C − ≤ u + v ≤ C in Ω, C = const > ,u + v ∈ C α (Ω) . (7)Moreover, we assume that the supports of u and v are separated by a smooth simple-connectedhypersurface Γ , Γ = ∂ { x ∈ Ω : v ( x ) > } , Γ ∩ ∂ Ω = ∅ , which means that the domain Ω is split into two parts: the annular domain Ω + , bounded by ∂ Ωand Γ (where v = 0, u > − (where u = 0, v > f ± ( q, r ) are assumed to satisfy the conditions ( f + (0 , r ) = 0 , f + ( q,
0) is locally Lipschitz-continuous for q ≥ ,f − ( q,
0) = 0 , f − (0 , r ) is locally Lipschitz-continuous for r ≥ , (8)an example of admissible f ± is furnished by the functions f + ( q, r ) = q ( α + − β + q − γ + r ) , f − ( q, r ) = r ( α − − β − q − γ − r ) ,α ± , β ± , γ ± = const >
0. Our aim is to construct a segregated solution of problem (6). To thisend we consider the initial and boundary value problem for function w = u + v . If problem (6)admits a segregated solution such that u + v > u · v = 0 everywhere in D , it is necessarythat w satisfies the conditions 3. Galiano, S. Shmarev, J. Velasco w t = div ( a w ∇ w ) + f ( w ) in D = Ω × (0 , T ] ,w = h on ∂ Ω × (0 , T ] ,w ( x,
0) = w := u + v in Ω (9)with the coefficient a and the right-hand side f defined by a = ( a + if u > ,a − if v > , f ( w ) = ( f + ( w, u > ,f − (0 , w ) if v > . (10)Problem (9) is regarded as the initial and boundary value problem for a parabolic equation withdiscontinuous data. If there is a continuous in D solution w , and if there exists a continuousbijective transformation Γ Γ t of the initially given surface Γ , we may try to define a solutionof the original problem (6) by the equalities w ( x, t ) = ( v ( x, t ) in the domain Ω − ( t ) bounded by Γ t , t ∈ [0 , T ] ,u ( x, t ) in the complement Ω + ( t ) of Ω − ( t ) in Ω . Definition 2.1.
A pair ( w, Γ) is called weak solution of problem (9) if1. Γ is a C hypersurface, the mapping Γ Γ t = Γ ∩ { t = const } is a bijection for t ∈ [0 , T ],2. ∀ t ∈ [0 , T ] the surface Γ t is the common boundary of the domains Ω ± ( t ),Ω = Ω + ( t ) ∪ Γ t ∪ Ω − ( t ) , where Ω + ( t ) is an annular domain bounded by ∂ Ω and Γ t , Ω − ( t ) is the complement ofΩ + ( t ) in Ω,3. w ∈ C ( D ) ∩ L (0 , T ; H (Ω ± ( t ))),4. for every φ ( x, t ) ∈ C ( D ), such that φ ( x, T ) = 0, φ = 0 on ∂ Ω × [0 , T ], Z D ( wφ t − aw ∇ w · ∇ φ + f ( w ) φ ) dxdt + Z Ω w φ ( x, dx = 0 . (11)To construct a solution of problem (9) we proceed in two steps. The first step consistsin the direct construction of the surface Γ and the corresponding solution w in a vicinity ofΓ. This is done by means of a special coordinate transformation similar to introduction ofa system of Lagrangian coordinates frequently used in continuum mechanics. Once the localsolution is constructed, we continue it to the rest of the problem domain and then check thatthis continuation is the thought solution of problem (9). Theorem 2.2 (Local in time existence-1) . Let conditions (7) , (8) be fulfilled. Assume that thedata of problem (9) satisfy the following conditions:1. ∂ Ω , Γ ∈ C α , w ∈ C α (Ω) with some α ∈ (0 , ,2. Γ is a level surface of w ,
4n a cross-diffusion system h ( x, t ) > on ∂ Ω × [0 , T ] , h ( x, t ) and w ( x ) satisfy the first-order compatibility conditionson ∂ Ω × { t = 0 } .Then for every Φ( t ) ∈ C [0 , T ]
1. there exists T ∗ ≤ T such that in the cylinder Ω × (0 , T ∗ ] problem (9) has a solution w ( x, t ) in the sense of Definition 2.1, which satisfies the condition w = Φ( t ) on Γ t ,2. the solution w represents the segregated solution ( u, v ) of system (6) : w = u + v , u ≡ in Ω − ( t ) × [0 , T ∗ ] , v ≡ in Ω + ( t ) × [0 , T ∗ ] . The method of construction allows us to present the surface Γ explicitly and to derive theequation of motion of Γ t , which is similar to the Darcy law in filtration theory. Theorem 2.3 (The interface equation) . Under the conditions of Theorem 2.2 there exists anannular domain ω + (0) , bounded by Γ and a smooth hypersurface ∂ω + (0) , ∂ω + (0) ∩ ∂ Ω = ∅ , ∂ω + (0) ∩ Γ = ∅ , and a function U ( y, t ) such that U ∈ W q ( ω + (0) × [0 , T ∗ ]) , U t ∈ W q ( ω + (0) × [0 , T ∗ ]) , U ( y,
0) = 0 in ω + (0) with some q > n + 2 , and Γ is parametrized by the equalities Γ = { ( x, t ) : x = y + ∇ U ( y, t ) , y ∈ Γ } , t ∈ [0 , T ∗ ] . Moreover, the velocity of advancement of the surface Γ t in the normal direction n x is defined bythe equation v · n x = ( − a + ∇ u + ∇ p ) · n x | Γ t ( the modified Darcy law ) , (12) where p is a solution of the elliptic equation ( div ( u ∇ p ) = f + ( u ) in ω + ( t ) = { x ∈ Ω : x = y + ∇ U, y ∈ ω + (0) } ,p = 0 on Γ t and ∂ω + ( t ) . Corollary 1.
The components u and v of the solution w = u + v to problem (9) constructed inTheorem 2.2 can be characterized in the following way:1. u, v ∈ L ∞ ( D ) , u ≥ , v ≥ in D ,2. u + v ∈ C ( D ) , u + v ∈ L (0 , T ; H (Ω)) ,3. for every test-function φ ∈ C ( D ) , φ ( x, T ) = 0 , φ = 0 on ∂ Ω × [0 , T ] , Z D ( uφ t − a + u ∇ ( u + v ) · ∇ φ + f + ( u ) φ ) dxdt + Z Ω u φ ( x, dx = 0 , (13) Z D ( vφ t − a − v ∇ ( u + v ) · ∇ φ + f − ( v ) φ ) dxdt + Z Ω v φ ( x, dx = 0 (14) (cf. with Definition 3.1 in [4]). The proof of this assertion is given in the end of Section 5.
5. Galiano, S. Shmarev, J. Velasco
Theorem 2.4 (Nonuniqueness) . Under the conditions of Theorem 2.2 the segregated solutionof probem (6) is not unique.
The assertion of Theorem 2.4 is an immediate byproduct of Theorem 2.2. Indeed: given u , v and a level surface Γ of the function w = u + v , for every smooth Φ( t ) such thatΦ(0) = w | Γ we obtain a new solution of problem (6) corresponding to the same initial dataand satisfying the condition w = Φ( t ) on Γ t .The assumptions that w = Φ( t ) on Γ t and that Γ is a level surface of w = u + v are notessential for the proof of Theorem 2.2 and were included in order to make evident nonuniquenessof segregated solutions of problem (9). Theorem 2.5 (Local in time existence-2) . Let conditions (7) , (8) be fulfilled. Assume that thedata of problem (9) satisfy the following conditions:1. ∂ Ω , Γ ∈ C α , w ∈ C α (Ω) with some α ∈ (0 , ,2. h ( x, t ) > , h and w satisfy the first-order compatibility conditions on ∂ Ω × { t = 0 } .Then there exists T ∗ ≤ T such that in the cylinder Ω × (0 , T ∗ ] problem (9) has a solution in thesense of Definition 2.1. The solution w of problem (9) represents the segregated solution ( u, v ) of system (6) : w = u + v , u ≡ in Ω − ( t ) × [0 , T ∗ ] , v ≡ in Ω + ( t ) × [0 , T ∗ ] . Moreover, for theinterface of the constructed solution Theorem 2.3 holds. Remark 1.
It is worth noting here that the choice of the Dirichlet boundary condition in (6)is mostly the question of convenience. The assertions of Theorems 2.2-2.5 remain true if theboundary condition in (6) is substituted by any other condition which allows one to guaranteethat the auxiliary problem (54) below has a regular solution. In particular, we may pose theno-flux conditions for u + v on ∂ Ω × [0 , T ].The proofs of the main results are based on a special nonlocal coordinate transformationwhich is similar to introduction of the system of Lagrangian coordinates in continuum mechanics.The change of independent variables allows us to reduce the construction of the moving boundaryΓ (the interface) to a problem posed in a time-independent domain. We follow the ideas of [7, 8],see also [22, 21, 23] where the method of Lagrangian coordinates was applied to the study offree boundary problems for nonlinear parabolic equations with degeneracy on the interface.Organization of the paper. In Section 3 we introduce a local system of Lagrangian coordi-nates. In the new coordinate system the problem of finding the surface Γ and the solution ofproblem (9) in a vicinity of Γ transforms into an equivalent problem posed in a time-independentcylinder. In the new formulation the interface Γ becomes a vertical surface. The new problem isa system of nonlinear evolution equations which is solved in Section 4. In Section 5 we give theproofs of the main theorems. Finally in Section 6 we give an account of the available results onthe problems of the type (5) without the contact inhibition assumption and present some resultson the numerical simulation of solution to system (6) which correspond to the segregated initialdata. Let us consider the following auxiliary problem: to find a strictly positive function w ( x, t ), afamily of annular domains { ω ± ( t ) } t> , and the surface6n a cross-diffusion systemΓ = [ t> Γ t , Γ t = ω + ( t ) ∩ ω − ( t ) , satisfying the conditions ∂ t w − div( a w ∇ w ) = f ( w ) in C ± = S t> ω ± ( t ) , [ w ] | Γ t = 0 ,w ( x,
0) = w ( x ) in ω ± (0) , Z ω ± ( t ) w ( x, t ) dx = Z ω ± (0) w ( x ) dx ∀ t ∈ (0 , T ] (15)Here and throughout the rest of the paper the symbol [ φ ] γ means the jump of the function φ across the surface γ . The surface Γ is the common boundary of the annular domains ω ± (0).The exterior boundary of ω + (0) is denoted by ∂ω + (0), ∂ω − (0) stands for the interior boundaryof ω − (0). Notice that problem (15) includes three unknown boundaries: the interface Γ and S t> ∂ω ± ( t ).We will use the notations ω ( t ) = ω + ( t ) ∪ Γ t ∪ ω − ( t ) and C = C + ∪ C − . Definition 3.1.
A pair ( w, C ) is called weak solution of problem (15) if(i) w ∈ C ( C ) ∩ L (0 , T ; H ( ω ( t ))),(ii) ∀ φ ∈ C ( C ), such that φ ( x, T ) = 0 and φ = 0 on ∂ω ± ( t ) × [0 , T ], Z C ( w φ t − a w ∇ w · ∇ φ + φ f ) dxdt + Z ω (0) φ ( x, w dx = 0 . (16) Let us consider the problem of defining the family of transformations X ( y, t ) : S (0) S ( t ) ofan open annular set S (0) ⊂ R n and a function w ( x, t ) according to the following conditions:a) for every t > X ( y, t ) : S (0) S ( t ) ⊂ R n is a diffeomorphism , (17)that is S ( t ) = X ( S (0) , t ), S (0) = X − ( S ( t ) , t ), ∂S ( t ) = X ( ∂S (0) , t ),b) the deformation of S ( t ) is governed by the differential equation ( div x ( w ( X t ( y, t ) − v ( X ( y, t ) , t ))) = 0 for a.e. y ∈ S (0) , t > ,X ( y,
0) = y ∈ S (0) , (18)with a given vector-field v ( x, t ) : S ( t ) × [0 , T ] R n in the sense that for every φ ∈ C (0 , T ; C ( S ( t ))) Z S ( t ) w ∇ φ · ( X t ( y, t ) − v ( X ( y, t ) , t )) dx = 0 , t > ,
7. Galiano, S. Shmarev, J. Velascoc) for every subset σ (0) ⊂ S (0) its image σ ( t ) at the instant t ≥ w ( x, t ) by the formula Z σ (0) w ( x, dx = Z σ ( t ) w ( x, t ) dx. (19)Let J be the Jacobian matrix of the mapping y → X ( y, t ), | J | 6 = 0 because of (17). By agreementwe always denote e g ( y, t ) = g ( x, t ) | x = X ( y,t ) , so that e w ( y, t ) ≡ w [ X ( y, t ) , t ] ≡ w ( x, t ). Take an arbitrary set σ (0) ⊆ S (0) and denote σ ( t ) = X ( σ (0) , t ). For a.e. t >
00 = ddt Z σ ( t ) w ( x, t ) dx ! = ddt Z σ (0) e w ( y, t ) | J | dy ! = Z σ (0) ddt ( e w ( y, t ) | J | ) dy, (20)provided that | J | is continuous as a function of y . Since σ (0) is arbitrary and | J ( y, | = 1, it isnecessary that e w ( y, t ) | J ( y, t ) | = w ( y,
0) for a.e. y ∈ S (0), t > . (21) Lemma 3.2.
Assume that1. X satisfy (17) , | J ( y, t ) | ∈ C ( S (0)) and | J ( y, t ) | 6 = 0 in S (0) for a.e. t ∈ (0 , T ) ,2. equations (18) and (21) are fulfilled a.e. in the cylinder S (0) × (0 , T ) ,3. e w · ( X t − v ( X, t )) ∈ ( L ∞ ( S (0))) n for a.e. t ∈ (0 , T ) ,4. ∂S ( t ) ∈ Lip for a.e. t ∈ (0 , T ) . Then the function w ( x, t ) | x = X ( y,t ) = e w ( y, t ) defined by (21) satisfies the conditions ( w t − div ( w v ( x, t )) = 0 in D ≡ S t ∈ (0 ,T ) X ( S (0) , t ) ,w ( x,
0) = w ( x ) in S (0) (22) in the following sense: ∀ φ ∈ C (0 , T ; C ( R n )) , φ ( x, T ) = 0 , φ = 0 on ∂S ( t ) × [0 , T ] , Z S (0) w ( x ) φ ( x, dx + Z D w ( φ t + ∇ x φ · v ( x, t )) dxdt = 0 . (23) Proof.
By [10, Th.2.2] for a.e. t > F := w ( X t − v ( X, t )) has the normal traceson every Lipschitz-continuous surface in S ( t ) and the Green-Gauss formulas hold: for every φ ∈ C (0 , T ; C ( S ( t ))) 8n a cross-diffusion system0 = Z S ( t ) φ div F dx = − Z S ( t ) ∇ φ · F dx.
Let us denote D T = S (0) × (0 , T ]. Notice that for every test-function φ ∈ C ∞ (0 , T ; C ( S ( t ))),vanishing as t = T , ddt φ ( x, t ) (cid:12)(cid:12)(cid:12)(cid:12) x = X ( y,t ) = e φ t ( y, t ) + ^ ∇ x φ ( x, t ) · X t . Then − Z S (0) w ( y ) φ ( y, dy = Z T ddt Z S ( t ) w ( x, t ) φ ( x, t ) dx ! dt = Z T ddt Z S (0) e w ( y, t ) e φ ( y, t ) | J | dy ! dt = Z D T ddt (cid:16) e w ( y, t ) e φ ( y, t ) | J | (cid:17) dydt = Z D T (cid:20) ddt ( e w ( y, t ) | J | ) e φ + e w ( e φ t + g ∇ x φ · X t ) | J | (cid:21) dydt. Using (21) we obtain − Z S (0) w ( y ) φ ( y, dy = Z D T e w ( φ t + g ∇ x φ · X t ) | J | dydt = Z D T e w (cid:16) e φ t + g ∇ x φ · v ( X, t ) (cid:17) | J | dydt = Z T Z S ( t ) w ( φ t + ∇ x φ · v ( x, t )) dxdt. (24) Theorem 3.3.
Assume that the domain ω (0) is split into two annular domains ω ± (0) by theLipschitz-continuous surface Γ such that Γ ∩ ∂ω ± (0) = ∅ . If the conditions of Lemma 3.2 arefulfilled in each of the domains ω ± (0) and if(i) lim ω + (0) ∋ y → y ∈ Γ | J ( y, t ) | = lim ω − (0) ∋ y → y ∈ Γ | J ( y, t ) | ∀ y ∈ Γ , t ∈ [0 , T ] ,(ii) v ∈ C ( ω + ( t ) ∪ ω − ( t )) × [0 , T ]; R n ) ,then w ( x, t ) defined by (21) satisfies conditions (22) in the sense of (23) .Proof. By Lemma 3.2 problem (22) has solutions w ± in each of the domains C ± . By virtue ofcondition (ii) the images of the surface Γ under the mappings X + and X − coincide, whichmeans that Γ t = C + ∩ C + . The function w ( x, t ) = e w ( y, t ) defined by (21) in each of the domains C ± is continuous across the surface Γ t because of assumption (i). Finally, to get (22) we gatherrelations (24), corresponding to the domains ω ± (0).9. Galiano, S. Shmarev, J. VelascoTheorem 3.3 will be used in the proof of Theorem 2.5. In the proof of Theorem 2.2 we relyon the following version of Theorem 3.3. Theorem 3.4.
The assertion of Theorem 3.3 remains true if condition (ii) is substituted by theconditions(iii) w ( x, t ) = Φ( t ) on Γ t , [ v · n x ] Γ t = 0 ,where n x denotes the unit normal vector directed inward ω − ( t ) , and Φ( t ) is a given strictlypositive function.Proof. The assertion follows from Lemma 3.2: although the tangential component of the velocityis no longer continuous across Γ t , the assumption w = Φ( t ) on Γ t provides continuity of the flux w ( v · n x ) across Γ t . Let us now search for the fields X ( y, t ) and v ( X, t ) in the potential form: X ( y, t ) = y + ∇ y U in ω ± (0) × [0 , T ] , v ( x, t ) = − a ± ∇ x w + ∇ x p in ω ± (0) × [0 , T ] ,U = 0 on the parabolic boundaries of ω ± (0) × [0 , T ] , (25)where U ( y, t ) and w ( x, t ) = e w ( y, t ) are scalar functions related by (21) and p ( x, t ) is the newunknown. The parabolic boundary of a cylinder means “the lateral boundaries and the bottom”.For every φ ∈ C (0 , T ; C ( ω ( t ))), φ ( x, T ) = 0, φ = 0 on ∂ C , Z ω (0) w φ ( x, dx + Z C ( w φ t − a w ∇ x φ · ∇ x w ) dx + Z C w ∇ x φ · ∇ x p dx = 0 . (26)Let us take for p a solution of the elliptic equation endowed with the Dirichlet boundary condi-tions on ∂ω ± ( t ) and satisfying the additional condition on Γ t , which provides continuity of theflux Φ( t ) ( v · n x ) across the moving boundary: − div x ( w ∇ x p ) = f ( w ) in ω ± ( t ) ,p = 0 on ∂ω ± ( t ) , [ ∇ x p · n x ] Γ t = [ a ∇ x w · n x ] Γ t . (27)Then for every smooth φ , such that φ ( x, T ) = 0, φ = 0 on ∂ C , Z ω (0) w φ ( x, dx + Z C ( w φ t − a w ∇ x φ · ∇ x w + f φ ) dx = 0 . (28)Let us formulate the conditions for U and P = e p in the time-independent annular cylinders Q ± T = ω ± (0) × [0 , T ] . Denote by J the Jacobian matrix of the mapping x = y + ∇ U . Applying Lemma 3.2 we havethat for every test-function φ ( x, t ) = φ ( X ( y, t ) , t ) = e φ ( y, t ), φ ∈ C (0 , T ; C ( ω ( t ))), φ = 0 on ∂ω ( t ), 10n a cross-diffusion system0 = Z ω ± ( t ) φ div x ( w ( ∇ y U t + a − ] ∇ x w − g ∇ x p ) dx = − Z ω ± (0) h w (( J − ) · ∇ y e φ ) · (cid:0) J · ∇ y U t + a ∇ y (cid:0) w | J | − (cid:1) − ∇ y e p (cid:1)i dy. In particular, if w , J ij ∈ C α ( ω ± (0)), and if | J | is separated away from zero, we may take for φ a solution of the problem ( div y (cid:16) w ( J − ) · ∇ y e φ (cid:17) = ∆ y e ψ in ω ± (0) ,φ = 0 on ∂ω ± (0)with an arbitrary ψ ∈ C (0 , T ; C ( ω (0))), whence Z ω ± (0) ψ div y (cid:0) J · ∇ y U t + a + ∇ y (cid:0) w | J | − (cid:1) − ∇ y e p (cid:1) dy = 0and L ( U, e p ) ≡ div y (cid:0) J · ∇ y U t + a ∇ y (cid:0) w | J | − (cid:1) − ∇ y e p (cid:1) = 0 in Q ± T , ( a ) U = 0 on the parabolic boundaries of Q ± T , ( b ) Φ( t ) | J | Γ = Φ(0) for all t ∈ [0 , T ] . (29)The boundary condition (29) (b) follows from (21) and the condition w = Φ( t ) on Γ t . (If weassume the conditions of Theorem 2.5, this condition is omitted). Proceeding in the same waywe transform the problem for e p ( y, t ) = p ( x, t ) into the problem posed in the time-independentdomains ω ± (0): M ( e p, U ) ≡ − div y (cid:0) w ( J − ) ∇ y e p (cid:1) + f ( w | J − | ) | J | = 0in ω ± (0) for a.e. t ∈ (0 , T ] , ( a ) e p = 0 on ∂ω ± (0) , ( b ) [ J − · ∇ y e p · e n x ] Γ = [ a J − · ∇ y ( w | J − | ) · e n x ] Γ . (30)Condition (30) (b) provides continuity of the normal component of the velocity v across themoving boundary Γ t . Theorem 3.5.
Let us assume that problem (29) - (30) has a solution ( U, e p ) such that the condi-tions of Lemma are fulfilled with X = y + ∇ y U and v = − a ∇ x w + ∇ x p. (31) Then the function w ( x, t ) defined by the formulas ( C ± = (cid:8) ( x, t ) : x = y + ∇ y U ( y, t ) , ( y, t ) ∈ Q ± T (cid:9) ,w ( x, t ) = w ( y ) | J − | , (32) is a solution of problem (15) in the sense of Definition 3.1. The moving boundaries of C andthe interface Γ are parametrized by the equations
11. Galiano, S. Shmarev, J. Velasco x | Γ t = y | Γ + ∇ y U ( y, t ) | Γ , x | ∂ω ± ( t ) = y | ∂ω ± (0) + ∇ y U ( y, t ) | ∂ω ± (0) . The proof is an immediate byproduct of Theorem 3.4. Q ± T The next step is to split the nonlinear system (29)-(30) into two similar systems in the annularcylinders Q ± T which can be solved sequentially. Let us consider first the following problem fordefining ( U + , P + ): L ( U + , P + ) = 0 in Q + T , M ( P + , U + ) = 0 in ω + (0) ,U + = 0 on the parabolic boundary of Q + T , ( ∗ ) | J | = Φ(0)Φ( t ) ≡ Ψ( t ) on Γ × [0 , T ] ,P + = 0 on ∂ω + (0) and Γ for all t ∈ [0 , T ] . (33)Let us assume that problem (33) has a solution ( U + , P + ) which satisfies the regularity assump-tions of Lemma 3.2. The function P + automatically satisfies then the boundary condition (30)(a) on the lateral boundaries of Q + T . Given a pair ( U + , P + ), we may formulate the problem for( U − , P − ) in Q − T , which should include the conditions of zero jumps of density and the normalvelocity across the interface Γ t . The problem in Q − T is formulated as follows: L ( U − , P − ) = 0 in Q − T , M ( P − , U − ) = 0 in ω − (0) ,U − = 0 on the parabolic boundary of Q − T , ( ∗∗ ) | J − | = Ψ( t ) on Γ × (0 , T ] ,P − = 0 on ∂ω − (0) , [ J − · ∇ y P · f n + x ] Γ = [ a J − · ∇ y ( w | J − | ) · f n + x ] Γ for t ∈ [0 , T ] , (34)where the upper index “+” indicates that the corresponding magnitudes are already defined bythe functions ( U + , P + ). By f n + x we denote the exterior normal vector to the hypersurface Γ t parametrized by the formula X = ( y + ∇ U + ) | y ∈ Γ . The vector f n + x is well-defined if Γ ∈ C α - see Remark 4 below. Once problems (33), (34) are solved, the functions U = ( U + in Q + T ,U − in Q − T , e p = ( P + in Q + T ,P − in Q − T define a solution of problem (29)-(30).Due to Theorem 3.5, to solve problem (15) it suffices to construct functions U ± , P ± thatsatisfy the assumptions of Theorem 3.4 (or Theorem 3.3). Remark 2.
Let the conditions of Theorem 2.5 be fulfilled. In order to construct a solution ofproblem (9) we omit condition ( ∗ ) in (33) and substitute condition ( ∗∗ ) in (34) by | J − | = | J + | on Γ × [0 , T ] (condition ( i ) of Theorem 3.3) (35)12n a cross-diffusion systemCondition ( ii ) of Theorem 3.3 has to be checked a posteriori . Q + T Nonlinear problems similar to (33), (34) were already studied in [7, 8]. By this reason we confineourselves to presenting the main ideas of the proofs and omit the technical details.We begin with problem (33) posed in Q + T . To decouple the system of equations for U + and P + we solve first the nonlinear equation L ( U + , P ) = 0 considering P as a given function from asuitable function space, and then solve the linear elliptic equation M ( P + , U ) = 0 with a given U . The solutions of these equations generate an operator χ : ( U, P ) ( U + , P + ). We show thatthe operator χ has a fixed point, which is the sought solution of system (33). Let q > n + 2. We introduce the Banach spaces Z + = (cid:26) U : U ∈ W q ( Q + T ) , U t ∈ W q ( Q + T ) ,U = 0 on the parabolic boundary of Q + T (cid:27) , Y + = (cid:8) f : f ∈ W q ( Q + T ) (cid:9) , X + = { φ : φ ∈ W , q ( Q + T ) , φ ( y,
0) = 0 in ω + (0) } with the norms k u k ( k ) q,Q + T := k u k W kq ( Q + T ) = X ≤| γ |≤ k k D γy u k q,Q + T , k U k Z + = k U k (4) q,Q + T + k U t k (2) q,Q + T , k f k Y + = k f k (2) q,Q + T , k φ k X + = k φ k W , q ( Q + T ) . By C α ( Q + T ), α ∈ (0 , (cid:10) v (cid:11) ( α ) Q + T = sup Q + T | v | + sup ( x,t ) , ( y,τ ) ∈ Q + T | v ( x, t ) − v ( y, τ ) || x − y | α + | t − τ | α/ . The embedding theorems yield that since D y U ∈ W , q ( Q + T ) with q > n + 2, then ∀ U ∈ Z + X | γ | =2 , (cid:10) D γy U (cid:11) ( α ) Q + T ≤ C k U k Z + (36)with some α ∈ (0 ,
1) (see, e.g., [18, Ch.2, Lemma 3.3]). Since U ( y,
0) = 0, it follows that X | γ | =2 , sup Q + T | D γy U | ≤ CT α/ k U k Z + . (37)Denote by J the Jacobi matrix of the transformation y y + ∇ U and represent it in the form J = I + H ( U ), where H ( U ) is the Hessian of U , H ij ( U ) = D ij ( U ). Estimate (37) allows us13. Galiano, S. Shmarev, J. Velascoto choose T so small that for every U ∈ Z + , k U k Z + ≤
1, the elements of the Jacobi matrix J = I + H ( U ) and the Jacobian satisfy the estimatessup Q + T | J ij | ≤ δ ij + C T α/ , sup Q + T || J | − | ≤ C T α/ k U k Z + (38)with an independent of U constant C . Let P ∈ Y + be given. Denote H ( U ) = ( H ( U ) , H ( U )) , ( H ( U ) = L ( U, P ) in Q + T , H ( U ) = | J | − Ψ( t ) on Γ × [0 , T ] . The solution of the nonlinear problem H ( U ) = 0 , U ∈ Z + (39)is constructed by means of the modified Newton’s method. Theorem 4.1. [17, Ch. X] Let X , Y be Banach spaces. Assume that1. the operator H ( U ) : X 7→ Y has the strong differential H ′ ( · ) in a ball B r (0) ⊂ X ,2. the operator H ′ ( V ) is Lipschitz-continuous in B r (0) , kH ′ ( U ) − H ′ ( U ) k ≤ L k U − U k , L = const,
3. there exists the inverse operator [ H ′ (0)] − and (cid:13)(cid:13)(cid:13)(cid:2) H ′ (0) (cid:3) − (cid:13)(cid:13)(cid:13) = M, (cid:13)(cid:13)(cid:13)(cid:2) H ′ (0) (cid:3) − hH (0) i (cid:13)(cid:13)(cid:13) = Λ . Then, if λ = M Λ L < / , the equation H ( U ) = 0 has a unique solution U ∗ in the ball B Λ t (0) ,where t is the least root of the equation λ t − t + 1 = 0 . The solution U ∗ is obtained as thelimit of the sequence U n +1 = U n − [ H ′ (0)] − hH ( U n ) i , U = 0 . (40)Item (2) of Theorem 4.1 means that the strong and weak defferentials of H coincide and canbe found by means of linearization of the operator H at the initial state U = 0. Let us denote J = I + H ( U ), where H ( U ) is the Hessian matrix of U , H ij ( U ) = D ij ( U ). We have to compute ddǫ H ( ǫU ) = ddǫ div (cid:0) ǫ ( I + ǫ H ( U )) ∇ U t + ∇ (cid:0) w | I + ǫH ( U ) | − i − P (cid:1)(cid:1) at ǫ = 0. Since H ( U ) is symmetric, for every fixed ( y, t ) ∈ Q + T the matrix H ( ǫ U ( y, t )) isequivalent to the diagonal matrix with the eigenvalues λ i , i = 1 , . . . , n . It follows that | I + ǫH ( U ) | = n Q i =1 (1 + ǫλ i ) and 14n a cross-diffusion system ddǫ H ( ǫU ) (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 = ddǫ | J || ǫ =0 = n X i =1 trace H ( U ) = ∆ U. It is easy to see now that ddǫ H ( ǫ U ) (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 = ∆ ( U t − a + w ∆ U )and the linearized equation H ′ (0)( U ) = (∆ f, φ ) takes the form: given g ∈ Y + , φ ∈ X + , find afunction U ∈ Z + such that ( ∆ ( U t − a + w ∆ U ) = ∆ g ∈ L q ( Q + T ) , (∆ U − φ ) | Γ × [0 ,T ] = 0 . (41) Lemma 4.2.
For every ( g, φ ) ∈ Y + × X + problem (41) has at least one solution U ∈ Z + satisfying the estimate k U k Z + ≤ C ( k g k Y + + k φ k X + ) , C ≡ C (cid:16) n, q, sup w , inf w , k w k (2) q,ω + (0) (cid:17) . (42) Proof.
The proof follows [7, Th. 9] with obvious modifications due to the form of the equation:instead of dealing with the heat equation now we have to study problem (41) for a linearuniformly parabolic equation. Let U be a solution of the problem ( U t − a + w ∆ U = g + G ∈ L q ( Q + T ) ,U = 0 on the parabolic boundary of Q + T with a harmonic in ω + (0) function G to be defined. For every g, G ∈ L q ( Q + T ) this problem hasa unique solution U ∈ W , q ( Q + T ) which satisfies the estimate k U k W , q ( Q + T ) ≤ C ( k g k q,Q + T + k G k q,Q + T ) (43)with a constant C depending only on q , n , sup w and inf w (see [18, Ch.4, Sec.9]). Let us takefor G the solution of the Dirichlet problem ∆ G ( · , t ) = 0 in ω + (0) ,G ( · , t ) + g ( · , t ) = 0 on ∂ω + (0) ,G ( · , t ) + g ( · , t ) = a + w ( · ) φ ( · , t ) on Γ . (The boundary conditions are understood in the sense of traces). The function G is uniquelydefined and satisfies the estimate k G ( · , t ) k W q ( ω + (0)) ≤ C (cid:16) k g ( · , t ) k W q ( ω + (0)) + k φ ( · , t ) k W q ( ω + (0)) (cid:17) ∀ a . e . t ∈ (0 , T ) , which gives k G k W q ( Q + T ) ≤ C (cid:16) k g k W q ( Q + T ) + k φ k W q ( Q + T ) (cid:17) .
15. Galiano, S. Shmarev, J. VelascoBy construction ∆( U t − a + w ∆ U − g ) = ∆ G = 0 U ( y,
0) = 0 and U t on ∂ω + (0) × [0 , T ]. By the choice of G the function g + G has zero trace on ∂ω + (0) × [0 , T ], while g + G − a + w φ has zero trace on Γ × [0 , T ]. By virtue of the equation for U we have that ∆ U = 0 on ∂ω + (0) × [0 , T ] and ∆ U = φ on Γ × [0 , T ]. It follows that V = ∆ U solves the problem V t − a + ∆( w V ) = ∆ g ∈ L q ( Q + T ) in Q + T ,V = 0 on ∂ω + (0) × [0 , T ] ,V − φ = 0 on Γ × [0 , T ]and satisfies the estimate k ∆ U k W , q ( Q + T ) = k V k W , q ( Q + T ) ≤ C (cid:16) k ∆ g k q,Q + T + k φ k W q ( Q + T ) (cid:17) with C depending also on k w k (2) q,ω + (0) (see [18, Ch.4, Sec.9]). Gathering this estimate with (43)we obtain (42). Corollary 2. (cid:13)(cid:13)(cid:13)(cid:2) H ′ (0) (cid:3) − (cid:13)(cid:13)(cid:13) = M ≤ C, (cid:13)(cid:13)(cid:13)(cid:2) H ′ (0) (cid:3) − hH (0) i (cid:13)(cid:13)(cid:13) = Λ ≤ C (cid:16) a + T /q k ∆ w k q,ω + (0) + k P k Y + (cid:17) + T /q | ω + (0) | (cid:18) max [0 ,T ] | − Ψ( t ) | + max [0 ,T ] | Ψ ′ ( t ) | (cid:19) with the constant C from (42) .Proof. The estimates follow from (42) and the equalities H (0) = a + ∆ w − ∆ P , H (0) =1 − Ψ( t ).To prove the existence of a unique solution of the equation H ( U ) = 0 in Z + amounts tochecking Lipshitz-continuity of the linearized operator H ′ ( V )( U ) = div (cid:16) H ( U ) ∇ V t + ( I + H ( V )) ∇ U t (cid:17) − a + ∇ (cid:0) trace (cid:2) ( I + H ( V )) − H ( U ) (cid:3)(cid:1) , H ′ ( V )( U ) = trace (cid:2) ( I + H ( V )) − H ( U ) (cid:3) , which can be done exactly as in [7] with the use of formulas (37): (cid:13)(cid:13) ( H ′ i ( V ) − H ′ i ( V ) (cid:1) h U ik ≤ L k V − V k Z + k U k Z + . (44) Theorem 4.3.
Let P ∈ W q ( Q + T ) with q > n + 2 and Ψ( t ) ∈ C [0 , . Then there exists T ∗ ∈ (0 , so small that λ = M L Λ < / with the constants Λ , M and L from Corollary ,and problem (39) has a unique solution U ∈ B r with r < . (45)16n a cross-diffusion system Remark 3.
Under the conditions of Theorem 3.3 problem (39) transforms into the problem H ( U ) = 0, U ∈ Z + , and the linearized problem (41) takes the form: find U ∈ Z + such that∆ ( U t − a + w ∆ U ) = ∆ g ∈ L q ( Q + T ) . We may take for a solution the solution of (39) with φ ≡
0. The estimates of Corollary 2 changein the obvious way,: (cid:13)(cid:13)(cid:13)(cid:2) H ′ (0) (cid:3) − hH (0) i (cid:13)(cid:13)(cid:13) = Λ ≤ C (cid:16) a + T /q k ∆ w k q,ω + (0) + k P k Y + (cid:17) . Given U ∈ Z + , we consider now the equation N ( P ) ≡ M ( P + , U ) = 0 in Q + T under thehomogeneous Dirichlet boundary conditions on ∂ω + (0) and Γ : N ( P ) ≡ − div y (cid:0) w ( J − ) ∇ y P (cid:1) + f + ( w | J − | ) | J | = 0in ω + (0), t ∈ [0 , T ] ,P = 0 on ∂ω + (0) and Γ , t ∈ [0 , T ] . (46) Lemma 4.4.
Let w , D x i w ∈ L q ( ω + (0)) and let f + be locally Lipschitz-continuous. Then forevery U ∈ Z + with k U k Z + ≤ problem (46) has a unique solution P ( · , t ) ∈ W q ( ω + (0)) suchthat k P ( · , t ) k (2) q,ω + (0) ≤ C k f + ( w | J | − ) k q,ω + (0) for a.e. t ∈ (0 , T ) (47) and k P k Y + ≤ C T /q sup (0 ,T ) k f + ( w | J | − ) k q,ω + (0) (48) with a constant C depending on n , q , sup w , inf w , k∇ w k q,ω + (0) .Proof. Using (38) we choose T be so small that || J | − | ≤
12 , which entails the inequalities12 ≤ | J | ≤ , ≤ | J − | ≤ Q + T . Moreover, by virtue of (38) J is strictly positive definite for small t . For every fixed t theexistence of a solution to problem (46) follows immediately from the standard elliptic theory -see, e.g., [19, Ch. 3, Sec. 5, 15]) or [14]. The second estimate follows upon integration of (47)over the interval (0 , T ).For t = 0 problem (46) takes the form ( − div y ( w ∇ y P ) + f + ( w ) = 0 in ω + (0) ,P = 0 on ∂ω + (0) and Γ . (49) Lemma 4.5.
Under the conditions of Lemma 4.4 k P ( · , t ) − P k (2) q,ω + (0) ≤ C t α/ k U k Z + .
17. Galiano, S. Shmarev, J. Velasco
Proof.
The function P − P solves the problem − div y (cid:0) w ( J − ) ∇ y ( P − P ) (cid:1) = F in ω + (0) , P − P = 0 on ∂ω + (0) and Γ with the right-hand side F = − ( f + ( w | J − | ) | J | − f + ( w )) − div y (cid:0) w ( I − ( J − ) ) ∇ y P (cid:1) = − ( f + ( w | J − | ) − f + ( w )) + f + ( w )( | J | − − div y (cid:0) w ( I − ( J − ) ) ∇ y P (cid:1) Since f is locally Lipschitz-continuous, it follows from (37) and (38) that k F ( · , t ) k q,ω + (0) ≤ C T α/ k U k Z + with a constant C depending also on the Lipshitz constant of f ( s ) on the interval | s | ≤ w .The required estimate follows now from (47). (33) Following [7] we consider the sequences { U k } , { P k } defined as follows: U = 0, P is the solutionof problem (49), for every k ≥ U k is the solution of (33) with P = P k , P k +1 is the solution ofproblem (46) with U = U k . Gathering the estimates on the solutions of problems (33), (46) wefind that independently of k k U k k Z + ≤ C (cid:16) a + T /q k ∆ w k q,ω + (0) + k P k k Y + (cid:17) , k P k k Y + ≤ C R T /q with R = sup {| f ( s ) | : | s | ≤ w } , provided that T is sufficiently small. It follows that, upto subsequences, U k ⇀ U in Z + , P k ⇀ P in Y + ,D i P k → D i P , D ij U k → D ij U in C α ′ ,α ′ / ( D + T ) (50)with some α ′ ∈ (0 , J k = ( I + H ( U k )) , v k = J − k ∇ (cid:0) a + w | J k | − − P k (cid:1) . By the method of construction Z ω + (0) η div ( w ( J k ∇ U k,t − v k )) dy = 0for every smooth test-function η . Passing to the limit as k → ∞ we find that ( U, P ) is thesolution of problem (33). Moreover, the constructed solution possesses the regularity propertiesrequired in Lemma 3.2. 18n a cross-diffusion system
Theorem 4.6.
Let w ∈ W q ( ω + (0)) be strictly positive in ω + (0) , f be Lipschitz-continuous onthe interval | s | ≤ w , and let ∂ω + (0) , Γ ∈ C β with some β ∈ (0 , . There exists T ∗ ,depending on k w k (2) q,ω + (0) , n , q , a + , β and the Lipschitz constant of f such that in the cylinder ω + (0) × (0 , T ∗ ] problem (33) has a unique solution U ∈ Z + , P ∈ Y + . Remark 4.
The normal vector n x is well-defined because Γ ∈ C α and v = J − ∇ (cid:0) a + w | J | − − P (cid:1) is continuous in t due to (37) and Lemma 4.5.By the method construction, the obtained solution satisfies all the conditions of Lemma 3.2except bijectivity of the mappings ∂ω + (0) X ( ∂ω + (0) , t ) = ∂ω + ( t ), Γ X (Γ , t ) = Γ t ,which has to be checked independently. Lemma 4.7.
Under the conditions of Theorem 4.6 the value of T ∗ can be chosen so small thatfor every points y, z ∈ ω + (0) , Γ | X ( y, t ) − X ( z, t ) | ≥ µ | y − z | with an independent of y, z constant µ ∈ (0 , .Proof. Let us fix an arbitrary pair of points y, z ∈ Γ and connect them by a Lipschitz-continuouscurve l ( y, z ) ⊂ ω + (0). Since Γ is smooth, we can choose l ( y, z ) in such a way that its length | l ( y, z ) | satisfies the estimates κ | y − z | ≤ | l ( y, z ) | ≤ κ | y − z | with finite constants κ i dependingonly on module of continuity of the parametrization of Γ . By the definition X ( y, t ) − X ( z, t ) = ( y − z ) + ∇ ( U ( y, t ) − U ( z, t )) = ( y − z ) + Z l ( y,z ) ddl ( ∇ U ) ds and by virtue of (37) | X ( y, t ) − X ( z, t ) | ≥ | y − z | − X | γ | =2 sup Q + T | D γ U || l ( y, z ) | ≥ | y − z | (cid:16) − C κ T α/ (cid:17) . Q − T and a local solution of the free-boundaryproblem To construct a solution of problem (34) we follow the same scheme that was used to find asolution of problem (33). The only difference is that now the solution P − of the linear ellipticproblem has to satisfy the Neumann boundary condition on Γ . Let us define the functionspaces Z − , Y − , X − , where the upper index means that we consider the functions defined on Q − T = ω − (0) × [0 , T ]. Problem (34) is split into the problems for defining U − and P − . The firststep is to find a solution U − of the problem L ( U − , P − ) = 0 in Q − T ,U − = 0 on the parabolic boundary of Q − T , | J − | = Ψ( t ) on Γ × (0 , T ] (51)with a given P − ∈ W q ( Q − T ). The boundary condition for | J − | is substituted (35) in case ofTheorem 2.5. Repeating the proof of Theorem 4.3 we arrive at the following assertion.19. Galiano, S. Shmarev, J. Velasco Lemma 4.8.
Let P − ∈ W q ( Q − T ) with q > n +2 and Ψ( t ) ∈ C [0 , . Then there exists T ∗ ∈ (0 , so small that problem (51) has a unique solution U − ∈ Z − such that k U − k Z − ≤ r ′ < and r ′ → as T ∗ → . The second step is to solve the problem ( M ( P − , U − ) = 0 in ω − (0) ,P − = 0 on ∂ω − (0) , ( J − ) − · ∇ y P − · f n + x = S on Γ (52)with given U ± ∈ Z ± , P + ∈ W q ( Q + T ) and S = ( J + ) − · ∇ y P − · f n + x + [ a J − · ∇ y ( w | J − | ) · f n + x ] Γ . Lemma 4.9.
Let U ± ∈ Z ± , P + ∈ W q ( Q + T ) . If f − is locally Lipschitz-continuous, then for a.e. t ∈ (0 , T ) problem (52) has a solution P ( · , t ) ∈ W q ( ω − ) which satisfies the estimates k P k Y − ≤ C (cid:18) k U k Z + + k U k Z − + k P + k Y + + T /q max [0 ,T ] | Ψ ′ ( t ) | (cid:19) (53) with an absolute constant C .Proof. The existence of a solution of problem (52) satisfying (53) follows from the classicalelliptic theory - see, e.g., [19, Ch. 3, Sec. 5-6, 15]) or [14].Recall that in the case of Theorem 2.5 the corresponding estimate (53) is independent ofΦ( t ).The next step consists in checking the convergence of the iteratively defined sequences { U − k } , { P − k } : U − = 0, P − is the solution of problem (52) with U − = 0, for every k ≥ U − k is thesolution of (51) with P = P − k , P − k +1 is the solution of problem (52) with U − = U − k . This is doneexactly as in the proof of Theorem 4.6. Lemma 4.10.
Let w ∈ W q ( ω ± (0)) be strictly positive in ω (0) , f be Lipschitz-continuous onthe interval | s | ≤ w , and let ∂ω ± (0) , Γ ∈ C β with some β ∈ (0 , . There exists T ∗ ,depending on k w k (2) q,ω (0) , n , q , a ± , β and the Lipschitz constant of f such that problems (33) , (34) have unique solutions ( U ± , P ± ) ∈ Z ± × Y ± . Finally, we repeat the proof of Lemma 4.7 to ensure the bijectivity of the mapping y X ( t, t ) := y + ∇ U for y ∈ ω − (0). The assertion of Theorem 3.5 follows now if we define U = ( U + in Q + T ,U − in Q − T , P = ( P + in Q + T ,P − in Q − T . Let us denote by Σ ± the images of the surfaces ∂ ± ω (0) under the mapping y X ( y, t ). Ac-cording to Theorem 3.5 the pair ( w, C ) defined by formulas (32) is a solution of problem (15)in the sense of Definition 3.1. Let us take a smooth simply connected surface γ ⊂ ω + (0) such20n a cross-diffusion systemthat γ ∩ Γ = ∅ and γ ∩ ∂ω + (0) = ∅ . By continuity of the mapping y y + ∇ U , there is T + such that Σ + and Γ t do not touch the vertical surface S = γ × [0 , T + ], so that S ⊂ C + . Since w > ω + (0), the function w constructed in Theorem 3.5 is strictly positive in C + and is aweak solution of the uniformly parabolic equation. The local regularity results for the solutionsof uniformly parabolic quasilinear equations [18, Ch. 6, Sec. 4] imply that w ∈ C β, (2+ β ) / x,t in avicinity of S . Let us set ψ = w | S ∈ C β, (2+ β ) / ( S ), denote by A the annular cylinder with thelateral boundaries ∂ Ω × [0 , T + ] and S , and consider the following problem: u t = div( a + u ∇ u ) + f + ( u ) in A ,u = ψ on S, u = h on ∂ Ω × [0 , T + ] ,u ( x,
0) = w in A ∩ { t = 0 } . (54)This problem has a unique solution u ∈ C β, (2+ β ) / ( A ), that is, D κx D st ( u − w ) | Σ = 0 for 0 ≤ | κ | + 2 s ≤ . The required continuation to the exterior of C + is now given by the formula W = (cid:26) w ( x, t ) in C + \ A ,u ( x, t ) in A , The continuation from C − is constructed likewise. The proof if a byproduct of the proof of Theorem 2.2. Items (1)-(2) follows directly fromTheorem 2.2. By Lemma 3.2 u is obtained as the solution of problem (22) in the moving domain C + and then continued across the exterior boundary of C + up to the lateral boundary of D bythe solution of problem (54). Recall that by construction u satisfies the equation u t + div( u v ) = 0 for a.e. ( x, t ) ∈ C + with v = − a + ∇ u + ∇ p (see (31)). Let S and A be the sets chosen in the proof Theorem 2.2, A = A ∩ { t = 0 } . By Lemma 3.2, for every φ ∈ C ( Q \ C − ), φ = 0 on ∂ Ω × [0 , T ], u satisfies(23): − Z S φa + u ∇ u · n + dS + Z C + \A u ( φ t + ∇ x φ · v ( x, t )) dxdt + Z ω (0) \A u ( x ) φ ( x, dx = 0 . Continuing u to A by the classical solution e u of problem (54) we have Z S φa + u ∇ u · n + dS + Z A ( e uφ t + a + e u ∇ x φ · ∇ e u − f + ( e u ) φ ) dxdt + Z A u φ ( x, dx = 0 . Gathering these equalities and taking into account the definition of p , we obtain (13). Relation(14) follows by the same arguments. 21. Galiano, S. Shmarev, J. Velasco The assertion is an immediate byproduct of the method of construction of the solution to problem(9).
The assertion of Theorem 2.5 will follow if we prove that the velocity given by formula (12) iscontinuous on Γ t . The normal component of velocity is continuous on Γ t by the definition. Letus fix an arbitrary point y ∈ Γ and denote by x +0 = X + ( y , t ) its image under the mapping X + = y + ∇ U + in Q + T . By the definition, for every t > v + ( x +0 , t ) = X + t ( y , t ) = ∇ y U + t ( y , t ) . Let τ ( y ) be an arbitrary unit vector in the tangent plane to Γ at the point y . Since U + t = 0on Γ × [0 , T ], we have ∇ y U + t ( y , t ) · τ ( y ) = 0 and v + ( x +0 , t ) · τ ( y ) = 0 for all t > , which means that for all t > v + ( X + ( y ) , t ) coincides with n x ( y ).Repeating this argument we find that the direction of v − ( X − ( y ) , t ) is also given by n x ( y )for all t >
0. Thus, the images X ± ( y , t ) of the point y ∈ Γ move along the same line withthe direction vector n x ( y ). Since [ v ( x )] · n x ( x ) = 0 by construction, it is necessary that thetangent component of v is also continuous at x : every tangent vector τ ( y ) can be representedin the form τ ( y ) = ατ ( x ) + β n x ( x ) with α = 0 (for small t ), whence0 = [ v ( x )] · τ ( y ) = α [ v ( x )] · τ ( x ) + β [ v ( x )] · n x ( x ) = α [ v ( x )] · τ ( x ) . In this section, we review special cases of system (6) available in the literature. The first exampleconcerns the possibility to construct a solution assuming that neither the contact inhibitionassumption (3) on the initial data is fulfilled, nor that the matrix A in (4) is positive definite.The second example is an explicit solution that corresponds to specific initial data generated bythe self-similar Barenblatt solution of the porous medium equation. Finally we provide examplesof numerical simulations. Given a fixed
T > ⊂ R n , with ∂ Ω ∈ C , , find u i : Ω × (0 , T ] = Q T → R , i = 1 ,
2, such that ∂ t u i − div J i ( u , u ) = u i F i ( u , u ) in Q T ,J i ( u , u ) · n = 0 on Γ T = ∂ Ω × (0 , T ] ,u i ( · ,
0) = u i in Ω , (55)with the flows given by J i ( u , u ) = au i ∇ ( u + u ) , (56)22n a cross-diffusion systemand the Lotka-Volterra terms of the special type F ( u , u ) = 1 − u − αu , F ( u , u ) = γ (1 − βu − u /k ) (57)with positive constants α , β , γ and k . Theorem 6.1 ([4]) . For i = 1 , , let u i ∈ C ( ¯Ω) such that u i ≥ and B ≤ u + u ≤ B − ,for some constant B . Then there exist a solution u i ∈ C , ([0 , ∞ ) × Ω) of (55) with J i givenby (56) and F i by (57) . The requirement of the strong regularity of the initial data is due to method of proof. Initially,the following formally equivalent system is solved for u = u + u and v = u /u : ∂ t u − div( u ∇ u ) = G ( u, v ) in Q T ,∂ t v − ∇ u · ∇ v = G ( u, v ) in Q T ,u ∇ u · n = 0 on Γ T ,u ( · ,
0) = u + u , v ( · ,
0) = u /u in Ω , (58)with some smooth functions G , G . The proof of existence of solutions of (58) is based onthe Schauder fixed point theorem. In order to obtain the required compactness for the fixedpoint operator, the authors pass to the system of Lagrangian coordinates related to the flow −∇ u ( t, x ), and claim the strong regularity assumptions on the initial data.A similar problem was studied in [13] under weaker assumptions on the initial data and witha more general flow of the type J i ( u , u ) = au i ∇ ( u + u ) + bqu i + c ∇ u i . (59)The existence was proved with a different method. Theorem 6.2 ([13]) . Assume the following conditions: for i = 1 ,
1. the flows J i ( u , u ) are given by (59) with constant a > , c ≥ and b ∈ R , q ∈ L ( Q T ) and div q ≥ a.e. in Q T ,2. u i ∈ L ∞ (Ω) with u i ≥ , u = u + u ∈ H (Ω) with u > δ for some constant δ > , ∇ u · n = 0 on ∂ Ω (the compatibility condition),3. F ( u , u ) = F ( u , u ) = F ( u + u ) with F ∈ C ( R + ) , ∀ s ≥ , F ( · , · , s ) ≤ Cs with C > .Then problem (55) has a weak solution ( u , u ) understood in the following sense:(i) u i ≥ , u i ∈ L ∞ ( Q T ) ∩ H (0 , T ; ( H (Ω)) ′ ) ,(ii) for all ϕ ∈ L (0 , T ; H (Ω)) Z T h ∂ t u i , ϕ i + Z Q T J i ( u , u ) · ∇ ϕ = Z Q T u i F ( u + u ) ϕ, (60) where h· , ·i denotes the duality product of ( H (Ω)) ′ × H (Ω) ,
23. Galiano, S. Shmarev, J. Velasco (iii) the initial conditions in (55) are satisfied in the sense lim t → k u i ( · , t ) − u i k ( H (Ω)) ′ = 0 as t → . The proof of this theorem is based on the following two observations. Firstly, note that if aweak solution of (55) does exist, then the addition of its components, u = u + u satisfies theequation ∂ t u − div J ( u ) = uF ( u ) in Q T (61)with the flow J ( u ) = u (cid:0) a ∇ u + bq (cid:1) + c ∇ u, (62)together with non-flow boundary conditions and the initial datum satisfying u > L ∞ ( Q T ) ∩ L (0 , T ; H (Ω)) positive solutions to this uniformlyparabolic problem is a well-known issue, see, e.g., [18]. Then, the non-negativity of the solutions u i of problem (55) results in u i ∈ L ∞ ( Q T ), i = 1 ,
2, which is a property difficult to obtaindirectly from the analysis of system (55).As a second observation, let us note that the usual approach to the proof of existence ofsolutions to cross-diffusion systems in the most conflicting case c i = 0 is based on justifyingthe use of log u i as a test-function in (60) in order to obtain estimates from the addition of theresulting identities Z Ω h ( u i ( T, · )) + Z Q T (cid:0) |∇ u i | + ∇ u · ∇ u (cid:1) ≤ C, (63)with h ( u i ) = u i (log u i −
1) + 1. However, in the present case the singularity of the diffusionmatrix corresponding to (55) prevents us from obtaining the L estimates for ∇ u i from (63).To circumvent this difficulty and keep at the same time the good properties derived for theaddition of the components of a solution, the following perturbation of the original problem isintroduced: ∂ t u i − div J ( δ ) i ( u , u ) = u i F ( u ) in Q T with J ( δ ) i ( u , u ) = J i ( u , u ) + δ u i u ) , (64)subject to the non-flow boundary conditions. Using results of [12] one may deduce the existenceof a sequence of non-negative functions ( u ( δ )1 , u ( δ )2 ). Moreover, it turns out that the sum u ( δ )1 + u ( δ )2 is uniformly bounded in L ∞ ( Q T ). This fact allows one to pass to the limit, which leads to theassertion of Theorem 6.2. The difficulties in identifying the limit of the sequence of solutions tothe approximated problems are delivered by the diffusive and the Lotka-Volterra terms Z Q T u ( δ ) i ∇ ( u ( δ )1 + u ( δ )2 ) · ∇ ϕ, Z Q T u ( δ ) i F i ( u ( δ )1 , u ( δ )2 ) ϕ.
24n a cross-diffusion systemSince the L ∞ ( Q T ) weak- ∗ convergence is the only convergence for the independent components u ( δ ) i obtained from the approximated problems, stronger conditions on the data of the problemsfor u ( δ )1 + u ( δ )2 are required in order to pass to the limit. To be precise, one needs the strictpositivity and H (Ω) regularity of the initial data. Notice, however, that if a strong convergenceof u ( δ ) i in, for instance, L ( Q T ) is proven, then the assumptions on u + u may be weakenedin such a way that just the usual L ( Q T ) weak convergence of ∇ ( u ( δ )1 + u ( δ )2 ) holds. In addition,in this case some other restrictions on the coefficients, such as the equality of the diffusive terms a = a , or the restriction on the form of the Lotka-Volterra terms, can be removed. In the one-dimensional case Bertsch et al. [3] proved BV ( Q T ) uniform estimates for the vanishing viscosityapproximation to (58), which allowed one to get strong convergence in L ( Q T ). However, theseestimates depend on the L ( Q T )-norm of the Laplacian of the sum, thus leading to similarregularity assumptions on the initial data. Let us finally notice that, due to the discontinuitiesarising in the limit problem, the uniform estimate for u ( δ )1 + u ( δ )2 in SBV ( Q T ) is the strongestestimate that can be expected. We consider a particular situation of the contact-inhibition problem in which an explicit solutionof (55) may be computed in terms of a suitable combination of the Barenblatt explicit solution ofthe porous medium equation, the Heavyside function and the trajectory of the contact-inhibitionpoint. To be precise, we construct a solution to the problem ∂ t u i − ( u i ( u + u ) x ) x = 0 in ( − R, R ) × (0 , T ) = Q T , (65) u i ( u + u ) x = 0 on {− R, R } × (0 , T ) , (66)with u ( x ) = H ( x − x ) B ( x, , u ( x ) = H ( x − x ) B ( x, . (67)Here, H is the Heavyside function and B is the Barenblatt solution of the porous mediumequation corresponding to the initial datum B ( x, − t ∗ ) = δ , i.e. B ( x, t ) = 2( t + t ∗ ) − / (cid:2) − x ( t + t ∗ ) − / (cid:3) + . For simplicity, we consider problem (65)-(67) for
T > R ( T ) < R , with R ( t ) = √ t + t ∗ ) / , so that B ( R, t ) = 0 for all t ∈ [0 , T ]. The point x is the initial contact-inhibitionpoint, for which we assume | x | < R (0), i.e. it belongs to the interior of the support of B ( · , Theorem 6.3.
The functions u ( x, t ) = H ( x − η ( t )) B ( x, t ) , u ( x, t ) = H ( η ( t ) − x ) B ( x, t ) , with η ( t ) = x ( t/t ∗ ) / , are a weak solution of problem (65) - (67) in the following sense: Z R − R (cid:0) ( u i ϕ )( · , T ) − u i ϕ ( · , (cid:1) − Z Q T u i ( ϕ t − ( u + u ) x ϕ x ) = 0 for all ϕ ∈ H ( Q T ) .
25. Galiano, S. Shmarev, J. VelascoLet H ǫ the regularization of the Heavyside function taking the values (cid:8) , (1 − x/ǫ ) , (cid:9) inthe intervals ( − R, − ǫ ), ( − ǫ, ǫ ) and ( ǫ, R ), respectively, for ǫ > Lemma 6.4.
Let u ǫi : [0 , T ] × [ − R, R ] , i = 1 , , be given by u ǫ ( x, t ) = H ǫ ( x − η ( t )) B ( x, t ) , u ǫ ( x, t ) = H ǫ ( η ( t ) − x ) B ( x, t ) (68) with η ( t ) = x ( t/t ∗ ) / . Then (cid:12)(cid:12)(cid:12)(cid:12)Z R − R (cid:0) ( u iǫ ϕ )( · , T ) − ( u iǫ ϕ )( · , (cid:1) − Z Q T u ǫi ( ϕ t − ( u ǫ + u ǫ ) x ϕ x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cǫ for all ϕ ∈ H (( t ∗ , T ) × ( − R, R )) .Proof. Observe that u ǫi are continuous and bounded in Ω × ( t ∗ , T ), and satisfy u ǫ + u ǫ = B .Therefore, u ǫ + u ǫ ∈ L ( t ∗ , T ; H ( − R, R )) uniformly in ǫ . Let ϕ ∈ H ( Q T ). Using ϕ ǫ ( x, t ) = ϕ ( x, t ) H ǫ ( x − η ( t )) as the test-function in the weak formulation of the problem satisfied by theBarenblatt solution in Q T we obtain Z R − R (cid:0) ( H ǫ Bϕ )( · , T ) − ( H ǫ Bϕ )( · , t ∗ ) (cid:1) − Z Q T H ǫ B ( ϕ t − B x ϕ x ) = I ǫ , with I ǫ = − Z Q T ϕ ( x, t ) B ( x, t ) H ′ ǫ ( x − η ( t )) (cid:0) η ′ ( t ) + B x ( x, t ) (cid:1) dxdt. Since | x | < R (0), we have η ( t ) < R − ǫ , for ǫ small enough and t ∈ (0 , T ), and then using theexplicit expression of B x and η ′ ( t ) we deduce I ǫ = 16 ǫ Z T Z ǫ − ǫ yϕ ( y + η ( t ) , t ) B ( y + η ( t ) , t ) dydt. Since ϕ and B are uniformly bounded in L ∞ , we obtain | I ǫ | ≤ Cǫ, (69)with
C > ǫ . The computation using ϕ ( x, t ) H ǫ ( η ( t ) − x ) as test function givessimilar results for some I ǫ satisfying the same estimate (69) than I ǫ . Observing that functions(68) satisfy u ǫ + u ǫ = B , we finish the proof. Proof of Theorem 6.3.
Since u ǫi are uniformly bounded in L ∞ ( Q T ) we may perform the limit ǫ → u i ∈ L ∞ ( Q T ) such that Z R − R (cid:0) ( u i ϕ )( · , T ) − u i ϕ ( · , (cid:1) − Z Q T u i ( ϕ t − ( u + u ) x ϕ x ) = 0 . On the other hand, taking the limit of expressions (68) we get u ( x, t ) = H ( x − η ( t )) B ( x, t ) , u ( x, t ) = H ( η ( t ) − x ) B ( x, t ) .
26n a cross-diffusion system
Remark 5.
The problem solved by η is related to B by the ODE problem (cid:26) η ′ ( t ) = − B x ( t, η ( t )) for t ∈ (0 , T ) ,η (0) = x , which ensures the mass conservation for each component. Indeed, defining M i ( t ) = Z R − R u i ( x, t ) dx = Z η ( t ) − R B ( x, t ) dx, we find, using the equation satisfied by B and its boundary conditions M ′ i ( t ) = Z η ( t ) − R B t ( x, t ) dx + η ′ ( t ) B ( η ( t ) , t )= B ( η ( t ) , t ) B x ( η ( t ) , t ) + η ′ ( t ) B ( η ( t ) , t ) = 0 . Remark 6.
It is not difficult to extend the above construction to other one-dimensional prob-lems. For instance, for problem (55) we may consider the solution u of (61)-(62) and the cor-responding approximations of the type (68). Then, to handle the integrals I iǫ , we first observethat for ǫ → I ǫ → − Z T ϕ ( η ( t ) , t ) B ( η ( t ) , t ) (cid:0) η ′ ( t ) + G ( η ( t ) , t ) (cid:1) dt, with G = au x + bq + c (log( u )) x . Therefore, if the ODE problem (cid:26) η ′ ( t ) = − G ( t, η ( t )) for t ∈ (0 , T ) ,η (0) = x , (70)is solvable, a solution for problem (55) may be constructed. Typical conditions on G for (70) tobe solvable are given in terms of Sobolev or BV regularity in space for G and L (0 , T ; L ∞ ( − R, R ))regularity for the divergence of G , G x in the one-dimensional case, see [9, 1] for further details. The discretization of (55) with the regularizing term given in (64) follows the standard FiniteElement methodology. To construct a solution we apply the semi-implicit Euler scheme in timeand a P continuous finite element approximation in space and then study the behavior ofsolutions as δ →
0, see [13] for the details.Let τ > t = t = 0, set u ǫi = u i . Then, for n ≥ u nǫi : (0 , T ) × Ω → R such that for, i = 1 , τ (cid:0) u nǫi − u n − ǫi , χ ) h + (cid:0) J ( δ ) i (Λ ǫ ( u nǫ ) , Λ ǫ ( u nǫ ) , ∇ u nǫ , ∇ u nǫ ) , ∇ χ (cid:1) h == (cid:0) α i u nǫi − λ ǫ ( u nǫi )( β i λ ǫ ( u n − ǫ ) + β i λ ǫ ( u n − ǫ )) , χ (cid:1) h , (71)for every χ ∈ S h , the finite element space of piecewise P -elements. Here, ( · , · ) h stands for adiscrete semi-inner product on C (Ω). The parameter ǫ > λ ǫ and Λ ǫ , which converge to the identity as ǫ → u u u +u −3 (u +u ) x u u u +u −3 (u +u ) x Figure 1:
A transient state of solutions of Experiments 1 (first row) and 2 (second row). Left panel:solutions ( u ( δ )1 , u ( δ )2 ). Center panel: The sum u ( δ ) = u ( δ )1 + u ( δ )2 . Right panel: the space derivative of thesum, u ( δ ) x . Since (71) is a nonlinear algebraic problem, we use a fixed point argument to approximateits solution, ( u nǫ , u nǫ ), at each time slice t = t n , from the previous approximation u n − ǫi . Let u n, ǫi = u n − ǫi . Then, for k ≥ u n,kǫi such that for i = 1 ,
2, and for all χ ∈ S h τ (cid:0) u n,kǫi − u n − ǫi , χ ) h + (cid:0) J ( δ ) i (Λ ǫ ( u n,k − ǫ ) , Λ ǫ ( u n,k − ǫ ) , ∇ u n,kǫ , ∇ u n,kǫ ) , ∇ χ (cid:1) h == (cid:0) α i u n,kǫi − λ ǫ ( u n,k − ǫi )( β i λ ǫ ( u n − ǫ ) + β i λ ǫ ( u n − ǫ )) , χ (cid:1) h . We use the stopping criteria max i =1 , k u n,kǫ,i − u n,k − ǫ,i k ∞ < tol, for empirically chosen values oftol, and set u ni = u n,ki .In the following experiments we take a uniform partition of Ω = (0 ,
1) in 10 subintervals andthe time step τ = 10 − . The drift and the linear diffusion coefficients are b i = c i = 0, and theLotka-Volterra terms, i.e. the right-hand side of (55) have the form f i ( u , u ) = u i ( α i − β i u − β i u ) with α = 1 , β = 1 , β = 0 . , α = 5 , β = 1,and β = 2. For the initial datawe take u i = exp(( x − x i ) / . f i = 0 for i = 1 , x = 0 . x = 0 .
6. Althoughthe initial data do not satisfy the condition u + u > − , and the perturbation parameter to δ = 10 − .We run two experiments according to different nonlinear diffusion matrices. In the firstexperiment, we set the same diffusion coefficient a = 1 for both equations, which is the situationstudied in Theorems 6.1 and 6.2. In the second experiment we take different diffusivities, a = 1and a = 3, in the equations for u and u (see (55)). The aim of these experiments is toconfirm numerically that, unlike the case of equal diffusivities, in our case the gradient of thesum u + u may develop discontinuity. This property can be checked on Figure 1. In thefirst row we show the results for a transient state of the equal-diffusivities case. Although theindependent components of the solution, u and u exhibit a discontinuity at the contact point,28n a cross-diffusion system x = 0 .
5, the sum u + u is continuous and, as it can be seen in the right panel of the first row,the derivative seems to be continuous as well. In the second row of Figure 1 we show the resultscorresponding to the different diffusivities case. The behavior is clearly different. Althoughthe continuity of u + u still holds, a discontinuity of ( u + u ) x at the contact point may beobserved. References [1] L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. math.158 (2004), 227–260.[2] M. Bertsch, M. E. Gurtin, D. Hilhorst, L. A. Peletier, On interacting populations thatdisperse to avoid crowding: preservation of segregation, J. Math. Biol. 23 (1985) 1–13[3] M.Bertsch, R.Dal Passo, M.Mimura A free boundary problem arising in a simplified tumourgrowth model of contact inhibition, Interfaces and Free Boundaries, 12 (2010) pp. 235–250.[4] M. Bertsch, D. Hilhorst, H. Izuhara, M. Mimura, A nonlinear parabolic-hyperbolic systemfor contact inhibition of cell-growth, Diff. Equ. Appl. 4 (2012) 137–157.[5] S. N. Busenberg, C. C. Travis, Epidemic models with spatial spread due to populationmigration, J. Math. Biol. 16 (1983) 181–198.[6] M. A. J. Chaplain, L. Graziano, L. Preziosi, Mathematical modelling of the loss of tissuecompression responsiveness and its role in solid tumour development, Math. Med. Biol.23(3) (2006) 197–229.[7] J.I.D´ıaz, S.Shmarev, Lagrangian approach to the study of level sets: application to a freeboundary problem in climatology. Arch. Ration. Mech. Anal. 194 (2009), no. 1, 75–103.[8] J.I.D´ıaz, S.Shmarev, Lagrangian approach to the study of level sets. II. A quasilinear equa-tion in climatology. J. Math. Anal. Appl. 352 (2009), no. 1, 475–495.[9] R. J. DiPerna, P. L. Lions, Ordinary differential equations, transport theory and Sobolevspaces, Invent. math. 98 (1989), 511–547.[10] G.-Q.Chen, H.Frid. Divergence-measure fields and hyperbolic conservation laws. Arch. Ra-tion. Mech. Anal. 147 (1999), no. 2, 89–118.[11] G. Galiano, On a cross-diffusion population model deduced from mutation and splittingof a single species, Comput. Math. Appl. 64(6) (2012) 1927-1936.[12] G. Galiano, M. L. Garz´on, A. J¨ungel, Semi-discretization in time and numerical conver-gence of solutions of a nonlinear cross-diffusion population model, Numer. Math. 93(4)(2003) 655–673.[13] G. Galiano, V. Selgas, On a cross-diffusion segregation problem arising from a model ofinteracting particles. To appear in Nonlinear Anal. Real World Appl.[14] P.Grisvard, Elliptic problems in nonsmooth domains. Monographs and Studies in Mathe-matics, 24. Pitman (Advanced Publishing Program), Boston, MA, 1985. xiv+410 pp.29. Galiano, S. Shmarev, J. Velasco[15] W. S. C. Gurtin, R. M. Nisbet, The regulation of inhommogeneous populations, J. Theor.Biol. 52 (1975) 441–457.[16] M. E. Gurtin, A. C. Pipkin, On interacting populations that disperse to avoid crowding,Q. Appl. Math. 42 (1984) 87–94.[17] A.Kolmogorov, S.Fomin. Elements of the theory of functions and functional analysis. Vol.2: Measure. The Lebesgue integral. Hilbert space. Translated from the first (1960) Russianed. by Hyman Kamel and Horace Komm Graylock Press, Albany, N.Y. 1961 ix+128 pp.[18] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural’ceva