On the global solvability of porous media equations with general (spatially dependent) advection terms
aa r X i v : . [ m a t h . A P ] M a y On the global solvability of porous media equationswith general (spatially dependent) advection terms
N. M. L. Diehl, L. Fabris and P. R. Zingano Instituto Federal de Educa¸c˜ao, Ciˆencia e TecnologiaCanoas, RS 92412, Brazil Coordenadoria AcadˆemicaUniversidade Federal de Santa Maria - Cachoeira do SulCachoeira do Sul, RS 96501, Brazil Departamento de Matem´atica Pura e AplicadaUniversidade Federal do Rio Grande do SulPorto Alegre, RS 91509, Brazil
Abstract
We show that advection-diffusion equations with porous media typediffusion and integrable initial data are globally solvable under verymild assumptions. Some generalizations and related results are also given. (primary), , Keywords: porous medium type equations, global solvability, weak solutions,Cauchy problem, spatially dependent advection flux, pointwise estimates
1. Introduction
In this note, we describe general results recently obtained by the authors con-cerning the solvability in the large of initial value problems for degenerate advection-diffusion equations of the type u t + div f ( x, t, u ) + div g ( t, u ) = µ ( t ) div ( | u | α ∇ u ) , (1.1 a ) u ( · ,
0) = u ∈ L ( R n ) ∩ L ∞ ( R n ) (1.1 b )and some generalizations (see Section 2). Here, α > µ ∈ C ([ 0 , ∞ ))is positive, and f = ( f , f , ..., f n ), g = ( g , g , ..., g n ), are given continuous advection1ux fields that are locally Lipschitz in u uniformly in x ∈ R n and bounded t ≥ f satisfying: f ( x, t,
0) = for all x, t and | f ( x, t, u) | ≤ F ( t ) | u | κ + 1 ∀ x ∈ R n , t ≥ , u ∈ R (1.2)for some F ∈ C ([ 0 , ∞ )) and some constant κ ≥
0, where | · | denotes the absolutevalue (in case of scalars) or the Euclidean norm (in case of vectors), as in (1.1 a ).By a (bounded) solution of the problem (1.1) in some time interval [ 0 , T ∗ ) is meantany function u ( · , t ) ∈ C ([ 0 , T ∗ ) , L ( R n )) ∩ L ∞ loc ([ 0 , T ∗ ) , L ( R n ) ∩ L ∞ ( R n )) having | u ( · , t ) | α u ( · , t ) ∈ L ((0 , T ∗ ) , W , ( R n )) which satisfies (1.1 a ) in distributional sense(i.e., in D ′ ( R n × (0 , T ∗ ))) and takes the initial value u ( · ,
0) = u , see e.g. [5, 16, 17].This says, in particular, that u ( · , t ) → u in L ( R n ) as t →
0, and that, for every0 < T < T ∗ given, k u ( · , t ) k L ( R n ) ≤ M ( T ) , ∀ ≤ t ≤ T , (1.3 a ) k u ( · , t ) k L ∞ ( R n ) ≤ M ∞ ( T ) , ∀ ≤ t ≤ T , (1.3 b )for some bounds M ( T ) , M ∞ ( T ) depending on T (and the solution u considered).For the local (in time) existence of such solutions, which are typically obtained byparabolic regularization or Galerkin approximations, see e.g. [5, 9, 13, 16, 17]. Fromthe basic theory, many interesting solution properties are known; for example, onehas u ( · , t ) ∈ C ([ 0 , T ∗ ) , L ( R n )) and Z T Z R n | u ( x, t ) | α | ∇ u ( x, t ) | dx dt < ∞ (1.4)for every 0 < T < T ∗ , see e.g. [4, 9, 12, 16, 17]. More importantly to us here, solutions u ( · , t ) decrease monotonically in L ( R n ), so that, in particular, k u ( · , t ) k L ( R n ) ≤ k u k L ( R n ) , ∀ < t < T ∗ . (1.5)For all that is presently known, however, little has been obtained regarding the solv-ability for large t in the general framework (1.1), (1.2) above, except in very specialsituations. Thus, for example, when the flux f ( x, t, u) does not depend explicitlyon x , or, when it does, if it behaves so as to satisfy special conditions like n X i = 1 u ∂ f i ∂x i ( x, t, u) ≥ , ∀ x ∈ R n , t ≥ , u ∈ R , (1.6)then solutions are known to be globally defined, with k u ( · , t ) k L q ( R n ) monotonically2ecreasing for every 1 ≤ q ≤ ∞ , see e.g. [6, 9, 10, 12, 14, 16]. In the absence of (1.6),however, things get much more complicated to analyze. To see why, let us considerby way of illustration the simple example below. Taking f ( x, t, u) = b ( x, t ) | u | κ ufor some κ >
0, and (say) g = , µ ( t ) = 1, the equation (1.1 a ) becomes u t + b ( x, t ) · ∇ ( | u | κ u ) = div ( | u | α ∇ u ) + β ( x, t ) | u | κ u (1.7)with β ( x, t ) = − P ni = 1 ∂ b i /∂x i . In regions where β ( x, t ) >
0, it is clear that | u ( x, t ) | tends to grow, particularly if β ( x, t ) ≫
1, potentially leading to finite-time blow-up.In fact, solutions are known to increase quite substantially in size in many cases, asshown in the examples below (Figs. 1, 2). In view of the constraint (1.5), however,any substantial growth of | u | leads to the development of high frequency struc-tures, as illustrated below, which in turn tend to be efficiently dissipated by theever larger viscosity present in these regions (the local bulk viscosity is proportionalto | u | α ). Thus, although the basic ingredients for solution blow-up are clearly there,especially for large κ , the final outcome seems difficult to predict, be it on physicalor mathematical grounds. This interesting interaction between convection and dif-fusion due to β ( x, t ) > Fig. 1:
The solution u ( · , t ) at time t = 1000 (right) for some given initial statecompactly supported in the square | x | ≤ | y | ≤ n = 2, α = κ = 1 and b ( x, t ) = − | x | x / (10 − + | x | / b ( x, t ) = − | x | x / (10 − + | x | / ig. 2: The solution of (1.7) at time t = 1000 (right) for the same initial stateconsidered in Fig. 1, assuming n = 2, α = 2, κ = 1 and b ( x, t ) = ( b ( x ) , b ( x ))with b ( x ) = − / x / [ (16 + x ) (10 − + x )], b ( x ) = − / x / [ (16 + x ) · (10 − + x ) ], showing once again an 18-fold increase in solution size. A similar18,000-fold increase in size would happen with b ( x ) = − − x / [ (4 + x ) · (10 − + x )] and b ( x ) = − − x / [ (4 + x ) (10 − + x ) ] in this example. Going back to the general setting (1.1), (1.2), we now state our main results.Even when solutions are subject to considerable growth due to strong convectioninstabilities, the constraint (1.5) makes diffusion ultimately have the upper handin every case, thus preventing any finite time blow-up from happening (i.e., T ∗ = ∞ ): Theorem I.
Let κ ≥ . Then, all solutions to the problem (1 . , (1 . satisfying (1 . are globally defined ( i.e, defined for all t > . Theorem I is established after a series of technical lemmas in [11]. It significantlyimproves a previous result obtained in [9, 12], which was restricted to vanishingviscosity solutions and in addition required the extra assumption that κ < α + 1 /n .From the lemmata in [11] we also obtain an important pointwise estimate for thesolutions of (1.1), (1.2) involving the quantities U q ( t ) := sup < τ < t k u ( · , τ ) k L q ( R n ) (1 ≤ q ≤ ∞ ) (1.8 a )4nd F µ ( t ) := sup < τ < t F ( τ ) µ ( τ ) (1.8 b )where F is given in (1.2) above. Let a := n ( κ − α ). Taking p ≥ σ > p > n ( κ − α ) , σ ≥ max n p , κ − α ) − p o (1.9)it is shown in [11] the following fundamental estimate (see also [2, 3, 9, 12, 18]): Theorem II.
Let p ≥ , σ > satisfy (1 . . Then U ∞ ( t ) ≤ K · max (cid:26) k u k L ∞ ( R n ) ; F µ ( t ) np − a U p ( t ) pp − a (cid:27) (1.10) for every t > , and some constant K = K ( n, κ, α, p, σ ) > , where a = n ( κ − α ) .
2. Some generalizations
We note that the results discussed above apply to more general degenerateparabolic equations of the form u t + div f ( x, t, u ) + div g ( t, u ) = div A ( x, t, u, ∇ u ) . (1.1)where f , g are as before and A ∈ C ( R n × [ 0 , ∞ ) × R × R n ) satisfies the condition h A ( x, t, u , v ) , v i ≥ µ ( t ) | u | α | v | and | A ( x, t, u , v ) | ≤ M ( t ) | u | α | v | (2.2)for all x ∈ R n , t ≥ , u ∈ R , v ∈ R n , and some µ, M ∈ C ([ 0 , ∞ )), with µ ( t ) > k u ( · , t ) k L ( R n ) cannot blow up in finite time. Theresults also extend to the case of arbitrary initial states u ∈ L ( R n ) [ not necessarilybounded ], with condition (1.3 b ) then replaced by the assumption that u ( · , t ) ∈ L ∞ loc ((0 , T ∗ ) , L ∞ ( R n )), or even more generally to initial data u ∈ L p ( R n ) for somegiven 1 ≤ p < ∞ , with only minor changes in the statements, provided once morethat it can be shown that k u ( · , t ) k L q ( R n ) will not blow up in finite time for somesuitable p ≤ q < ∞ . 5 cknowledgements. This work was partially supported by
CAPES ( M inistryof E ducation, B razil), G rant . The computations in this re-search were performed by the SGI cluster
Altix
Centro Nacionalde Processamento de Alto Desempenho em S˜ao Paulo (CENAPAD-SP) , Brazil.
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Nicolau Matiel Lunardi Diehl
Instituto Federal de Educa¸c˜ao, Ciˆencia e TecnologiaCanoas, RS 92412, BrazilE-mail: [email protected]
Lucin´eia Fabris
Coordenadoria AcadˆemicaUniversidade Federal de Santa MariaCampus de Cachoeira do SulCachoeira do Sul, RS 96501, BrazilE-mail: [email protected]
Paulo Ricardo de Avila Zingano
Departamento de Matem´atica Pura e AplicadaUniversidade Federal do Rio Grande do SulPorto Alegre, RS 91509, BrazilE-mail: [email protected]@[email protected]@gmail.com