Global Regularity vs. Finite-Time Singularities: Some Paradigms on the Effect of Boundary Conditions and Certain Perturbations

0.29 MB·

by  Adam Larios

GLOBAL REGULARITY VS. FINITE-TIME SINGULARITIES:
SOME PARADIGMS ON THE EFFECT OF BOUNDARY
CONDITIONS AND CERTAIN PERTURBATIONS
4
1
ADAMLARIOSANDEDRISSS.TITI
0
2

n Abstract. In light of the question of finite-time blow-up vs. global well-
a posednessofsolutionstoproblemsinvolvingnonlinearpartialdifferentialequa-
J tions, we provide several cautionary examples which indicate that modifica-
tions to the boundary conditions or to the nonlinearity of the equations can
7
effect whether the equations develop finite-time singularities. In particular,

weaimtounderscoretheideathatinanalyticalandcomputationalinvestiga-
]
P tions of the blow-upof three-dimensional Euler andNavier-Stokes equations,
A the boundaryconditions mayneedtobetaken intogreater account. We also
examineaperturbationofthenonlinearitybydroppingtheadvectiontermin
.
h the evolution of the derivative of the solutions to the viscous Burgers equa-
t tion,whichleadstothedevelopmentofsingularitiesnotpresentintheoriginal
a
equation, and indicates that there is a regularizing mechanism in part of the
m
nonlinearity. This simple analytical example corroborates recent computa-
[ tionalobservations inthesingularityformationoffluidequations.

1
v
4 MSCClassification. 35B44,35Q35,76B03,76D03,35K55,35A01,35Q30,35Q31
3 Keywords: Finite-Time Blow-Up, Global Existence, Boundary Driven Flows,
5 Kuramoto-Sivashinsky,Viscous Hamilton-Jacobi, Kardar-Parisi-Zhang,
1
Burgers Equation, Navier-Stokes Equations, Euler Equations.
.
1
0 1. Introduction
4
1 A fundamental goal in the study of non-linear initial boundary value problems
: involvingpartialdifferential equations is to determine whether solutions to a given
v
equation develop a singularity in finite time. Resolving the issue of finite-time
i
X blow-upis important,inpartbecauseitcanhavebearingonthe physicalrelevance
r and validity of the underlying model. However, determining the answer to this
a
questionisnotoriouslydifficultforawiderangeofequations;the 3DNavier-Stokes
and Euler equations for incompressible fluid flow being perhaps the most well-
known examples. Given that attacking the question directly is so challenging,
many researchers have looked for other routes. One route is to try to simplify
or modify the boundary conditions in an attempt to gain evidence for or against
the occurrenceoffinite-time blow-up. Asecondrouteis to modify the equationsin
someway,andtostudythemodifiedequationswiththehopeofgaininginsightinto
the blow-up of solutions to the original equations. In this paper, we will examine
several case studies related to such approaches. A major aim of the present work
is to provide examples which demonstrate that one must be extremely cautious in
generalizing claims about the blow-up of problems studied in idealized settings to
claims about the blow-up of the original problem. A second aim is to demonstrate
Date:January7,2014.
2 ADAMLARIOSANDEDRISSS.TITI
a phenomenon which has been observed computationally in the difficult setting of
fluid flowsin 3D, by means of a simple 1D example, whichis amenable to analysis;
namely, that a seemingly harmless alteration (from the perspective of enstrophy
balance) to the nonlinearity of a problem can cause the formation of a singularity,
where no such singularity is present in the unaltered equation.
Wewillfocusonthreemajorcases. Thefirstcaseexaminestheeffectofreplacing
Dirichlet boundary conditions with periodic boundary conditions. This is often
done in both analyticalandnumericalstudies of, e.g., the Navier-StokesandEuler
equations. The original,physicalequationscome equipped withphysicalboundary
conditions, such as, e.g., Dirichlet boundary conditions in the case of the Navier-
Stokesequations. However,manysuchstudies havetriedto searchfor singularities
ofthesolutionsoftheequationsinthesettingofperiodicboundaryconditions(see,
e.g., [23, 40, 42, 43, 47]; in particular, see the surveys [34, 35], and the references
therein). With this in mind, in section 3, we provide an example of an equation
which develops a singularity in finite time when Dirichlet boundary conditions are
imposed, and yet is globally well-posed in the case of either periodic boundary
conditions or the case where the domain is the full space (i.e., in the absence
of physical boundaries). Therefore it may be the case that physical boundary
conditionsneedtobetakenintogreaterconsiderationinanalyticandcomputational
searches for blow-up of the solutions. Indeed, in a recent computational study,
the authors of [44, 50] observe the formation of a finite-time singularity near the
boundary in the 3D Euler equations, of axi-symmetric flow confined in a physical
cylinder, subject to no-normal flow boundary conditions. Notably, a new blow-up
criterionforthe3DEulerequationsinboundeddomains,subjecttono-normalflow
boundaryconditions,hasbeenestablishedin[36]. Itisworthstressingthatthisnew
criteriondoes apply for the periodic boundary conditions caseor when the domain
is full space, i.e. in the absence of physical boundaries. For other issues regarding
boundarybehaviorofthe Navier-StokesandEulerequationsseethe recentsurveys
[4, 5] and the references therein.
The above discussion is particularly relevant due to the notion of “boundary
driven” mechanisms for possible blow-up. To illustrate how such a mechanism
might work, we give a heuristic scenario in the context of the Navier-Stokes equa-
tions for fluid flow. It was shown in the celebrated work [6] that blow-up of
the Euler equations occurs if and only if the vorticity becomes infinite (see also
[6, 13, 14, 18, 19, 55, 59] for additional blow-up criteria). Infinite vorticity would
alsocausetheNavier-Stokessolutionstobecomesingular. Now,inthesettingofvis-
cousincompressiblefluids,physicalboundariesarethesourceofvorticityshedding.
Indeed, near the physical boundary of a fluid, the “no-slip” (Dirichlet) boundary
conditions can cause the development of boundary layers, where the vorticity is
large. Iftheviscousdiffusionofthefluidvelocityissufficientlysmallincomparison
to the advection, then large magnitudes of the gradient and the vorticity can be
propagated from the boundary layer to the interior of the domain by the nonlin-
ear advection term. The vorticity can then be further intensified by the nonlinear
vorticity stretching term, which may thus lead to blow-up of the solution. Such a
physicalmechanism does not exist in the periodic setting, nor in the full space R3.
It may therefore be illuminating to pay greater attention to the effect of boundary
conditionsinthesearchfortheblow-upofsolutionstotheNavier-StokesandEuler
equations. We do not explore these ideas in greater detail as they are only meant
EFFECT OF BOUNDARY CONDITIONS 3
to give motivation. Instead we examine a different, simpler equation in section 3,
for which we can provide a definite answer.
In section 4, we examine the Kuramoto-Sivashinsky equation in a bounded do-
main with two different types of boundary conditions. The question of global
well-posedness of this equation, when equipped with certain physically relevant
boundary conditions, is still open. Recently, in [54], it was shown that, by apply-
ing a different (non-physical) set of third-order boundary conditions, a singularity
develops in finite time. In contrast to this, we provide a different set of (also
non-physical) third-order boundary conditions for which the equation is globally
well-posed. Therefore, we maintain that it is difficult to obtain information about
the blow-up or global well-posedness of an equation by altering its boundary con-
ditions.
We note that such questions relating boundary conditions to blow-up can be
highly relevant to applied and computational problems in science. Indeed, we re-
callherethatsuchanissueoccurredinthestudyoftheplanetarygeostrophicmodel
used in ocean dynamics. The model is derived asymptotically by keeping only the
hydrostatic balance of the vertical momentum and the leading order geostrophic
balance of the horizonal momentum, where the latter is damped by the friction
with the continental shelf, while retaining the relevant physical boundary condi-
tions. In [15], it was observed that this model is over determined and hence is
ill-posed (it has more boundary conditions than needed for the underlying PDEs).
This observation explains the numerical instabilities that had been observed near
the boundary in simulations of this model. The resulting oscillations had proven
difficult to eliminate, and were dealt with in [15] by adding artificial higher-order
diffusion corresponding to the additional boundary conditions in the model.
It is commonly believed that adding hyper-viscosity into a numerical scheme
enhancesthestabilityoftheunderlyingscheme. Insection5weprovideinexample
which questions the validity of this claim. That is, even though the hyper-viscous
term enhances the dissipation of small scales, it destroys the maximum principle,
which is an essential property for the global stability in certain physical systems.
Finally,insection6weconsideracertaintypeofperturbationofthenonlinearity.
In particular, in the context of the Navier-Stokes or Euler equations, by removing
the advection term in the vorticity formulation, several recent works [23, 40, 41,
44, 50] have observed computationally that the solutions of the altered equations
seem to blow up in finite time, naming this phenomenon, “advection depleting
singularity”. We give an analogous simple example based on a similar alteration
of the 1D viscous Burgers equation, and we show analytically that a singularity
develops in finite time, which adds credence to the numerical observations of the
aforementioned works. Indeed, since the viscous Burgers equation is globally well-
posed, the development of a singularity in the altered model indicates that the
removed portion of the nonlinearity has a regularizing effect. However, it is worth
stressingthatthesealterationsturnouttobenon-localinnature,andinthecontext
ofthehydrodynamicsequationstheytranslatetomodificationintherepresentation
of the pressure term.
Many of the results and proofs are not completely new, but, for the sake of
being somewhat self-contained, are collected, compared, and contrasted here. We
also aim to state specific, as opposed to general results, whenever doing so simpli-
fies the exposition. The reason for this approach is that our goal is to lay out a
4 ADAMLARIOSANDEDRISSS.TITI
simple set of examples and counter-examples for the use of the reader in consider-
ing potential mechanisms for singularity formulation or prevention, in particular,
in computational studies.
2. Preliminaries
Inthissection,wesetsomenotationandrecallbasicresultsusedbelow. Wedenote
by Lp, Ws,p the usualLebesgue andSobolev spaces. We denote by C, C′, C , etc.
Ω
generic constants which may vary from line to line.
We recall some basic facts about the Laplace operator := n ∂2 in the
△ i=1 xi
setting of either periodic or homogeneous Dirichlet boundary conditions (see, e.g.,
[24] for proofs and further discussion). Recall that the operator (P )−1, subject
−△
tothe appropriateboundaryconditions,isapositive-definite,self-adjoint,compact
operator from L2 into itself, and therefore it has an orthonormal basis of positive
eigenfunctions ϕ ∞ (which are also eigenfunctions of ), corresponding to a
{ k}k=1 −△
sequence of positive eigenvalues. Since the eigenvalues of ( )−1 can be ordered
−△
to be non-increasing, we can label the eigenvalues of , which we denote by λ ,
k
−△
to be such that 0<λ λ .
1 2
≤ ≤···
We will pay special attention to the first eigenfunction of ,subject to homo-
−△
geneous Dirichlet boundary condition, namely ϕ , corresponding to λ . We recall
1 1
Hopf’sLemma,whichstatesthat ∂ϕ1 >0on∂Ω,whereν istheoutward-pointing
− ∂ν
normal of Ω. It can also be shown that ϕ is strictly positive on Ω. For proofs of
1
these facts, see, e.g., [24].
We denote the distance function to the boundary by
dist(x,∂Ω):=inf x y :y ∂Ω .
{| − | ∈ }
Let Ω Rn be a domain which is bounded in at least one direction. For all
u W1,p(⊂Ω), p 1, the following Poincar´e inequality holds
∈ 0 ≥
(2.1) u C u ,
Lp Ω Lp
k k ≤ k∇ k
with C =λ−1/2 if p=2.
1
We next recall the Gevrey classes of spatially analytic functions.
Definition 2.1. We define the Gevrey classes Gs/2(Tn) of spatially analytic func-
σ
tions on the torus Tn := Rn/(2πZ)n, to be the set of all u L2(Tn) such that
∈
u < , where
k kGsσ/2(Tn) ∞
1/2
(2.2) u := u 2(1+ k2)se2σ(1+|k|2)1/2 ,
k kGsσ/2(Tn) k∈Zn| k| | | !
X
where u are the Fourier coefficients of u, and where σ >0.
k
Such functions are called Gevrey regular. Note that formally setting σ = 0, we
recover the usual Sobolev spaces Hs(Tn). Furthermore, it can be shown that for
σ >0, σ is comparable to the minimal radius of analyticity.
3. Periodic Vs. Dirichlet Boundary Conditions
Consider the Cauchy problem for the following viscous Hamilton-Jacobiequation,
(3.1a) u u= up, in Ω (0,T),
t
−△ |∇ | ×
(3.1b) u(0)=u , in Ω,
0
EFFECT OF BOUNDARY CONDITIONS 5
equippedwitheitherperiodicboundaryconditionsorhomogeneousDirichletbound-
ary conditions. Many authors have studied the cases p [0, ) (see, e.g., [1, 2, 8–
∈ ∞
10, 37, 38]), but in this work, we will focus on the case p=4 for simplicity. In the
case p = 2, is an integrated version of the viscous Burgers equation, and is some-
times referred to as the Kardar-Parisi-Zhangequation, which is used to model the
growthandrougheningof certainsurfaces,as derivedin [46]. Furthermore,(3.1) is
an important test equation, since it is one of the simplest examples of a parabolic
PDE with non-linear dependence on the gradient.
In the case of periodic boundary conditions, (3.1) with p 2 is well-posed,
≥
globally in time. However, in the Dirichlet case, and for p > 2, a singularity will
develop in finite time, for certain initial data. We give a relatively simple proof of
the well-posedness in the periodic case with p=4. The proof for p>2 is given in
[38]. Forthe proofofblow-upinthe Dirichletcase,choosingp=4doesnotappear
to make things significantly simpler than allowing p > 2, so we give the proof for
p>2. We follow closelythe proofin[61] to show thata singularityoccursin finite
time in the Dirichlet case, at least for sufficiently large initial data in the sense
given in (3.7), below.
It is worth noting that the following identity holds for sufficiently smooth func-
tions u=u(t,x):
(∂ )eu =(∂ u u u2)eu,
t t
−△ −△ −|∇ |
Thus, (3.1) can be solved explicitly in the case p = 2, by making the change of
variablesv =eu(knownastheCole-HopftransformationfortheBurgersequation),
and noting that if u solves (3.1), then v solves the linear heat equation, with the
corresponding boundary conditions.
3.1. Global Well-Posedness in the Periodic Case. We prove the global exis-
tence of solutions to (3.1), for p = 4 under the assumption of periodic boundary
conditions. We begin by stating a special case of a theorem in [25] (which follows
ideas from [30]), that gives short-time existence, uniqueness, and regularity.
Theorem 3.1 ([25]). Let u0 Hs(Tn), with s > n/2, and u0 Hs(Tn) M0 for
∈ k k ≤
someM >0. ThenthereexistsaT >0dependingonlyuponM suchthatequation
0 0
(3.1), with pandpositive even integer, has a uniquesolution uontheinterval [0,T)
withtheinitial valueu , which satisfies u C([0,T);Hs(Tn)) L2((0,T);H2(Tn)),
0
∈ ∩
du L2((0,T);L2(Tn)). Moreover, u(,t) Gs/2(Tn) for t [0,T).
dt ∈ · ∈ t ∈
With this theorem in hand, we now state and prove a global existence theorem
for (3.1) with p = 4 and n = 1. For global well-posedness in the general case, see
[37, 38].
Theorem 3.2. Suppose u H1(T), and consider (3.1) in the one-dimensional
0
∈
casewithperiodic boundary conditions, andp=4. Then theunique, Gevreyregular
solution given by Theorem 3.1 can be extended to an arbitrarily large time interval
[0,T].
Proof. First note that, since p = 4, the right-hand side of (3.1) is real analytic in
u ,andwehaveshort-timeexistenceanduniqueness(say,onatime interval[0,T])
x
of (3.1) under periodic boundary conditions by using, e.g., the Galerkin method.
Furthermore,asshownin[25],thesolutionisGevreyregularinspace. Inparticular,
ithascontinuousderivativesofallorders. Itremainstoshowthatthesolutionexists
6 ADAMLARIOSANDEDRISSS.TITI
globally in time. Suppose [0,T∗) is the maximal interval of existence. If T∗ =
∞
there is nothing to prove. Therefore, we assume by contradiction that T∗ < .
From the above regularity, we infer in particular, that u(,T∗) H2(T). We u∞se
· 2 ∈
a technique of E. Hopf and G. Stampacchia (cf. [48, 64]) to prove a maximum
principle for u . Write v := u and v () := u (,T∗). For any function f H1,
x x ∗ · x · 2 ∈
we use the standard notation f+ :=max f,0 . It is a standard exercise (see, e.g.,
{ }
[24], section 5.10) to show that f H1 implies f+ H1. Taking the derivative of
∈ ∈
(3.1a), we have v v =4v3v . Let us denote
t xx x
−
θ(x,t):=v(x,t) v∗ L∞.
−k k
Since v∗ L∞ is a constant, θx =vx and θt =vt, so that
k k
(3.2) θ θ 4v3θ =0.
t xx x
− −
TakingtheinnerproductinL2 of (3.2)withθ+,weintegratebypartsseveraltimes
and use the fact that θ+θ =(θ+)2 to find
1 d
θ+ 2 + θ+ 2 = 4v3θ θ+dx= 2v3((θ+)2) dx
2dtk kL2 k xkL2 T x T x
Z Z
= 6v2v (θ+)2dx= 2v2((θ+)3) dx
x x
− T − T
Z Z
= 4vv (θ+)3dx= v((θ+)4) dx
x x
T T
Z Z
1
= v (θ+)4dx= ((θ+)5) dx=0.
x x
− T − T 5
Z Z
Thus, integrating in time, for a.e. t [T∗,T∗) we have
∈ 2
θ+(t) 2 θ+(T∗) 2 =0.
k kL2 ≤k 2 kL2
Thus, θ+(t) ≡ 0, and so, v(x,t) ≤ kv∗kL∞, for t ∈ [T2∗,T∗). Similarly, one can
show that v(x,t) v∗ L∞, and thus we have
− ≤k k
kux(t)kL∞ ≤kux(T2∗)kL∞,
for t [T∗,T∗). Next, taking the inner product of (3.1a) with u and using the
Lions-∈Mag2enes Lemma, we have, for t [T∗,T∗),
∈ 2
1 d u 2 + u 2 = (u )4udx u (T∗) 4 u C v 4 u .
2dtk kL2(T) k xkL2(T) T x ≤k x 2 kL∞k kL1 ≤ k ∗kL∞k kL2
Z
Integrating the above inequality now yields
ku(t)kL2(T) ≤ kux(T2∗)k4L∞(T)T +ku0k1L/22(T) 2 <∞,
fort [T∗,T∗). Thus,fromt(cid:16)heaboveandTheorem3.1one(cid:17)canextendthesolution
beyon∈d T2∗, which leads into a contradiction. Consequently, T∗ = . (cid:3)
∞
3.2. Finite-Time Blow-Up in the Dirichlet Case. In this section, we investi-
gate the existence, uniqueness, and the finite-time blow-up, of solutions to (3.1),
under the assumption of Dirichlet boundary conditions. The short time existence
and uniqueness of solutions to (3.1) can be proven by using, e.g., Duhamel’s prin-
ciple and the Schauder fixed point theorem, see, e.g. [31]. We state the theorem
without proof here.
EFFECT OF BOUNDARY CONDITIONS 7
Theorem 3.3 (Short time existence). Let Ω Rn be a bounded C2 domain. Sup-
⊂
pose u C1+α(Ω) for some α (0,1) and u 0 on ∂Ω. Then there exists a
0 0
∈ ∈ ≡
T > 0 such that (3.1) has a unique solution in C1+α(Ω [0,T]) C(Ω [0,T]),
× ∩ ×
with u 0 on ∂Ω.
≡
In [1] it is shown that (3.1) under homogeneous Dirichlet boundary conditions,
cannot have a global solution if u is very irregular (namely, if u is a positive
0 0
measure satisfying certain conditions). In [61] it is shown that for smooth, but
sufficiently largeinitialdata (in a sense of (3.7), below), the solutionu blows upin
finite time for p > 2. In particular, it is shown that so-called “gradient blow-up”
occurs; that is, u remains uniformly bounded, but limsupt→T∗k∇u(t,·)kL∞ = ∞,
whereT∗ < isthemaximalexistencetimeforu. Theideaistoexploitproperties
∞
of the first eigenvalue of the negative Laplacian operator, subject to homogeneous
Dirichlet boundary conditions. Let λ >0 be the smallest eigenvalue of , with
1
−△
homogeneousDirichletboundaryconditions,andϕ acorrespondingeigenfunction,
1
chosenasinsection2. We beginwithtwolemmas. The firstis usedto supportthe
second,andthesecondisthat (ϕ (x))−αdx< forα (0,1). Thismeansthat
Ω 1 ∞ ∈
wehaveacertaingrowthofϕ neartheboundary,andwillbeneededinsubsequent
1
R
calculations.
Lemma 3.4. Assume that Ω Rn is a bounded domain with C2 boundary. Then
⊂
there exists a constant C >0 such that for all x Ω,
∈
ϕ (x) C dist(x,∂Ω).
1
≥ ·
The proof is a fairly straight-forward application of Hopf’s Lemma, but we
include it here for completeness. Note that in the one-dimensional case, with
Ω = (0,π), we have ϕ (x) = π sin(x), and the results of Lemmas 3.4 and 3.5
1 2π
are trivial in this case.
Proof. Since Ω is C2, it satisfies the interior sphere condition. Therefore for x Ω
∈
sufficiently close to ∂Ω, we may write x = x sν for some x ∂Ω, s > 0 and
0 0
− ∈
where ν is the exterior normal of Ω. Then dist(x,∂Ω)=dist(x,x )=s. Note that
0
byHopf’sLemma,lim s−1ϕ (x sν)>0. Since ∂ΩisC2,andΩisbounded,
s→0+ 1 0
−
we have sufficient regularity on ϕ to conclude that this limit is uniform in x .
1 0
Thus(choosingxcloserto∂Ωifnecessary),thereexistsaconstantC˜ >0suchthat
s−1ϕ (x sν) C˜,thatis,ϕ (x) C˜ dist(x,∂Ω)forallxsufficientlyclosetothe
1 0 1
− ≥ ≥ ·
boundary,saywithin ǫ neighborhood,forsomeǫ (0,1). Since Ω is bounded, the
− ∈
set E := x Ω:dist(x,∂Ω) ǫ is compact. By the elliptic maximum principle,
{ ∈ ≥ }
ϕ C˜ǫ on E. Setting C =C˜ǫ/(1+diam(Ω)), we have ϕ (x) C dist(x,∂Ω), as
1 1
des≥ired. ≥ · (cid:3)
Usingthislemma,wenextshowthatϕ satisfiesacertaingrowthconditionnear
1
its zeros (that is, near the boundary). This is the main property that is exploited
in[61]toshowfinite-timeblow-up. Acruciallemma,provedin[61],butalsostated
(without proof) in [1] and [26], is the following.
Lemma 3.5. Assume that Ω is a bounded domain in Rn with C2 boundary, and
let α (0,1). Then
∈
ϕ−α(x)dx< .
1 ∞
ZΩ
8 ADAMLARIOSANDEDRISSS.TITI
Proof. We follow closely the proof in [61]. We can use a partition of unity to
reducetoalocalargument. SinceΩisboundedwithC2 boundary,itcanbelocally
represented as the graph of a C2 function, say f : U ( ǫ,ǫ), for some ǫ > 0,
0
where x=(x′ ,x ) Rn−1 R, and → −
n n ∈ ×
U := x Rn : x′ ǫ, x <ǫ , U := x′ Rn−1 : x′ ǫ ,ǫ>0,
{ ∈ | n|≤ | n| } 0 n ∈ | n|≤
and f(0)=0. Furthermore, we define (cid:8) (cid:9)
ω := x U :x <f(x′ ) , Γ:= x U :x =f(x′ ) .
{ ∈ n n } { ∈ n n }
By projecting the vector (~0,f(x′ ) x ) onto the inward-pointing normal of the
n − n
graph of f, it follows that
f(x′ ) x
dist(x,∂Ω) n − n .
≥ 1+ f 2
k∇ kL∞
Using this and Lemma 3.4, we estimate
p
f(x′)
n
(ϕ (x))−αdx c−α (dist(x,Γ))−αdx=c−α (dist(x,Γ))−αdx dx′
1 ≤ n n
Zω Zω ZU0Z−ǫ
f(x′)
c−α 1+ f 2 α/2 n (f(x′ ) x )−αdx dx′ < ,
≤ k∇ kL∞ n − n n n ∞
ZU0Z−ǫ
since α<1. (cid:0) (cid:1) (cid:3)
Next, we seek a lower bound on upϕ (x)dx.
Ω|∇ | 1
Lemma 3.6. Let p>2, and supposRe Ω Rn is a C2 bounded domain. Then there
exists a constant C >0 such that for⊂any function v W1,p(Ω),
Ω,p ∈ 0
p
(3.3) C v(t,x)ϕ (x)dx v pϕ dx.
Ω,p 1 1
≤ |∇ |
(cid:12)ZΩ (cid:12) ZΩ
Furthermore, no such con(cid:12)(cid:12)stant exists for 1(cid:12)(cid:12) p 2.
(cid:12) (cid:12)≤ ≤
Proof. For the proof we follow closely [61]. Using Ho¨lder’s inequality, we have
1/p 1−1/p
v dx= v ϕ1/pϕ−1/pdx v pϕ dx ϕ−1/(p−1)dx
|∇ | |∇ | 1 1 ≤ |∇ | 1 1
ZΩ ZΩ (cid:18)ZΩ (cid:19) (cid:18)ZΩ (cid:19)
1/p
(3.4) C′ v pϕ dx ,
≤ Ω,p |∇ | 1
(cid:18)ZΩ (cid:19)
where, due to Lemma 3.5,
1−1/p
C′ := ϕ−1/(p−1)dx < ,
Ω,p 1 ∞
(cid:18)ZΩ (cid:19)
since p>2 implies 1 (0,1).
p−1 ∈
Since Ω is bounded, we have by Poincar´e’s inequality (2.1), that there exists a
constant C such that
Ω
(3.5) v(x,t) dx C v(x,t) dx.
Ω
| | ≤ |∇ |
ZΩ ZΩ
EFFECT OF BOUNDARY CONDITIONS 9
Therefore, (3.4) and (3.5) give
p p
(3.6) v(t,x)ϕ1(x)dx ϕ1 L∞ v(t,x) dx
≤ k k | |
(cid:12)ZΩ (cid:12) (cid:18) ZΩ (cid:19)
(cid:12) (cid:12) p
(cid:12) (cid:12)
(cid:12) (cid:12) ϕ1 L∞CΩ v(x,t) dx
≤ k k |∇ |
(cid:18) ZΩ (cid:19)
p
≤ kϕ1kL∞CΩCΩ′,p |∇v(x,t)|pϕ1(x)dx,
(cid:16) (cid:17) ZΩ
Setting CΩ,p = kϕ1kL∞CΩCΩ′,p −p yields (3.3).
Wenextgiveacounterexampletoshowthat(3.3)cannotholdforallv W1,p(Ω)
(cid:0) (cid:1) ∈
inthecasep=2. Here,forsimplicity,weonlyshowtheone-dimensionalcase,with
Ω=(0,2π),sincesimilarcounterexamplescanbeconstructedinhigherdimensional
cases based on the one-dimensional case, using the fact that the domain satisfies
the interior ball condition and comparing to the distance function, as in the proof
of Lemma 3.5.
For ǫ (0,π/4), consider the function v defined on [0,π], given by
ǫ
∈
0 for x [0,ǫ2] [π ǫ2,π],
∈ ∪ −
log(x/ǫ2) for x [ǫ2,ǫ],
v (x):= ∈
ǫ log(1/ǫ) for x∈[ǫ,π−ǫ],
log((π x)/ǫ2) for x [π ǫ,π ǫ2].
− ∈ − −
We calculate the derivative
0 for x (0,ǫ2) (ǫ,π ǫ) (π ǫ2,π),
∈ ∪ − ∪ −
v′(x)= 1/x for x (ǫ2,ǫ),
ǫ  ∈
1/(x π) for x (π ǫ,π ǫ2).
− ∈ − −
Notice that vǫ ∈ W01,2((0,π)). Furthermore, using the fact that sin(x) ≥ π2x on
[0,2/π] and that 0<ǫ<π/4 we have
π π/2 4 π/2
v (x)sin(x)dx=2 v (x)sin(x)dx v (x)xdx
ǫ ǫ ǫ
≥ π
Z0 Z0 Z0
4 π/2 1 2 1 π2 3π 1
log xdx= log ǫ2 > log ,
≥π ǫ π ǫ 4 − 8 ǫ
Zǫ (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
Thus,
π 2 9π2 1 2
v (x)sin(x)dx > log .
ǫ
64 ǫ
(cid:18)Z0 (cid:19) (cid:18) (cid:18) (cid:19)(cid:19)
On the other hand, notice that
π ǫ 1 2 ǫ 1 2 1
(v′(x))2sin(x)dx=2 sin(x)dx 2 xdx=2log .
ǫ x ≤ x ǫ
Z0 Zǫ2 (cid:18) (cid:19) Zǫ2 (cid:18) (cid:19) (cid:18) (cid:19)
Taking ratios of the above inequalities, we observe
0πvǫ(x)sin(x)dx 2 96π42 log 1ǫ 2 = 9π2 log 1
π(v′(x))2sin(x)dx ≥ 2log 1 128 ǫ →∞
(cid:0)R0 ǫ (cid:1) (cid:0) (cid:0)ǫ (cid:1)(cid:1) (cid:18) (cid:19)
as ǫ 0+, aRnd therefore no finite number C(cid:0) >(cid:1) 0 can be chosen to make (3.3) true
→
for all functions v W1,2((0,π)). (cid:3)
∈ 0
10 ADAMLARIOSANDEDRISSS.TITI
Remark 3.7. The counterexample for p = 2 we believe to be new. A counterex-
ample, based on a piecewise linear function, was given in the case 1 p < 2, for
≤
n=1, in [7].
Withtheabovelemmasinhand,wearenowreadytoprovethemaintheoremfor
blow-upof (3.1) under homogeneous Dirichlet boundary conditions. As mentioned
earlier, the proof is very similar to the one given in [61], where it is also given in
greater generality. The proof is given here for the sake of completeness.
Theorem 3.8. Let p> 2 and suppose u C2(Ω) L∞(Ω). There exists K >0,
0
∈ ∩
given by equation (3.10) below, such that if
(3.7) u (x)ϕ (x)dx K,
0 1
≥
ZΩ
then any solution to (3.1), taken with homogeneous Dirichlet boundary conditions
and initial data u , develops a singularity in finite time.
0
Proof. By Theorem 3.3, we know that there exists a time T > 0 and a unique
solution u C1+α([0,T] Ω) C([0,T] Ω) to (3.1) satisfying u 0 on ∂Ω. Let
∈ × ∩ × ≡
T∗ > 0 be the maximal time of existence of the solution to (3.1). If T∗ < then
∞
there is nothing to prove. Therefore, we assume by contradiction that T∗ = .
∞
Following [61], let
(3.8) z(t)= u(t,x)ϕ (x)dx.
1
ZΩ
The use of z(t) will allow us to use standard non-existence results for ODEs, ex-
ploiting properties of ϕ . Integrating by parts, we calculate in (0,T∗),
1
(3.9) z′(t)+λ z(t)= u (t,x)ϕ (x)dx u(t,x) ϕ (x)dx
1 t 1 1
− △
ZΩ ZΩ
= u (t,x) u(t,x) ϕ (x)dx= upϕ (x)dx.
t 1 1
−△ |∇ |
ZΩ ZΩ
Applying Lemma 3.6 to (3.9)(cid:0)gives (cid:1)
z′(t)+λ z(t) C (z(t))p.
1 Ω,p
≥
Now, if
(3.10) z(0) (2λ /C )1/(p−1) =:K,
1 Ω,p
≥
thenthe aboveestimateimplies thatz′(t) 0forashortintervaloftime,andthus
≥
z(t) K for all t 0. Let y(t):=eλ1tz(t). Notice that y(0)=z(0). We then have
≥ ≥
y′(t) C eλ1(1−p)t(y(t))p
Ω,p
≥
Integrating, we obtain
(y(0))1−p (y(t))1−p C λ−1(1 eλ1(1−p)t).
− ≥ Ω,p 1 −
Thus,
−1
(3.11) (y(t))p−1 (z(0))1−p C λ−1(1 eλ1(1−p)t) .
≥ − Ω,p 1 −
(cid:16) (cid:17)
Now, since z(0) K, then the right-hand side of (3.11) become infinite at finite
time t = T∗∗, w≥here eλ1(1−p)T∗∗ = 1/2. Hence T∗ T∗∗, which contradicts the
≤
assumption that T∗ = . In particular, we have shown that a singularity of u
develops in finite time. ∞ (cid:3)