ANALYTICAL BEHAVIOR OF 2-D INCOMPRESSIBLE FLOW IN POROUS MEDIA

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1 ANALYTICAL BEHAVIOR OF -D INCOMPRESSIBLE FLOW IN POROUS MEDIA DIEGO CÓRDOBA, FRANCISCO GANCEDO AND RAFAEL ORIVE Abstract In this paper we study the analytic structure of a two-dimensional mass balance equation of an incompressible fluid in a porous medium given by Darcy s law We obtain local existence and uniqueness by the particle-trajectory method and we present different global existence criterions These analytical results with numerical simulations are used to indicate nonformation of singularities Moreover, blow-up results are shown in a class of solutions with infinite energy 1 Introduction The dynamics of a fluid through a porous medium is a complex and not thoroughly understood phenomenon [3, 16] The purpose of the paper is to study a nonlinear two-dimensional mass balance equation in porous media and the conditions of formation of singularities using analytical results and numerical calculations In real applications, one might be interested in the transport of a dissolved contaminant in porous media where the contaminant is convected with the subsurface water [17] For example, one is usually interested in the time taken by the pollutant to reach the water table below Such flows also occur in artificial recharge wells where water and or) chemicals from the surface are transported into aquifers In this case, the chemicals may be the nutrients required for degradation of harmful polluting hydrocarbons resident in the aquifer after a spillage We use Darcy s law to model the flow velocities, yielding the following relationship between the liquid discharge flux per unit area) v = v 1, v ) and the pressure gradient v = k p + gγρ), where k is the matrix position-independent medium permeabilities in the different directions respectively divided by the viscosity, ρ is the liquid density, g is the acceleration due to gravity and the vector γ =, 1) While the Navier-Stokes equation and the Stokes equation are both microscopic equations, Darcy s law gives the macroscopic description of a flow in a porous medium [3] The free boundary problem given by an incompressible -D fluid through porous media with two different constant densities and viscosities at each side of the interface is studied in [18, 1] see references therein) Here, we analyze the dynamics of the density function ρ = ρx 1, x, t) with a regular initial data ρ = ρx 1, x, ) The mass balance equation is given by φ Dρ Dt = φ ) ρ t + v ρ =, where φ denotes the porosity of the medium To simplify the notation, we consider k = g = φ = 1 Thus, our system of a two-dimensional mass balance equation in porous media DPM) is written as 11) Dρ Dt =, 1) v = p + γρ) We close the system assuming incompressibility, ie, 13) divv =, Date: May 31, 6 Key words and phrases Flows in porous media Mathematics Subject Classification 76S5, 76B3, 65N6 The authors was partially supported by the grant MTM5-598 of the MEC Spain) and S-55/ESP/158 of the CAM Spain) The third author was partially supported by the grant MTM5-714 of the MEC Spain) 1

2 D CÓRDOBA, F GANCEDO AND R ORIVE therefore there exists a stream function ψx, t) such that 14) v = ψ ψ, ψ ) x x 1 Computing the curl of equation 1), we get the Poisson equation for ψ 15) ψ = ρ x 1 A solution of this equation is given by the convolution of the Newtonian potential with the function x1 ρ 16) ψx, t) = 1 ln x y ρ y, t)dy, x R π R y 1 Thus, the velocity v can be recovered from ψ by the operator by the two equivalent formulas 17) vx, t) = Kx y) ρy, t)dy, x R, R 18) vx, t) = PV Hx y) ρy, t)dy 1 R, ρx)), x R, where the kernels K ) and H ) are defined by 19) Kx) = 1 π x 1 x and Hx) = 1 π x 1x x 4, x 1 x ) x 4 Differentiating the equation 11), we obtain the evolution equation for ρ ρ, ρ ) 11) x x 1 which is given by 111) D ρ Dt Taking the divergence of the equation 1), we get = v) ρ p = ρ x and the pressure can be obtained as in 16) The objective of this work is to analyze the behavior of the solutions of the system DPM 11)- 13) First, we present the existence of singularities in a class of solutions with infinite energy in DPM see Proposition and Remark 3) In the case of solutions with regular initial data and finite energy, we get local well-posedness using the classical particle trajectories method We illustrate a criterion of global existence solutions via the norm of the bounded mean oscillation space of 11) A similar result is known in the three dimensional Euler equation 3D Euler) [] Also, using the geometric structure of the level sets of the density where ρ is constant) and the nonlinear evolution equations of the gradient of the arc length of the level sets, we establish that no singularities are possible under not very restrictive conditions This result is comparable to the 3D Euler equations [7] and to the two-dimensional quasi geostrophic equation DQG) [8] Applying these criterions, we find no evidence of formation of singularities in our numerical simulations The paper is organized as follows In Section we study the analytical behavior of solutions with infinite energy In Section 3 we prove the existence and uniqueness for the DPM, show a characterization of formation of singular solutions and we present geometric constraints on singular solutions Finally, in Section 4 we illustrate two numerical examples in which the analytical results are applied to show nonsingular solutions

3 ANALYTICAL BEHAVIOR OF DPM 3 Let the stream function ψ be defined by Singularities with infinite energy 1) ψx 1, x, t) = x fx 1, t) + gx 1, t) Note that under this hypothesis the solution of 11) 13) has infinite energy We reduce the equations 11) 13) to other system with respect to the functions f and g From 15) the density, apart from a constant, satisfies ) ρt, x 1, x ) = x f x 1 x 1, t) g x 1 x 1, t) = x f x1 g x1, and, by 14), v verifies 3) vt, x 1, x ) = f fx 1, t), x x 1, t) + g ) x 1, t) = f, x f x1 + g x1 ) x 1 x 1 Therefore, the system 11) 13) under the hypothesis 1) is equivalent to 4) 5) f x ) t = ff xx f x ), g x ) t = fg xx f x g x Here and in the sequel of the section, we denote with subscript the derivatives with respect to x) We note the non-linear character of the first equation Thus, our study of formation of singularities is concentrated in the solutions of 4) The function g depends implicitly on f in equation 5) Now, we show that the system 4) and 5) is local well posed in the Sobolev spaces H k, 1) Lemma 1 Let f = fx, ) and g = gx, ) satisfy f x, g x H k, 1) with k 1 Then, there exists T > such that f x, g x C 1 [, T ]; H k, 1)) are the unique solution of 4) 5) Proof By 4) and integrating by parts, we have Analogously, dt f x L = dt f xx L = f x ff xx f xxf x We can repeat for all k 1 and we obtain Integrating in time, we get and f 3 x = 3 f xx ff xxx = 1 dt f x H k f x H k fx 3 C f x L f x L C f x 3 H 1 C f x 3 H k f x H k 1 Ct f x H k fxxf x C f x L f xx L C f x 3 H 1 On the other hand, by 4) and integrating by parts, we have for g x the following inequalities dt g x L = dt g xx L = g x g xx f g xx g xxx f Thus, we obtain using Gronwall s Lemma g xf x = 3 g xx g x f xx = 1 g xf x f x L g x H 1 g xxf x g xx g x f xx f x L g xx L + g xx L g x L f xx L f x H 1 g x H 1 g x H k g x H k exp C and we have existence up to a time T = T f x H k ) t ) f x H k ds

4 4 D CÓRDOBA, F GANCEDO AND R ORIVE In order to prove the uniqueness, let f x x, t) = h x x, t) k x x, t), with h x, k x two solutions of 4) with the same initial data fx Since h x, k x satisfy 4) and integrating by parts, we have Thus, we get dt f x L = = = 1 f x hh xx kk xx ) f x hf xx + f x ) h x + f x fk xx f x h x k x) f x fk xx f x ) h x + k x ) f x ) h x + k x ) dt f x L k xx L f x L f L + C h x L + k x L ) f x L C h x H 1 + k x H 1 ) f x L and using Gronwall s Lemma it follows h x = k x Finally, we conclude the uniqueness of g x since 5) is a linear differential equation The following result shows that the solution of 4) blows up in finite time under certain conditions on the initial data Proposition Let f x be a solution of 4) with initial data satisfies f x H, 1) and min x f x < Then, f x L blows up in finite time T = 1/ min x f x Proof By the local existence result, we have f x C 1 [, T ]; H ) C 1 [, T ] [, 1]) We consider the application m : [, T ] R defined by mt) = min x f x x, t) = f x x t, t) By Rademacher Theorem, it follows that m is differentiable at almost every point First, we calculate the derivative of m as in [5, 9] Let s be a point of differentiability of mt), then for τ > m ms + τ) ms) f x x s+τ, s + τ) f x x s, s) s) = lim = lim τ τ τ τ f x x s+τ, s + τ) f x x s, s + τ) = lim + f xx s, s + τ) f x x s, s) τ τ τ Since f x x, s + τ) reaches the minimum at the point x s+τ, we obtain m f x x s, s + τ) f x x s, s) s) lim = f xt x s, s) τ τ We compute the derivative with a sequence of negative τ < and, by the sign of τ, we get the opposite inequality and we conclude that We replace x for x s in 4) and yields m s) = f xt x s, s) due to f xx x s, s) =, and the proof follows almost everywhere m s) = f xx s, s) = ms)), Remark 3 There are other blow-up results with an initial data of lower regularity In particular, we consider f x H 1 and assuming that Thus, by 4), we have f x Defining d dt f x = ff xx f x ) = ct) = f x, ) f x ) f x

5 ANALYTICAL BEHAVIOR OF DPM 5 and integrating, we get ct) c) 1 + tc) Then, ct) blows up for c) < In the case c) =, we have c t) < for all t >, therefore, ct) also blows up Remark 4 Let x 1 = x t be the point such that and consider f x x t, t) = min f x x, t), x x = 1 g xx t, t) f x x t, t) Then, by ), ρx 1, x, t) = f x x t, t) blows up in finite time by Proposition Analogously, v defined in 3) blows up in finite time 3 Analysis of DPM with finite energy 31 Local existence of DPM We derive a reformulation of the system as an integro-differential equation for the particle trajectories Given a smooth field vx, t), the particle trajectories Φα, t) satisfy dφ 31) dt α, t) = vφα, t), t), Φα, t) t= = α The time-dependent map Φ, t) connects the Lagrangian reference frame with the variable α) to the Eulerian reference frame with the variable x) It is well known Section 5 in [15]) that the equation 111) implies the following formula ρφα, t), t) = α Φα, t) ρ α), where ρ is the orthogonal gradient of the initial density This last equality shows us that the orthogonal gradient of the density is stretched by α Φα, t) along particle trajectories We rewrite 17) as 3) vφα, t), t) = KΦα, t) Φβ, t)) α Φβ, t) ρ β)dβ R We consider 31) as an ODE on a Banach space and using Picard Theorem the local in time existence follows This is proved analogously like the existence and uniqueness of solutions to the inviscid Euler equation see Section 41 in [15]) In fact, we consider ρ C δ R ), δ, 1) Let B be the Banach space defined by B = { Φ : R R such that Φ) + α Φ + α Φ δ < }, where is the L -norm and δ is the Hölder semi-norm Define O M, the open set of B, as { O M = Φ B inf det αφα) > 1 } α R and Φ) + αφ + α Φ δ < M The mapping vφ), defined by 3), satisfies the assumptions of the Picard theorem, ie, v is bounded and locally Lipschitz continuous on O M As a consequence, for any M > there exists T M) > and a unique solution Φ C 1 T M), T M)); O M ) to the particle trajectories 31, 3) Remark 31 The DPM 11) 13) has quantities conserved in time, the L p norm of ρ for 1 p, ie, 33) ρt) p = ρ p, t >, 1 p The velocity is obtained from ρ by 18) These operators are singular integrals with Calderón-Zygmund kernels see [19]) Then for 1 < p < the L p norm of the velocity is bounded for any time t >

6 6 D CÓRDOBA, F GANCEDO AND R ORIVE 3 Blow-up criterion In order to estimate the growth of the Sobolev norms we use the space of functions of bounded mean oscillation BMO) see Chapter IV in [19] for an introduction of this function space) Theorem 3 Let ρ be the solution of equation 11) 13) with initial data ρ H s R ) with s > Then, the following are equivalent: A) The interval [, ) is the maximal interval of H s existence for ρ B) The quantity 34) T ρ BMO t) dt < T > Proof We denote the operator Λ s by Λ s ) s/ Since the fluid is incompressible, we have for s > dt Λs ρ L = Λ s ρλ s v ρ) dx = Λ s ρλ s v ρ) vλ s ρ)) dx, R R C Λ s ρ L Λ s v ρ) vλ s ρ) L Using the following estimate see [13]) Λ s fg) fλ s g) L p c f L Λ s 1 g L p + Λ s ) f L p g L we obtain for p = 35) dt Λs ρ L C v L + ρ L ) Λs ρ L Integrating, we get for any t T 36) Λ s ρ L Λ s ρ L exp C T v L + ρ L ) ) 1 < p <, Now, we use the following inequality given in [14]: Let f W s,p with 1 < p < and s > /p, then, there exists a constant C=Cp,s) such that 37) f L C1 + f BMO 1 + ln + f W s,p)), where ln + x) = max, lnx)) Therefore, for s > we have and from 36), we obtain that ρ L C1 + ρ BMO 1 + ln + ρ H s 1)), T 38) ρ L C1 + ln + ρ H s) ρ BMO v L + ρ L ) dt) On the other hand, applying 37) for v H s R ), we have v L C1 + v BMO 1 + ln + v H s 1)) Since v satisfies 18) and the singular integrals are bounded operators in BMO see [19]), we get and, using 36), we obtain v L C1 + ρ BMO 1 + ln + ρ H s 1)), T 39) v L C1 + ln + ρ H s) ρ BMO v L + ρ L ) dt) From 38) and 39), follows T v L + ρ L C1 + ln + ρ H s) ρ BMO v L + ρ L ) dt) Applying Gronwall s inequality, we have T v L + ρ L ) dt CT exp and so A) is a consequence of B) ln + ρ H s) T ρ BMO dt ),

7 ANALYTICAL BEHAVIOR OF DPM 7 Finally, due to the inequality we conclude that A) implies B) Remark 33 Using that ρ BMO ρ H 1, ρ BMO C ρ L, we get a blow-up characterization for numerical simulations 33 Geometric constraints on singular solutions From equation 11) it follows that the level sets, ρ = constant, move with the fluid flow Then ρ, defined in 11), is tangent to these level sets For the DPM, the infinitesimal length of a level set for ρ is given by ρ and from 111), the evolution equation for the infinitesimal arc length is given by 31) The factor Lx, t) is defined through by D ρ Dt 311) Lx, t) = where the direction of ρ is denoted by = L ρ { Dη η, η,, η = 31) η = ρ ρ and Dx, t) is the symmetric part of the deformation matrix defined by [ 1 vi 313) D = D ij ) = + v )] j x j x i Now, we show a singularity criterion of DPM using the geometric structure of the level sets and mild hypotheses of the solutions The theorem states below is analogous to 3D Euler [7] and to DQG [8] Recall that η is the direction field tangent to the level sets of ρ defined 31) Analogously to [8], a set Ω is smoothly directed if there exists δ > such that 314) sup x Ω where T η, t) L B δ Φx,t))) dt <, B δ x) = {y R : x y < δ}, Ω = {x Ω; ρ x) = }, and Φ is the particle trajectories map We define Ωt) = ΦΩ, t) and O T Ω) the semi-orbit, ie, O T Ω) = {t} Ωt) Theorem 34 If Ω is smoothly directed and 315) T t T R j ρ L t)dt <, j = 1,, T >, where R j denotes the Riesz transform in the direction x j, then sup ρx, t) < O T Ω) Remark 35 Using the Remark 33, the previous theorem illustrates that finite-time singularities are impossible in smoothly directed sets Proof We show a similar formula of the level set stretching factor L defined in 311) We start by computing the full gradient of the velocity v From formula 17) vx) = Ky) ρx y)dy, R

8 8 D CÓRDOBA, F GANCEDO AND R ORIVE we have vx) = Ky) y y ρ)x y)dy R Take the integral as a limit as ε of integrals on y > ε and integrate by parts In this fashion, we obtain the formula 316) vx) = 1 π PV y ρx y) ) ỹ dy R y 1, where ỹ is the unit vector defined by ỹ = y 1y y, y 1 y ) y ρ x 1 x) ρ x x) By definition of η in 31), we have η ρ = Thus, computing we get 317) Lx) = 1 π PV ỹ ) ηx) ηx y) η x) ) } ρx y) dy R y Let be φ Cc R), φ, suppφ) include in [ 1, 1] and φs) = 1 in s [ 1/, 1/] Consider r > and decompose Lx) = I 1 + I, with I 1 1 π 1 φ y /r )) ỹ ηx) ) ρx y) η x) ) dy R y Integrating by parts and using Cauchy-Schwartz inequality, we get We have for any y < r I 1 C r ρ L C r ρ L ηx y) η x) y η L B rx)) Applying this in the integral I, we get I ρx y) φ y /r ) dy y η L B rx)) We integrate by parts and decompose ρx y) φ y /r ) dy y = where J 1 = J = J 3 = obtaining the following estimates ρx y) ηx y)φ y /r ) 1 ) dy = J 1 + J + J 3, y ρx y)ηx y) φ y /r )) dy y, ρx y) ηx y))φ y /r ) dy y, ρx y)ηx y)φ y /r ) y, y 1 ) y 3 dy, J 1 c ρ L and J cr ρ L η L B rx)) The J 3 term can be bounded using the identity J 3 = ηx) R ρ)x), R 1 ρ)x)) + J 4, getting the following estimate for J 4 in a similar way J 4 r ρ L η L B rx)) + r 1 ρ L Thus, we conclude the following estimate for the factor L: [ ] Lx) c η L B rx)) max R jρ) + r η L B j=1, rx)) + 1) η L B rx)) ρ + r ρ )

9 ANALYTICAL BEHAVIOR OF DPM 9 Using 31), we obtain by Gronwall s lemma sup O T Ω) ρx, t) sup Ω ρ exp sup y Ω T Mt)dt ), where Mt) is defined by [ ] Mt) = c η L B rx)) max R jρ) + r η L B j=1, rx)) + 1) η L B rx)) ρ + r ρ ), with x = Φy, t) This concludes the proof of Theorem 34 Remark 36 The condition 315) depending on the Riesz transform is different that in DQG see [8]) This appears because the integral kernels 19) in DPM are different to the Kernels in DQG Now, we present a geometric conserved quantity that relates the curvature of the level sets and the magnitude ρ in a similar way as in [6] see references therein for more details) In particular, if we define the curvature of the level sets κ by 318) κx, t) = η η) η x, t), where η is the direction of ρ see 31)), the following identity is satisfied 319) with Dκ ρ ) Dt = ρ β 3) βx, t) = η v) η x, t) Indeed, we now prove the identity 319) Since ρ and ρ satisfies 111) and 31) respectively, we get Dη = v)η L η Dt Using 316), we obtain Dη Dt = β η, with β defined in 3) By the definition of κ 318) and the previous formula, we have Dκ Dt = v η L η) η) η + η β η + η β η v)) η βη η) η and, simplifying, Dκ = β)η L κ Dt Using this identity and 31), 319) is satisfied Remark 37 The integral of the quantity κ ρ over a region given by two different level sets is conserved along the time, ie d ) 31) κ ρ dx = dt {x: C 1 ρx,t) C } This can be showed using the equation 319) and integrating by parts Thus, in the case that ρ is large by 31) the curvature κ is small if the level sets do not oscillate In all of our numerical experiments, we find no evidence of level set oscillations On the contrary, we observe that the level sets are flattering where the gradient of ρ is growing 4 Numerical simulations Here we present two examples of numerical simulations for solutions of the -D mass balance in porous media with initial data in a period-cell [, π] Although periodic boundary conditions are rather unphysical, which does not matter because we are interested in the small-scale structures

10 1 D CÓRDOBA, F GANCEDO AND R ORIVE Figure 1 Evolution of the density in the Case 1 for times t =, 3, 6, 85 The numerical method We solve the equations 11) 13) numerically on a π-periodic cell with a spectral method with smoothing This numerical method is similar to the scheme developed by E and Shu [1] for incompressible flows and also used for the quasi-geostrophic active scalar in [8] This algorithm is the standard Fourier-collocation method see [4]) with smoothing Roughly speaking, the differentiation operator is approximated in the Fourier space, while the nonlinear operations such as v ρ are done in the physical space We smooth the gradients adding filters to the spectral method in order for the numerical solutions do not degrade catastrophically A way of adding the filters is to replace the Fourier multiplier ik j by ik j ϕ k j ), where ϕk) = e ak/n)b, for k N Here N is the numerical cutoff for the Fourier modes and a, b is chosen with the machine accuracy see []) For the temporal discretization, we use Runge-Kutta methods of various order In our case, we have no explicit temporal dependence and we get a Runge-Kutta method of order 4 that requires at most three levels of storage, see in [4] p 19 We present the numerical approximation with an initial resolution of 56) Fourier modes We refine this resolution when the growth of ρ L is substantial to give additional insight preserving the relation space-time We conclude our numerical simulations with a resolution of 819) Fourier modes The computational part of this work was performed on HPC3 cluster of 8 SMP servers with 3 processors Alpha EV68 1 GHz) of the Centro de Supercomputación de Galicia CESGA) We used the MATLAB routines to obtain the calculations

11 ANALYTICAL BEHAVIOR OF DPM 11 Figure Evolution of the level sets -999, -99 on the left of the figure) and 99, 999 on the right) of the density in the Case 1 for times t =, 3, 6 Figure 3 Evolution of the L -norms of ρ, the velocity v and the Riesz transforms R 1 ρ, R ρ) for the Case 1 Case 1 We consider the initial datum ρx 1, x ) = sinx 1 ) sinx ) The time step is t = 5 from t = to t = 4 with a = 45 and b = 3, stopping the experiment with t = 15 and a = 91, b = 71 During the simulation the ratio h/ t is preserved getting a finer resolution as the gradients are growing In this way the method conserves in time the L -norm of the density Figure 1 presents the density at times t =, 3, 6, 85 with a numerical resolution of 56), 56), 14), 496), respectively The initial data has a hyperbolic saddle point that does not present a nonlinear behavior as time evolves In all our numerical simulations we do not find any saddle point structures that present stronger front formation than in case see Figure 5) Nevertheless, the case 1

12 1 D CÓRDOBA, F GANCEDO AND R ORIVE Figure 4 Evolution of the density in Case for times t =, 3, 6, 9 shows a front formation in the elliptic plots and approaching the maximum with the minimum see Figure ) as time evolves If we choose two level sets that are approaching each other we have the following scenario where δ = δx, t) is the distance between the two counters From previous work see [1]) it is easy to check that in order for the two graphs f l, f r to collapse at time T in any interval x [a, b], ie, it is necessary that 41) lim [f r x, t) f l x, t)] = x [a, b] t T T v s)ds =

13 ANALYTICAL BEHAVIOR OF DPM 13 Figure 5 Evolution of around contour ρx 1, x ) = 1 in Case for times t =, 15, 3, 45, 6, 75 In figure 3 we see no evidence for quantity T v s)ds to blow up in finite time Nevertheless, we can obtain an estimate of how close the two graphs approach each other without any assumption on the velocity From figure it seems reasonable to assume that the minimum and maximum of δ are comparable that means there exists and constant c > such that min δx, t) c max δx, t) x [a, b] Then we can obtain an evolution equation for the area A = At) At) = 1 b a b a [f r x, t) f l x, t)]dx in between the two graphs that satisfies see [11] for more details) 4) da dt t) C b a sup ψf r x, t), x, t) ψf l x, t), x,, t) a x b

14 14 D CÓRDOBA, F GANCEDO AND R ORIVE Figure 6 Level contour 98,99,1,11,1) of ρ in Case for times t = 75, 8, 85, 9 where ψ is the stream function 16) ψx, t) = 1 π Using this formula we obtain that for any x, y R x 1 z 1 R x z ρz, t)dz, x R 43) ψx, t) ψy, t) C ρ, ρ L ) x y 1 ln x y ), due to the following estimates ψx, t) ψy, t) = 1 x 1 z 1 π R x z y 1 z 1 )ρz, t)dz y z π π π where r = x y and I 1 C ρ x y, I C ρ x y I 3 C ρ L x y B rx) = I 1 + I + I 3 r B x) B rx) B x) s s C ρ x y ln x y ), Then, using 43) in 4) we get that the area At) is bounded by At) A e Cet

15 ANALYTICAL BEHAVIOR OF DPM 15 Figure 7 Evolution of L -norms of ρ, the velocity v and the Riesz transforms R 1 ρ, R ρ) for Case In order to apply the global solution criterion Theorem 3), in Figure 3 is plotted the logarithm of the L -norms of 1 ρ and ρ showing an exponential growth The hypothesis 315) of the Theorem 34) is checked in this example see Figure 3), so that using this result no singularity is possible due to the variation of the direction field tangent to the level sets is smooth Case In this case, the initial datum is ρx 1, x ) = sinx 1 ) cosx ) + cosx 1 ) The time step is t = 5 from t = to t = 45 with a = 45 and b = 3, stopping the experiment with t = 1 and a = 114, b = 8 The L -norm is preserved during the simulation of this case Figure 4 presents the density at times t =, 3, 6, 9 with a numerical resolution of 56), 56), 14), 819), respectively In Figure 7, we show log plots of max x1 ρ and max x ρ, where we obtain an analogous exponential growth as in the case 1 This growth is not sufficient to guarantee a singularity The initial data for the density scalar clearly has a hyperbolic saddle and the numerical solution develops a front as time evolves see Figures 5 and 6) We observe that the front does not develop nonlinear or potentially singular structure as time evolves Where η is smoothly directed we observe the highest growth of ρ L Where η changes rapidly we obtain less growth of ρ L Using theorem 34 we show no evidence of singularities References [1] D Ambrose, Well-posedness of Two-phase Hele-Shaw Flow without Surface Tension, Euro Jnl of Applied Mathematics 15 4), [] J T Beale, T Kato and A Majda, Remarks on the Breakdown of Smooth Solutions for the 3-D Euler Equations, Commun Math Phys ), [3] J Bear, Dynamics of Fluids in Porous Media, American Elsevier, New York, 197 [4] C Canuto, M Y Hussaini, A Quarteroni and T A Zang, Spectral methods in fluid dynamics, Springer Series in Computational Physics Springer-Verlag, New York, 1988 [5] A Constantin and J Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math ), no, 9 43 [6] P Constantin, Geometric statistics in turbulence, SIAM Rev ), [7] P Constantin, C Fefferman and A Majda, Geometric constraints on potential singularity formulation in the 3-D Euler equations, Commun Partial Diff Eq ), [8] P Constantin, A Majda and E Tabak, Formation of strong fronts in the -D quasigeostrophic thermal active scalar, Nonlinearity ), no 6, [9] A Córdoba and D Córdoba, A maximum principle applied to Quasi-geostrophic equations, Comm Math Phys 49 4), no 3, [1] D Córdoba and C Fefferman, Scalars convected by a two-dimensional incompressible flow, Comm Pure Appl Math 55 ), no, 55 6

16 16 D CÓRDOBA, F GANCEDO AND R ORIVE [11] D Córdoba and C Fefferman, Growth of solutions for QG and D Euler equations J Amer Math Soc 15 ), no 3, [1] W E and C-W Shu, Small-scale structures in Boussinesq convection, Phys Fluids ), no 1, [13] T Kato and Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Communications on Pure and Applied Mathematics ), [14] H Kozono and Y Taniuchi, Limiting case of the Sobolev inequality in BMO, with application to the Euler equations, Communications in Mathematical Physics 14 ), 191 [15] A J Majda and A L Bertozzi, Vorticity and incompressible flow, Cambridge Univ Press, Cambridge, [16] DA Nield and A Bejan, Convection in porous media, Springer-Verlag, New York, 1999 [17] M Price, Introducing Groundwater, nd ed Chapman & Hall, London, 1996 [18] M Siegel, R E Caflisch and S Howison, Global existence, singular solutions, and ill-posedness for the Muskat problem, Comm Pure Appl Math 57 4), no 1, [19] EM Stein, Harmonic Analysis, Princeton University Press, Princeton, NJ, 1993 [] H Vandeven, Family of spectral filters for discontinuous problems, J, Sci Comput ), Diego Córdoba and Francisco Gancedo Departamento de Matemáticas Instituto de Matemáticas y Física Fundamental Consejo Superior de Investigaciones Científicas Serrano 13, 86 Madrid, Spain address: dcg@imaffcfmaccsices and fgancedo@imaffcfmaccsices Rafael Orive Departamento de Matemáticas Facultad de Ciencias Universidad Autónoma de Madrid Crta Colmenar Viejo km 15, 849 Madrid, Spain address: rafaelorive@uames