Lack of uniqueness for weak solutions of the incompressible porous media equation
LLACK OF UNIQUENESS FOR WEAK SOLUTIONS OFTHE INCOMPRESSIBLE POROUS MEDIA EQUATION
DIEGO CORDOBA, DANIEL FARACO AND FRANCISCO GANCEDO
Abstract.
In this work we consider weak solutions of the incom-pressible 2-D porous media equation. By using the approach of DeLellis-Sz´ekelyhidi we prove non-uniqueness for solutions in L ∞ inspace and time. Introduction
The incompressible 2-D porous media equation (IPM) is describedby ρ t + ∇ · ( vρ ) = 0where the scalar ρ ( x, t ) is the density of the fluid. The incompressiblevelocity field ∇ · v = 0is related with the density by the well-known Darcy’s law [1] µκ v = −∇ p − (0 , gρ )where µ represents the viscosity of the fluid, κ is the permeability ofthe medium, p is the pressure of the fluid and g is acceleration due togravity. Without lost of generality we will consider µ/κ = g = 1.In this paper we study the weak formulation of this system and weconstruct non trivial solutions with ρ, v ∈ L ∞ ( T × [0 , T ]) from initialdata ρ ( x,
0) = 0. Here T is the two dimensional flat torus.We define a weak solution of IPM ( ρ, v, p ) if ∀ ϕ, χ, λ , λ ∈ C ∞ c ([0 , T ) × T ) with λ = ( λ , λ ) the following identities hold T (cid:90) (cid:90) T ρ ( x, t ) ( ∂ t ϕ ( x, t ) + v ( x, t ) · ∇ ϕ ( x, t )) dxdt (1) + (cid:90) T ρ ( x ) ϕ ( x, dx = 0 , (2) T (cid:90) (cid:90) T v ( x, t ) · ∇ χ ( x, t ) dxdt = 0 , Date : December 16, 2009. a r X i v : . [ m a t h . A P ] D ec DIEGO CORDOBA, DANIEL FARACO AND FRANCISCO GANCEDO (3) T (cid:90) (cid:90) T ( v ( x, t ) + ∇ p ( x, t ) + (0 , ρ ( x, t ))) · λ ( x, t ) dxdt = 0 . For initial data in the Sobolev class H s ( T ) ( s >
2) there is local-existence and uniqueness of solutions in a classical sense and globalexistence is an open problem [4]. It is known the existence of weaksolutions, where the motion takes place in the interface between flu-ids with different constant densities, modeling the contour dynamicsMuskat problem [3]. The existence of weak solutions for general initialdata is not known. In this context we emphasize that the solutions weconstruct satisfy lim sup t → + (cid:107) ρ (cid:107) H s ( t ) = + ∞ for any s > v ( x, t ) = P V (cid:90) R Ω( x − y ) ρ ( y, t ) dy −
12 (0 , ρ ( x )) , x ∈ R , where the kernel is a Calderon-Zygmund typeΩ( x ) = 12 π (cid:18) − x x | x | , x − x | x | (cid:19) . The integral operator is defined in the Fourier side by (cid:98) v ( ξ ) = ( ξ ξ | ξ | , − ( ξ ) | ξ | ) (cid:98) ρ ( ξ ) . This system is analogous to the 2-D surface Quasi-geostrophic equa-tion (SQG) [4], in the sense that is an active scalar that evolves bya nonlocal incompressible velocity given by singular integral opera-tors. It follows that, for Besov spaces, if the weak solution ρ is in L ([0 , T ] × B s, ∞ ) with s > then the L norm of ρ is conserved [27].This result frames IPM in the theory of Onsager’s conjecture for weaksolutions of 3-D Euler equations [2],[16]. However there is an extracancelation, for SQG, due to the symmetry of the velocity given by (cid:98) v ( ξ ) = i ( − ξ | ξ | , ξ | ξ | ) (cid:98) ρ ( ξ )that provides global existence for weak solution with initial data in L ( T ) [20]. Furthermore, one can find a substantial difference be-tween both systems for weak solutions of constant ρ in complementarydomains, denoted in the literature as patches [13]. For IPM the Muskatproblem presents instabilities of Kelvin-Helmholtz’s type [3] and thereis no instabilities for SQG ([21],[8]). OROUS MEDIA EQUATION 3
The first results of non-uniqueness for the incompressible Euler equa-tions are due to Scheffer [22] and Shnirelman [23] where the velocityfield is compacted supported in space and time with infinite energy.The method of the proof we use in this paper is based on understand-ing the equation as a differential inclusion in the spirit of Tartar [25, 26].In a ground breaking recent paper De Lellis-Sz´ekelyhidi [6] showed thatwith this point of view the modern methods for solving differential in-clusions could be reinterpreted and adapted to construct wild solutionsto the Euler equation with finite energy. Our plan was to investigatethe scope of this approach in the context of the porous media equation.However it turned out that the situation is different and we have totake different routes at several places which might be of interest forthe theory of differential inclusions. We describe them shortly. Firstly,using the terminology of this area that will be recovered in the firstsection, the special role of the direction of gravity yields certain lack ofsymmetry in the wave cone Λ. Moreover the set K describing the nonlinear constraint belongs to the zero set of a Λ convex function. Thenit follows that, opposite to Euler, the Λ convex hull does not agree withthe convex hull and more relevant K ⊂ ∂K Λ . This is an obstructionfor the available versions of convex integration, the ones based on Bairecategory [5, 10, 15] and direct constructions [9, 11, 14]. Thus we donot see anyway to use the method to produce solutions to the equationwith constant pressure and velocity for some time. Surprisingly, aneasy argument shows that the states in int(K Λ ) can still be used toproduce periodic weak solutions starting with ρ = v = 0. This leavesthe difficulty of choosing a proper subset ˜ K ⊂ K with sufficient largeΛ hull. In general the computation of Λ hull might be rather compli-cated. In this work we argue differently to suggest a more systematicmethod: Instead of fixing a set and computing the hull, we pick a rea-sonable matrix A and compute ( A + Λ) ∩ K . Then by [10, Corollary4.19] it is enough to find a set ˜ K ⊂ ( A + Λ) ∩ K such that A ∈ ˜ K c to find what are called degenerate T K c .To our knowledge this is the first time they are used to produce exactsolutions. Then, we are able to choose the set ˜ K carefully so that theconstruction is stable and allow us to solve the inclusion. Our proof ofthis later fact is related, to the known ways of solving differential inclu-sions based on T DIEGO CORDOBA, DANIEL FARACO AND FRANCISCO GANCEDO analytic one. In this last section we include some remarks about thenature of our solution as well as why this approach seems to fail forconstructing weak solutions compactly supported in space and time.2.
General strategy and notation
The strategy to construct wild solutions to a non linear PDE , under-standing it as a differential inclusion, builds upon the following mainsteps. Consider the equation P ( u ) = 0 , u : R n → R m , where P is a general differential operator. Tartar framework:
Decompose the non linear PDE into a lin-ear equation (Conservation law) L (˜ u ) = 0 and a pointwise constraint˜ u ( x ) ∈ K . Possibly this requires to introduce new state variables ˜ u tolinearize the non linearities P ( u ) = 0 ⇐⇒ (cid:40) L (˜ u ) = 0˜ u ( x ) ∈ K .
Specially easy to bring to this framework are equations of the form L ( F i ( u )) = 0where F i : R m → R i = 1 , ..., l are some functions possibly nonlinear. Then one considers new states variables ˜ u i = F i ( u ) , whichclearly satisfy the conservation law. Then the pointwise constraintamounts to q i = F i ( u ) . Wave Cone:
As discovered by Tartar [25] in his development ofcompensated compactness to such conservation law one can associatea corresponding wave cone Λ L , defined as follows:Λ L = { A ∈ R m × R l : ∃ ξ such that for z ( x ) = Ah ( x · ξ ) , L ( z ) = 0 } , where h : R → R is an arbitrary function. That is, the wave conecorresponds to one dimensional solution to the conservation law. Potential: If A ∈ Λ L we can build z : R n → R m × R l such that L ( z ) = 0 and z ( R n ) ⊂ σ A where σ A = { tA : t ∈ R } is the line passing through A . The wildsolutions are made by adding one dimensional oscillating functions indifferent directions A . For that it is needed to localize the waves. Thisis done by finding a suitable potential, i.e. a differential operator D such that L ( D ) = 0 , Here n and m are suitable spaces dimensions that we keep general at this point. OROUS MEDIA EQUATION 5 and the plane wave solutions to the conservation law can be obtainedfrom the potential.
Pointwise constraint:
In the pointwise constraint one adds theadditional features desired for the weak solution (fixed energy, fixedvalues, etc). This describes a new set ˜ K ⊂ K . Then the solution isobtained by two additional steps. • Find a bigger set U so that there exists at least a solution tothe relaxed system U ∈ U , with the proper boundary conditions. In practice the set U ⊂ int( K Λ ), the so-called Λ convex hull of K . • Then one fixes a domain Ω compact in space and time andin properly chosen subdomains of Ω one adds infinitely manysmall perturbations to find a function U ∞ ∈ K a.e ( x, t ) ∈ Ωand U ∞ = U outside of Ω. If U is a solution to the conservationlaw, so is U ∞ . Notice that then U ∈ K and to be sure that U ∞ (cid:54) = U , we need that U / ∈ ˜ K .There is no standard way to produce the weak solutions from K Λ and it is not even always possible. In [6] (for Euler equations) it isenough that (0 , ∈ U = K c , which is open in the spaces where theimages of the potential lie and that for every state U ∈ U , A ˜ u ∈ Λ suchthat the segment which end points ˜ u ± A ∈ U and | A | ≥ C dist(˜ u, K ) . As discussed in the introduction, this strategy fails in our contextand needs to be modified.3.
Porous media equation
Here we start by recalling the 2-D IPM system(4) ∂ t ρ + ∇ · ( vρ ) = 0 ∇ · v = 0 v = −∇ p − (0 , ρ ) . Our task below is to adapt section 2 to the IPM system.3.1.
Porous media equation in the Tartar framework.
Givenfunctions ( ρ, v, q ) we define the differential operator L by(5) L ( ρ, q, v ) = ∂ t ρ + ∇ · q ∇ · v Curl( v + (0 , ρ )) (cid:82) T v (cid:82) T ρ DIEGO CORDOBA, DANIEL FARACO AND FRANCISCO GANCEDO
Proposition 3.1.
A pair ( ρ, v ) ∈ L ∞ ( T × [0 , T ] , R × R ) is a weaksolution to IPM if and only if we find u = ( ρ, v, q ) ∈ L ∞ ( T × [0 , T ] , R × R × R ) such that, (cid:40) L ( u ) = 0 q = ρv Proof.
The two first equations are obvious. The third follows fromthe Hodge decomposition in T . Namely consider a vector field f ∈ L ( T ) with π ) (cid:82) T f dx = 0 and curl f = 0. We claim that the Hodgedecomposition implies that f is a gradient field. Namely, every field f ∈ L ( T ) on the torus T has a unique orthogonal decomposition f = 1(2 π ) (cid:90) T f dx + w + ∇ p, such that div w = 0. But curl f = 0 = curl w . Now we have thatdiv w = 0 and curl w = 0, then for example by using the formulacurl curl g = − ∆ g + ∇ div g one finds that ∆ w = 0 and which yields w = 0. Thus, f = 1(2 π ) (cid:90) T f dx + ∇ p = ∇ p, It is convenient to write the differential part of L in a matrix form.For that to each state u = ( ρ, v, q ) we associate a matrix value function U ( u ) : R × R × R → M × by U ( u ) = − v − ρ v , v v q q ρ and the subspace U ⊂ M × by U = U ( R × R × R ).The following lemma is straightforward. Lemma 3.1.
Let U = U ( ρ, v, q ) ∈ L ( T × R ) with π ) (cid:82) T ρdx = π ) (cid:82) T vdx = 0 . Then the following statement holds in the sense ofdistributions: L ( u ) = 0 ⇐⇒ Div ( U ) = 0 , where Div ( a ij ) = ∂ j a ij OROUS MEDIA EQUATION 7
The wave cone.
We denote by Λ the wave cone related to (IPM).The reason to write the states in matrix form is that the wave coneis particularly easy to characterize. Namely, we want a plane wave z ( x ) = Ah ( x · ξ ) to satisfy Div ( z ) = 0 . We obtain that for each i , h (cid:48) ( x · ξ ) ξ j a ij = 0, that is A ( ξ ) = 0 . In other words A ∈ Λ if and only if A is a singular matrix. Thus A ∈ Λ ∩ U if − ρ ( v + v + ρv ) = 0 ⇐⇒ ρ = −| v | v or ρ | v | = 0 . Observe that fixed ρ this is v + v + ρv = v + ( v + ρ − ρ ρ the corresponding v ∈ S ((0 , − ρ ) , | ρ | ).3.3. Potentials.
The aim of this section is to find a differential oper-ator D such that L ( D ) = 0. We define for ψ, ϕ : T × R → R , D ( ϕ, ψ ) = ∂ x x ψ ∂ x x ψ ∂ x x ψ − ∂ x x ψ − ∂ tx ψ − ∂ x ϕ − ∂ tx ψ + ∂ x ϕ ∆ ψ − ∆ ψ Or more compactly written − ∂ x x ψ ∂ x x ψ ∂ x x ψ − ∂ x x ψ − ∂ tx ψ − ∂ x ϕ − ∂ tx ψ + ∂ x ϕ ∆ ψ Clearly D ( ϕ, ψ ) ∈ U . Lemma 3.2.
Let Ω ⊂ T × R be a bounded open set, ϕ ∈ W , ∞ (Ω) and ψ ∈ W , ∞ (Ω) . Then it holds that L D ( ϕ, ψ ) = 0 in Ω in the sense of distributions.Proof. In our notation we need to obtain that
Div ( D ( ϕ, ψ )) = 0distributionally. The claim follows since distributional partial deriva-tives commute. Namely, it holds that I: − ∂ x x x ψ + ∂ x x x ψ = 0, II: ∂ x x x ψ − ∂ x x x ψ = 0, DIEGO CORDOBA, DANIEL FARACO AND FRANCISCO GANCEDO
III: ( − ∂ tx x ψ − ∂ tx x ψ + ∂ t ∆ ψ ) + ( − ∂ x x ϕ + ∂ x x ϕ ) = 0 , as desired. Finally we observe that since the flat torus has no bound-ary then (cid:90) T D ( ϕ, ψ ) = 0and thus L D ( ϕ, ψ ) = 0. (cid:3) Remark 3.1.
Observe that we are solving ∆ ψ = ρ and then using thePDE to define the rest.3.4. Plane waves with potentials.
In this section we show that forany direction Λ with ρ (cid:54) = 0 we can obtain a suitable potential. It willbe more convenient for us to work with saw-tooth functions insteadof trigonometric functions. The following proposition is related to [10,Proposition 3.4]. Lemma 3.3.
Let U ∈ Λ with ρv (cid:54) = 0 , < λ < , (cid:15) > , Ω ⊂ T × R .Then there exists a sequence u N : Ω → U of piecewise smooth functionssuch that • L ( u N ) = 0 , • sup ( x,t ) ∈ Ω dist( u N ( x, t ) , [ − (1 − λ ) U, λU ]) ≤ (cid:15), • | ( x, t ) ∈ Ω : u N ( x, t ) = ( − − λ ) U | ≥ λ (1 − (cid:15) ) , • | ( x, t ) ∈ Ω : u N ( x, t ) = λU | ≥ (1 − λ )(1 − (cid:15) ) , • u N ∗ (cid:42) in L ∞ ( T × R ) . Proof.
Let S ∈ W , ∞ ( R , R ) and set ψ N ( x , x , t ) = 1 N S ( N ( − v (cid:112) | v | x + (cid:112) | v | x + ct ))and ϕ N ( x , x , t ) = 1 N S (cid:48) ( N ( − v (cid:112) | v | x + (cid:112) | v | x + ct )) . Then D ( dϕ N , ψ N ) = S (cid:48)(cid:48) ( N ( − v (cid:112) | v | x + (cid:112) | v | x + ct )) − v − ρ v v v c v √ | v | − d (cid:112) | v | − c (cid:112) | v | − d v √ | v | ρ . Now the matrix v √ | v | − (cid:112) | v |− (cid:112) | v | − v √ | v | OROUS MEDIA EQUATION 9 has determinant − ( v + v ) / | v | (cid:54) = 0 and hence defines a bijection of R . Thus for any q , q we can choose ( c, d ) so that( v (cid:112) | v | − d (cid:112) | v | , − c (cid:112) | v | − d v (cid:112) | v | ) = ( q , q ) . Therefore we have that for any direction in Λ with ρv (cid:54) = 0 we canobtain a suitable potential.Next we choose an appropriate function S . We consider the Lipschitz1-periodic functions S, s : R → R such that S (0) = 0, S (cid:48) = s , s (cid:48) ( x ) =1 − λ − χ [ λ/ , − λ/ ( x ) for x ∈ [0 , s (1 /
2) = 0, s (cid:48) (1 / x ) = s (cid:48) (1 / − x ) and s (1 / x ) = s (1 / − x ) if | x | < / Figure 1.
The graph of s . We then localize the wave using the potential in the standard wayi.e. we define a test functions ζ (cid:15) (cid:48) such that | ζ (cid:15) (cid:48) | ≤ , ζ (cid:15) (cid:48) = 1 on B − (cid:15) (cid:48) (0) , supp( ζ (cid:15) (cid:48) ) ⊂ B (0) . Let Ω = B (0) and for a suitable (cid:15) (cid:48) we define u BN = D ( ζ (cid:15) (cid:48) ( ϕ N , ψ N )).For a general Ω consider disjoint balls B r k ( x k , t k ) such that | Ω \ ∪ k B r k ( x k , t k ) | > − (cid:15) (cid:48) , and finally we define u N ( x, t ) = u BN ( x − x k r k , t − t k r k ) on x ∈ B r k ( x k , t k ) . (cid:3) The construction
Geometric setup.
In this section we identify U with R × R × R .For A ∈ U we will use coordinates ( ρ, w, z ) with ρ ∈ R and w, z ∈ R .To manipulate the Λ cone it will be helpful to introduce the followingnotation. Along this section we denote by S ρ = S ((0 , − ρ , | ρ | . Similarly, B ρ = B ((0 , − ρ , | ρ | . The set K = ( ρ, w, ρw ) ⊂ U defines our pointwise constraint. Thestrategy in Section 2 requires to find X ∈ ( K \ ˜ K ) ∩ int( ˜ K Λ ). HoweverProposition 5.1 shows that this is not possible. Instead we will findstates of the form (0 , , z ) ∈ int( ˜ K Λ ). In the next section it will beshown that this is enough to produce a weak solution.We choose a suitable ˜ K ⊂ K . For that we need to introduce the T Definition 4.1.
We say that C ∈ U is the center of T { T i ( C ) } i =1 ∈ U such thata) C ∈ ( { T i ( C ) } i =1 ) c , b) C − T i ( C ) ∈ Λ.We say that A belongs to the T A ∈ [ C, T i ( C )] forsome i . In this case we also set T i ( A ) = T i ( C ) .T Lemma 4.1.
Let z (cid:54) = (0 , − ) ∈ B . Then there exists δ = δ ( z ) such that every A ∈ U with | A − (0 , , z ) | ≤ δ is thecenter of mass of a T configuration in K . The T i ( A ) are ofthe form (1 , x , x ) , (1 , x , x ) , ( − , y , − y ) , ( − , y , − y ) ∈ K , where ( x , x , y , y )( A ) : R → R are differentiable submersions. Moreover A ∈ int { T i ( A ) } c .Proof. First we observe that by the definition of the cone Λ( ρ, w, z ) − (1 , x, x ) ∈ Λ ⇐⇒ x − w ∈ S − ρ and ( ρ, w, z ) − ( − , y, − y ) ∈ Λ ⇐⇒ y − w ∈ S − − ρ , if in addition we have( ρ, w, z ) ∈ int( { (1 , x , x ) , (1 , x , x ) , ( − , y , − y ) , ( − , y , − y ) } c )the lemma is proved. OROUS MEDIA EQUATION 11
Figure 2.
Convex hull of T i ( A ) . First, we notice that for each ( ρ, w, z ) with | ρ | < x, y : R × R × R → R such that( ρ, w, z ) ∈ [(1 , x, x ) , ( − , y, − y )] . Namely for t = ρ and x ( A ) = x ( ρ, w, z ) = z + w ρ , y ( A ) = y ( ρ, w, z ) = w − z − ρ , it holds that( ρ, w, z ) = ( t (1) + (1 − t )( − , tx + (1 − t )( y ) , tx + (1 − t )( − y )) . Now we consider the balls B x = w + B − ρ , B y = w + B − − ρ withcenters a x , a y .Thus, if δ ≤ δ ( z ) is sufficiently small it follows that(6) x = z + w ρ ∈ B x \ a x , y = z − w − ρ ∈ B y \ a y . In this case we can use the convexity of Euclidean balls to show that x ( A ) ∈ [ x , x ] , x , x ∈ w + S − ρ and similarly for y ( A ). Namely if | x − a | ≤ r then x = 12 (1 − | x − a | r ) ( a − r x − a | x − a | ) (cid:124) (cid:123)(cid:122) (cid:125) x + 12 (1 + | x − a | r ) ( a + r x − a | x − a | ) (cid:124) (cid:123)(cid:122) (cid:125) x . The same argument provides us the mapping y ( A ) , y ( A ) Thus wedeclare T ( A ) = (1 , x , x ) , T ( A ) = (1 , x , x ) ,T ( A ) = ( − , y , − y ) , T ( A ) = ( − , y , y ) . We have shown the existence of { λ i } i =1 such that 0 < λ i < (cid:80) i =1 λ i = 1 and A = (cid:88) i =1 λ i T i ( A ) . Since the T i ( A ) are linearly independent and λ i (cid:54) = 0 for i = 1 , , , A ∈ int( ∪ i T i ( A )) c . We turn to the regularity of the mappings T i ( A ) : U → R . Itis enough to consider the case T . Notice that from the convexityargument follows(7) a x = a ( w, ρ ) = w + (1 − ρ )(0 , −
12 ) and r = 1 − ρ. Thus, both are differentiable functions of A . Since x (cid:54) = a x the same istrue for x , x , y and y . It remains to see that they are submersions.It is enough to argue for the function x . We write x = F ◦ F with F : R → R , F ( r, z, w ) = ( r x , a x , x ) which are defined in (7,6) and F : R → R is given by F ( r, a, x ) = a − r x − a | x − a | . For small δ F is asubmersion and F is always a submersion. (cid:3) Remark 4.1.
It is instructive to realize that by definition if ( ρ, w, z )belongs to K ∩ Λ then z ≤ , , / ∈ int( K ∩ Λ) c . Thisshows that the above strategy never could work with z = 0 (comparewith Proposition 5.1).We will choose a set U adapted to the T Definition 4.2.
Let K ⊂ U . The first Lambda convex hull of K , K , Λ is defined by K , Λ = { A ∈ U : A = tB +(1 − t ) C, B, C ∈ K, B − C ∈ Λ , t ∈ [0 , } . Definition 4.3.
Let z (cid:54) = (0 , − ) ∈ B and the corresponding δ ( z ) fromlemma 4.1. We denote by B z ∈ U the euclidean ball centered in (0 , , z )and with radius δ ( z ) . Furthermore we declare U z = ∪ i =1 { B z ∪ T i ( B z ) } , Λ and K z = ∪ i =1 { T i ( B z ) } ∈ K. Lemma 4.2.
The set U z \ K is open in U . OROUS MEDIA EQUATION 13
Proof.
Let C ∈ U with | C | ≤ (cid:15) and A t ∈ U z \ K . Our task is to findan (cid:15) so that C t = A t + C ∈ U z \ K . By definition A t = tA + (1 − t ) X A ,where A ∈ B z , 0 < t < X A ∈ T i ( B z ) for some i = 1 , , , A − X A ∈ Λ.It will be enough to show that if (cid:15) is small enough, then there exists X ∈ T i ( B z ) such that C t − X ∈ Λ and X + 1 t ( C t − X ) ∈ B z . We use coordinates A = ( ρ, w, z ), X A = (1 , x A , x A ), A t = ( ρ t , w t , z t ) ,C = ( ρ C , w C , z C ). Since X A − A ∈ Λ it holds that X A − A t ∈ Λ as well.Thus, by the definition of Λ, x A − w t ∈ S − ρ t . Therefore there exists ξ with | ξ | = 1 such that X A = w t + (1 − ρ t )[(0 , −
12 ) + ξ ] . Next recall that C t − (1 , x, x ) ∈ Λ if x − w t − w C ∈ S − ( ρ t + ρ C ) . Set x = w t + w C + (1 − ρ t − ρ C )((0 , − ) + ξ ) and notice that | x − x A | ≤ | w C | + 2 | ρ C | ≤ | C | ≤ (cid:15). The function x : A → x ( A ) is a submersion by Lemma 4.1 andthus x ( B z ) is an open subset of R . Hence, by choosing 6 (cid:15) ≤ dist( x A , ∂x ( B z )), we obtain that (1 , x, x ) ∈ T i ( B z ).Now since A = X A + t ( A t − X A ) we have | X + 1 t ( C t − X ) − A | = | X − T i ( A ) | + 1 t | C t − A t | + | X − T i ( A ) |≤ (3 + 4 t ) (cid:15). Then if (cid:15) ≤ t · dist( A, ∂ B z ) it follows that A C = X + t ( C t − X ) ∈ B z and C t = tA C + (1 − t ) X ∈ U z . A final choice 8 (cid:15) ≤ min { t · dist( A, ∂ B z ) , dist( x A , ∂x ( B z )) } yields theclaim. (cid:3) From geometric structure to weak solutions
There are many ways to pass from solutions in U to solutions in K depending on the geometry of the sets. We have followed to present anapproach which in some sense axiomatize the arguments in [6]. Howeverit is by no means the only argument and for example one can verifythat our sets K and U verify the conditions in [10, Proposition 4.42].The argument is easier to explain via the following definition which isrelated to the notion of stability near K given in [10, Definition 3.15] Definition 5.1 (Analytic Perturbation Property, APP) . The sets K ⊂U have the analytic perturbation property if for each A ∈ U and forevery domain Ω ⊂ R x × R there exists a sequence Z j of piecewisesmooth function such thati) L ( Z j ) = 0.ii) Z j is supported in Ω.iii) A + Z j ∈ U a.e.iv) Z j (cid:42) c > (cid:82) Ω | Z j | ≥ c · dist K ( A ) | Ω | . Next we prove that our sets enjoy the analytic perturbation property.The reader that is familiar with convex integration will realize that inpractice our arguments are related to the concept of in approximation[14, 9] for Λ convexity (based on degenerated T Lemma 5.1.
The sets K z and U z have the APP property.Proof. We will obtain v) as a consequence of´v) There exists c , c > |{ ( x, t ) ∈ Ω : | Z j ( x, t ) | ≥ c dist K ( A ) }| ≥ c | Ω | . Let C ∈ B z . By lemma 4.1 is the center of a T { T i ( C ) } i =1 . Now observe that for s small enough C belongs alsoto a T T i,s ( C ) = sC + (1 − s ) T i ( C ) ∈ int K z,δ . Namely, since T i,s ( C ) → T i ( C ) as s → C ∈ int( { T i ( C ) } ) c implies that C ∈ { T i,s ( C ) } c . The other propertiesare obvious.Let T i,s ( C ) be as above. First notice that since | C − T i ( C ) | ≥ dist( C, K z ) by continuity we can choose s such that for s > s itholds that(8) 2 | C − T i,s ( C ) | ≥ dist( A, K z ) . Since C ∈ { T i,s ( C ) } c , there exist λ i > (cid:80) λ i = 1 such that C = (cid:80) i λ i T i,s ( C ). Since the T C (cid:15)i = C + (cid:15) (cid:80) ij =1 ( T i,s ( C ) − C ) and T (cid:15)i ( C ) = T i,s ( C ) + C (cid:15)i it follows that C (cid:15)i +1 = (1 − (cid:15) (cid:15) ) C (cid:15)i + (cid:15) (cid:15) T (cid:15)i +1 ( C ) , with T (cid:15)i +1 ( C ) − C (cid:15)i ∈ Λ, i.e T (cid:15)i is a non degenerate T s for simplicity).Fix a sequence η j such that Π ∞ r =1 (1 − η r ) ≥ and ε = (cid:15) (cid:15) >
0. Wewill obtain the sequence { Z j } ∞ j =0 recursively. We claim that given Z j there exists Z j +1 such thati) |{ ( x, t ) : | Z j +1 ( x ) | ≥ dist K ( C ) |}| ≥ |{ ( x, t ) : | Z j ( x ) | ≥ dist K ( C ) |}| + (1 − ε ) j ε . OROUS MEDIA EQUATION 15 ii) There exists i = i ( j ) such thatΠ jr =1 (1 − η r )(1 − ε ) j ≤ |{ ( x, t ) : C + Z j +1 ( x ) = C (cid:15)i }| . iii) Z j (cid:42) . iv) C + Z j ∈ U z . The proof follows by induction with Z = 0. We start by consideringthe open set Ω j = { ( x, t ) ∈ Ω : C + Z j = C (cid:15)i } . Since C (cid:15)i = (1 − (cid:15) (cid:15) ) C (cid:15)i − + (cid:15) (cid:15) T (cid:15)i ( C ) we apply the Lemma 3.3 in Ω j with η < η j +1 < η to obtain a new sequence { Z j,k } . • Z j,k (cid:42) . • |{ ( x, t ) ∈ Ω j : Z j,k + C (cid:15)i = T (cid:15)i ( C ) }| ≥ ( (cid:15) (cid:15) )(1 − η j +1 ) | Ω j | . • |{ ( x, t ) ∈ Ω j : Z j,k + C (cid:15)i = C (cid:15)i − }| ≥ ( (cid:15) )(1 − η j +1 ) | Ω j | . • sup ( x,t ) ∈ Ω j dist( Z j,k ( x, t ) + C (cid:15)i , [ T (cid:15)i ( C ) , C (cid:15)i − ]) ≤ (1 − η j ) . Now set Z j +1 ,k = Z j + Z j,k and ε = (cid:15) (cid:15) . Let us verify properties i ) and ii ) . Notice that if ( x, t ) ∈ Ω j and Z j,k ( x ) + C (cid:15)i = T (cid:15)i ( C ), then Z j +1 ,k ( x, t ) = T (cid:15)i ( C ) − C and thus by (8), for small (cid:15) , we get | Z j +1 ,k ( x, t ) | ≥
14 dist K ( C ) . For i ) we have |{ ( x, t ) : | Z j +1 ,k ( x, t ) | ≥ dist K ( C ) }| ≥ |{ ( x, t ) ∈ Ω \ Ω j : | Z j ( x, t ) | ≥ dist K ( C ) }| + |{ ( x, t ) ∈ Ω j : Z j,k ( x, t ) + C (cid:15)i = T (cid:15)i ( C ) }|≥ |{ ( x, t ) ∈ Ω \ Ω j : | Z j ( x, t ) | ≥ dist K ( C ) }| + ε (1 − η j +1 ) | Ω j |≥ |{ ( x, t ) ∈ Ω \ Ω j : | Z j ( x, t ) | ≥ dist K ( C ) }| + ε Π j +1 r =1 (1 − η r )(1 − ε ) j | Ω |≥ ε (1 − ε ) j | Ω | . As for ii ), we have |{ ( x, t ) ∈ Ω : Z j +1 ,k ( x, t ) + C = C (cid:15)i − }|≥ |{ ( x, t ) ∈ Ω j : Z j ( x, t ) + Z j,k ( x, t ) + C = C (cid:15)i − | ≥ (1 − ε )(1 − η j +1 ) | Ω j |≥ (1 − ε ) j +1 Π j +1 r =1 (1 − η j +1 ) | Ω | . Next, property iv) follows because the segments [ T (cid:15)i ( C ) , C (cid:15)i − ] arecompact subsets of U z and by a diagonal argument we choose a subse-quence Z j,k ( j ) such that Z j (cid:42) . It just remains to show that Z j satisfies property ´v in the definitionof analytic perturbation property (APP). Indeed by property i) in thedefinition of Z j yields |{ ( x, t ) ∈ Ω : | Z j +1 ( x, t ) | ≥
14 dist K ( C ) }| = | Ω | j (cid:88) k =1 (1 − (cid:15) ) j (cid:15)
12 = | Ω |
12 (1 − (cid:15) j +1 ) ≥ | Ω | for j large enough. Thus for C ∈ B z (APP) holds.Let A ∈ U z \ B z . By definition A = sC + (1 − s ) X C with C ∈ B z , X C ∈ T i ( B z ), 0 < s < C − X C ∈ Λ. Consider the sequence { Z j } given by Lemma 3.3. If s < { Z j } is the sequence required bythe (APP) property. Hence we may assume that s ≥ , obtaining that |{ x ∈ Ω : A + Z j ( x ) = C }| ≥ . Let Ω j = { ( x, t ) ∈ Ω : A + Z j ( x, t ) = C } . Since C ∈ B z we can use the argument above for the domain Ω j .We are provided with a sequence { Z k } such that Z k (cid:42) , Z k + C ∈ U z and |{ ( x, t ) ∈ Ω j : | Z k ( x, t ) | ≥
14 dist K ( C ) }| ≥ | Ω j | ≥ | Ω | . A proper subsequence W j = Z j + Z k ( j ) satisfies all the desired prop-erties in the definition of (APP). As a matter of fact just property ´vneeds verification. Notice that for C ∈ B z and X ∈ T i ( B z ) it holds that1 − δ < | C − X | ≤ δ + max C ∈ B z | T i ( C ) | = M ( δ ) . Thus dist K ( C ) > − δ . Next we notice that(9) dist K ( A ) ≤ | A − X C | ≤ s | C − X C | ≤ M ( δ ) . Let ( x, t ) ∈ Ω j be such that | Z k ( x, t ) | ≥ dist K ( C ). Then(10) | W j ( x, t ) | ≥
14 dist K ( C ) − | C − A |≥
14 (1 − δ ) − (1 − s ) | C − X C |≥
18 (1 − δ ) . Hence if we declare c ( δ ) = − δ M ( δ ) , and we put together (9) and (10)we obtain c ( δ )dist K ( A ) ≤ | W j ( x, t ) | for every ( x, t ) ∈ Ω j with | Z k ( x, t ) | ≥ dist K ( C ). We arrive to theestimate OROUS MEDIA EQUATION 17 |{ ( x, t ) ∈ Ω : | W j ( x, t ) | ≥ c ( δ )dist K ( A ) |} ≥ | Ω | . Finally (v’) holds with c ( δ ) = c and c = . (cid:3) Next we show that if a set has a perturbation property there aremany solutions for our inclusion. We formulate the existence theoremin the following way:
Theorem 5.1.
Let K, U be two sets that satisfy the Analytic Pertur-bation Property and Ω ⊂ T × R a bounded open set and A ∈ U . Thenthere are infinitely many solutions U such that (1) L ( U ) = 0(2) U ( x, t ) ∈ K a.e ( x, t ) ∈ Ω(3) U ( x, t ) = A for ( x, t ) ∈ T × R \ ΩWe shall prove the theorem using Baire category since it yields in-finitely many solutions. In addition we will provide a direct argumentto construct such solutions since it helps to understand its nature.
Definition 5.2 (The space of subsolutions X ) . Let A ∈ U . We saythat U ∈ L ∞ ( T × R ) belongs to X if(I) U is piecewise smooth. (Regularity)(II) For t ≤ t ≥ T U ( x, t ) = A. (Boundary conditions)(III) L ( U ) = 0 . (Conservation law)(IV) U ( x, t ) ∈ U a.e ( x, t ) ∈ T × R . (Relaxed inclusion)We endow X with the L ∞ weak star topology and declare X = X .If a set has the APP property then it holds the perturbation lemma. Lemma 5.2 (Perturbation Lemma) . There exists
C > such that forevery U ∈ X we can find a sequence U k ∈ X such that U k (cid:42) U but (cid:90) T × R | U k − U | ≥ C (cid:90) T × [0 ,T ] dist K ( U ) . Proof.
The sequence U k will differ from U only on the compact set T × [0 , T ]. Since U is piecewise smooth we can find a finite family ofpairwise disjoint balls B j (cid:98) T × [0 , T ] such that(11) (cid:90) T × [0 ,T ] dist K ( U ) ≤ (cid:88) dist K ( U ( x j , t j ) | B j | Next we use that U ( x j , t j ) ∈ U z to apply the APP property in thedomain B j . Let us called the corresponding perturbation sequence by { Z j,k } ∞ k =1 supported in B j . Thus we consider the sequence U k = U + (cid:88) Z j,k , which satisfies that U k ∈ U (a small continuity argument is needed). Then (cid:90) T × R | U k − U | = (cid:88) j (cid:90) B j | Z j,k | ≥ c (cid:88) dist K ( U ( x j , t j )) | B j |≥ c (cid:90) dist K ( U )where the last inequality follows from (11). (cid:3) Next we recall how to obtain a solution U ( x, t ) ∈ K a.e. from theperturbation lemma and the non emptiness of X .The solution U will be the strong limit of a sequence U k which isobtained from the perturbation lemma. Hence strong convergenceimplies that U is in K . The Baire category argument yields theexistence of such U as point of continuity of the Identity map andthen a sequence U k which converge to it. The direct argument buildsthe sequence U k iteratively and then show that indeed it convergesstrongly to some U . Baire Category:
We consider the Identity as a Baire-one map from (
X, weak ∗ ) to( X, L ( T × R )) (see [6, Lemma 4.5]). Lemma 5.3.
Let U ∈ X be a point of continuity of the identity. Then U ∈ K a.e ( x, t ) ∈ Ω × [0 , T ] . Suppose that U ∈ X is a point of continuity of the identity. Bydefinition of X , there exists { U j } ∈ X such that U j (cid:42) U in the weak star topology. As u is a point of continuity of the identityit holds that lim j →∞ (cid:90) T × R | U j − U | = 0and thus, lim j →∞ (cid:90) T × [0 ,T ] dist K ( U j ) dx = (cid:90) T × [0 ,T ] dist K ( U ) . For each U j we use a perturbation lemma. The corresponding se-quence is indexed by U j,k . By a diagonal argument we can choose asequence U j,k ( j ) such that U j,k ( j ) (cid:42) U and since U is a point of continuity yieldslim j →∞ (cid:90) T × R | U j,j ( k ) − U | = 0Now OROUS MEDIA EQUATION 19 (cid:90) T × [0 ,T ] dist K ( U ) = lim j →∞ (cid:90) T × [0 ,T ] dist K ( U j ) ≤ (cid:124)(cid:123)(cid:122)(cid:125) P.Lemma C lim j →∞ (cid:90) T × R t | U j − U j,k ( j ) |≤ C lim j →∞ (cid:90) T × R ( | U j − U | + | U − U j,k ( j ) | ) = 0 . Proof of Theorem 5.1: It is a well known fact that the points ofcontinuity of a Baire one map are a set of second category (see [17].Therefore there are infinitely many U ∈ X such that U ∈ K a.e. and( x, t ) ∈ Ω. This is equivalent to property (II) in Theorem 5.1. Since(I) and (III) hold for any element in X they hold as well for any U ∈ X . Direct Construction:Lemma 5.4.
To each function U ∈ X we can associate a sequence { U j } ∞ j =0 ∈ X such that there exists U ∞ ∈ X satisfying • U ∞ ∈ K a.e. ( x, t ) ∈ T × [0 , T ] • lim j →∞ (cid:82) T × R | U j − U ∞ | = 0 Proof.
We construct directly a function U ∞ in X and a sequence U k ∈ X which converges strongly to U as follows. We start with U in thelemma and obtain with U k +1 from U k by the perturbation lemma. Toeach U k we associate a mollifier η j ( k ) such that(12) (cid:90) T × R | U k − η j ( k ) ∗ U k | ≤ − k . Then we apply the perturbation lemma to the function U to obtaina corresponding sequence U s ∈ X . By the weak convergence propertywe can choose s ( k ) so that for each i < k (13) (cid:90) T × R | U k − U s ( k ) ∗ η j ( i ) ) | ≤ − k and finally we declare U k +1 = U s ( k ) . Properties (12) and (13) yieldstrong convergence. Now (cid:90) T × R dist K ( U ) = lim k →∞ (cid:90) T × R dist K ( U k ) ≤ (cid:124)(cid:123)(cid:122)(cid:125) P.Lemma C lim k →∞ (cid:90) T × R | U k − U k +1 | = 0 (cid:3) Theorem 5.2.
For every
T > there exists infinitely many non trivialweak solutions ( ρ, v ) ∈ L ∞ ( T × [0 , T ]) to the 2D IPM system (4) suchthat ρ ( x,
0) = 0 . Proof.
We fix z (cid:54) = (0 , − ) ∈ B and the corresponding B z , U z and K z given by definition 4.3. By Lemma 5.1 U z and K z enjoy the APPproperty. Thus we can apply Theorem 5.1 to these sets, the domain Ωequals T × (0 , T ) and A equals (0 , , z ).We claim that each of the functions U = ( ρ, v, q ) ∈ X with U ( x, t ) ∈ K z a.e ( x, t ) ∈ T × [0 , T ] from Theorem 5.1. yields a weak solution( ρ, v ) to the IPM system with initial values ρ ( x,
0) = 0. Indeed fromthe fact that L ( ρ, v, q ) = 0 it follows that div( v ) = 0 and that Curl( v +(0 , ρ )) = 0. Thus there exists a function p such that ∇ p = v + (0 , ρ ).The only delicate issue corresponds to the equation of conservation ofmass where we see the nonlinearity. By definition we have that for ψ ∈ C ∞ [ T × R ] 0 = (cid:90) T × R ( ∂ t ψρ + ∇ ψ · q ) . Notice that ϕ ∈ C ∞ c [ T × [0 , T )] can be extended to ˜ ψ ∈ C ∞ [ T × R ].Next we observe that q ( x, t ) = z for a.e. ( x, t ) ∈ T × R \ [0 , T ). Thusfor a.e. t ∈ R \ [0 , T ) it holds that(14) (cid:90) T ∇ ˜ ψ ( x, t ) · q ( x, t ) dx = (cid:90) T ∇ ˜ ψ ( x, t ) · zdx = 0 . Now we plug ˜ ψ in (5). Then (14) together with the fact ρ ( x, t ) = 0 fora.e. ( x, t ) ∈ T × R \ [0 , T ) yields0 = T (cid:90) (cid:90) T ( ∂ t ϕρ + ∇ ϕ · q ) = T (cid:90) (cid:90) T ( ∂ t ϕρ + ∇ ϕ · ρv ) , where in the last equality we have used that for ( x, t ) ∈ T × [0 , T ] , q = ρv . Hence ( ρ, v ) is a weak solution to the IPM system according to (1),(2), (3). (cid:3) Remark 5.1.
The boundary values (or initial data) are attained inthe weak sense. Namely we use the fact (see [7]) that a solution to L ( U ) = 0 can be redefined in a set of times of measure zero in a waythat the map t → (cid:90) T v ( x, t ) ϕ is continuous. Since for negative t (cid:82) T v ( x, t ) ϕ = 0 we obtain the desiredlim t → (cid:90) T v ( x, t ) ϕ = 0and we can argue similarly for the density. From this point ofview it is not surprising that lim t → ρ ( x, t ) = 0 = lim t → v ( x, t ) butlim t → ρ ( x, t ) v ( x, t ) = z (cid:54) = 0 since they are weak limits and weak limitsdo not commute with products. In fact from this we deduce that oursolutions do not attain the boundary values in the strong sense and OROUS MEDIA EQUATION 21 hence by the Frechet Kolmogorov theorem they are highly irregular.Another additional feature of the weak solution is that | ρ ( x, t ) | = 1 fora.e. ( x, t ) ∈ T × (0 , T ) which prevents the pointwise convergence tothe initial data.We conclude with a proposition pointing out to the fact that thismethod does not seem to yield a solution compact in space and time.For that we need some more standard terminology. We say that afunction f : U → R is Λ convex if for A, B, C ∈ U such that A − B ∈ Λthe function g ( t ) : f ( C + t ( A − B )) is convex. Let K ⊂ U . Then theLambda convex hull of K , K λ K Λ = { A ∈ U : f ( A ) ≤ sup f ( C ) , C ∈ K, f
Λ convex } . In all the methods to solve differential inclusions the canonical choicefor the set U is the interior of K Λ . However in our case the set K cannot be there. Proposition 5.1.
The set K ⊂ ∂K Λ Proof.
The proof relies on the fact that the function f ( A ) = z − ρv is Λ convex. In fact it is easy to see that det Λ is Λ linear, thus − det Λ is Λ convex. Hence | v | − det Λ = − ρv is Λ convex and f is Λ convex. Moreover f ( A ) = 0 for all A ∈ K andhence K Λ ⊂ f − ( −∞ ,
0] and K ⊂ ∂f − ( −∞ , ∩ K Λ ⊂ ∂K Λ . (cid:3) Remark 5.2.
The above argument also shows that if 0 ∈ K Λ (orany other element in K ) the elements of the splitting sequence in thelaminate [10] should belong to f − (0). Acknowledgments.
DC and FG were partially supported by
MTM2008-03754 project of the MCINN (Spain) and StG-203138CDSIF grant of the ERC. DF was partially supported byMTM2008-02568 project of the MCINN (Spain). FG was partiallysupported by NSF-DMS grant 0901810. The authors would like tothank L. Sz´ekelyhidi Jr for helpful conversations.
References [1]
Bear, J.
Dynamics of Fluids in Porous Media . American Elsevier, Boston,MA, 1972.[2]
Constantin, P., E, W., and Titi, E. S.
Onsager’s conjecture on the en-ergy conservation for solutions of Euler’s equation.
Comm. Math. Phys. 165 ,1 (1994), 207–209.[3]
C´ordoba, D., and Gancedo, F.
Contour dynamics of incompressible 3-Dfluids in a porous medium with different densities.
Comm. Math. Phys. 273 , 2(2007), 445–471. [4]
C´ordoba, D., Gancedo, F., and Orive, R.
Analytical behavior of two-dimensional incompressible flow in porous media.
J. Math. Phys. 48 , 6 (2007),065206, 19.[5]
Dacorogna, B., and Marcellini, P.
General existence theorems forHamilton-Jacobi equations in the scalar and vectorial cases.
Acta Math. 178 (1997), 1–37.[6]
De Lellis, C., and Sz´ekelyhidi, Jr., L.
The Euler equation as a differentialinclussion.
Ann. Math 170 , 3 (2009), 1417–1436.[7]
De Lellis, C., and Sz´ekelyhidi, Jr., L.
On admissibility criteria for weaksolutions of the Euler equations.
Arch. Rational Mech. Anal. 195 , 1 (2010),225–260.[8]
Gancedo, F.
Existence for the α -patch model and the QG sharp front inSobolev spaces. Adv. Math. 217 , 6 (2008), 2569–2598.[9]
Gromov, M.
Partial differential relations , vol. 9 of
Ergebnisse der Mathematikund ihrer Grenzgebiete (3) . Springer-Verlag, Berlin, 1986.[10]
Kirchheim, B.
Rigidity and Geometry of microstructures. Habilitation thesis,University of Leipzig, 2003.[11]
Kirchheim, B., M¨uller, S., and ˇSver´ak, V.
Studying nonlinear PDEby geometry in matrix space. In
Geometric analysis and Nonlinear partialdifferential equations , S. Hildebrandt and H. Karcher, Eds. Springer-Verlag,2003, pp. 347–395.[12]
Kirchheim, B., and Preiss, D.
Construction of Lipschitz mappings havingfinitely many gradients without rank-one connections. in preparation.[13]
Majda, A. J., and Bertozzi, A. L.
Vorticity and incompressible flow ,vol. 27 of
Cambridge Texts in Applied Mathematics . Cambridge UniversityPress, Cambridge, 2002.[14]
M¨uller, S., and ˇSver´ak, V.
Convex integration for Lipschitz mappings andcounterexamples to regularity.
Ann. of Math. (2) 157 , 3 (2003), 715–742.[15]
M¨uller, S., and Sychev, M. A.
Optimal existence theorems for nonhomo-geneous differential inclusions.
J. Funct. Anal. 181 (2001), 447–475.[16]
Onsager, L.
Statistical hydrodynamics.
Nuovo Cimento (9) 6 , Supplemento,2(Convegno Internazionale di Meccanica Statistica) (1949), 279–287.[17]
Oxtoby, J. C.
Measure and category , second ed., vol. 2 of
Graduate Texts inMathematics . Springer-Verlag, New York, 1980.[18]
Pedregal, P.
Laminates and microstructure.
European J. Appl. Math. 4 (1993), 121–149.[19]
Pedregal, P.
Parametrized measures and variational principles , vol. 30of
Progress in Nonlinear Differential Equations and their Applications .Birkh¨auser Verlag, Basel, 1997.[20]
Resnick, S.
Dynamical problems in nonlinear advective partial differentialequations. Dissertation, University of Chicago, 1995.[21]
Rodrigo, J. L.
On the evolution of sharp fronts for the quasi-geostrophicequation.
Comm. Pure Appl. Math. 58 , 6 (2005), 821–866.[22]
Scheffer, V.
An inviscid flow with compact support in space-time.
J. Geom.Anal. 3 , 4 (1993), 343–401.[23]
Shnirelman, A.
On the nonuniqueness of weak solution of the Euler equation.
Comm. Pure Appl. Math. 50 , 12 (1997), 1261–1286.[24]
Sz´ekelyhidi, Jr., L.
The regularity of critical points of polyconvex function-als.
Arch. Ration. Mech. Anal. 172 , 1 (2004), 133–152.[25]
Tartar, L.
Compensated compactness and applications to partial differentialequations. In
Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol.IV , vol. 39 of
Res. Notes in Math.
Pitman, Boston, Mass., 1979, pp. 136–212.
OROUS MEDIA EQUATION 23 [26]
Tartar, L.
The compensated compactness method applied to systems of con-servation laws. In
Systems of nonlinear partial differential equations (Oxford,1982) , vol. 111 of
NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.
Reidel, Dor-drecht, 1983, pp. 263–285.[27]
Wu, J.
The quasi-geostrophic equation and its two regularizations.
Comm.Partial Differential Equations 27 , 5-6 (2002), 1161–1181.
Diego C´ordoba, Instituto de Ciencias Matem´aticas, Consejo Supe-rior de Investigaciones Cientificas, Calle Serrano 123, 28006 Madrid,Spain.
E-mail address : [email protected] Daniel Faraco, Instituto de Ciencias Matem´aticas CSIC-UAM-UCM-UC3M and Department of Mathematics, Universidad Aut´onoma deMadrid, 28049 Madrid, Spain.
E-mail address : [email protected] Francisco Gancedo, Department of Mathematics, University ofChicago, 5734 University Avenue, Chicago, IL 60637, U.S.A.
E-mail address ::