J. Evol. Equ. (2023) 23:57
© 2023 The Author(s)
1424-3199/23/030001-28, published online August 5, 2023
https://doi.org/10.1007/s00028-023-00900-3
Journal of Evolution
Equations
Blow-up for a stochastic model of chemotaxis driven by conservative
noise on R2
Avi Mayorcas and Milica Tomaševi´c
Abstract. We establish criteria on the chemotactic sensitivity χfor the non-existence of global weak
solutions (i.e., blow-up in finite time) to a stochastic Keller–Segel model with spatially inhomogeneous,
conservative noise on R2. We show that if χis sufficiently large then blow-up occurs with probability 1. In
this regime, our criterion agrees with that of a deterministic Keller–Segel model with increased viscosity.
However, for χin an intermediate regime, determined by the variance of the initial data and the spatial
correlation of the noise, we show that blow-up occurs with positive probability.
1. Introduction
In this work, we present criteria for non-existence of global solutions (that we will
frequently refer to as finite time blow-up) to a stochastic partial differential equation
(SPDE) model of chemotaxis on R2. The model we consider,
⎧
⎪
⎨
⎪
⎩
dut=(ut−χ∇·(ut∇ct))dt+√2γ∞
k=1∇·(σkut)◦dWk
t,on R+×R2,
−ct=ut,on R+×R2,
u|t=0=u0∈P(R2), on R2,
(1.1)
is based on the parabolic-elliptic Patlak–Keller–Segel model of chemotaxis (γ=0)
with the addition of a stochastic transport term (γ>0), where {Wk}k≥1is a family of
i.i.d. standard Brownian motions on a filtered probability space, (, F,(Ft)t≥0,P),
satisfying the usual assumptions. Here P(R2)denotes the set of probability measures
on R2. We will give detailed assumptions on the vector fields σk:R2→R2below
(see (H1)–(H3)), but for now simply stipulate that they are assumed to be divergence
free and such that σ:= {σk}k≥1∈2(Z;L∞(R2)).
The noiseless model (γ=0) is a well-known PDE system modeling chemotaxis:
the collective movement of a population of cells (represented by its time-space density
u) in the presence of an attractive chemical substance (represented by its time-space
concentration c). The chemical sensitivity is encoded by the parameter χ>0. The
Mathematics Subject Classification: Primary 60H15, 35R60; Secondary 35B44, 35Q92, 92C17
Keywords: Blow-up criteria for SPDE, Keller–Segel equations of chemotaxis, SPDE with conservative
noise.
57 Page 2 of 28 A. Mayorcas and M. Tomaševi´c J. Evol. Equ.
main particularity of the model is that solutions may become unbounded in finite time
even though the total mass is preserved. This is the so-called blow-up in finite time,
and it occurs depending on the spatial dimension of the problem and the size of the
parameter χ. In particular, on R2blow-up occurs in finite time for χ>8π,att=∞
for χ=8πsee [2] and global existence holds for χ<8π, see for example the survey
by [32]. For results in other dimensions, we refer to [4,20,30,32].
Since the scenario described by the noiseless model often occurs within an external
environment, it is natural to take into account additional environmental effects. In
some cases, this can be done by coupling additional equations into the system, such as
the Navier–Stokes equations of fluid mechanics [27,37,38]. With particular relevance
to our work, we note the results of [22,23] where it was shown that transport by
sufficiently strong relaxation enhancing flows can have a regularizing effect on the
Keller–Segel equation. However, for both modeling and analysis purposes it is also
relevant to study the effect of random environments. These either model a rough
background, accumulated errors in measurement or emergent noise from micro-scale
phenomena not explicitly considered.
The noise introduced in (1.1) is related to stochastic models of turbulence, [6,8,24,
26] and we refer to the monograph by [12] for a broader overview of its relevance to
SPDE models. Noise satisfying either our assumptions, or closely related ones, has
been applied in a number of related settings; interacting particle systems, [7,11,15];
regularization, stabilization and enhanced mixing of general parabolic and transport
PDE, [14,17,19], and with particular applications to the Keller–Segel and Navier–
Stokes equations among others in [13,16,18].
The motivation of the present work is to understand the persistence of blow-up in
the case of stochastic chemotactic models driven by conservative noise. Our main
result is that if χ> (1+γ)8πthen finite time blow-up occurs almost surely, while
if χ>(1+γV[u0]Cσ)8πthen finite time blow-up occurs with positive probability.
Here V[u0]denotes one half the spatial variance of the initial data and Cσindicates a
type of Lipschitz norm of the vector fields σand measures the spatial decorrelation of
the noise. We refer to (2.2) for a precise definition. Furthermore, if χsatisfies either of
the above conditions and blow-up does occur then it must do so before a deterministic
time T∗>0, (see Theorem 2.8).
Note that when γ=0, we recover the usual conditions for blow-up of the deter-
ministic equation, see [32].
Three interesting regimes emerge from our criteria, on the one hand, if we let Cσ
increase to infinity, the second condition becomes the first and blow-up must occur
almost surely, albeit for larger and larger χ. On the other hand, when Cσis arbi-
trarily small (which is the case for spatially homogeneous noise) one again recovers
the deterministic criterion. However, in the third regime, where the noise and initial
variance are reciprocally of the same order, i.e., V[u0]Cσ<1, we are only able to
show blow-up with positive probability. It is an interesting question, that we leave for
J. Evol. Equ. Blow-up for a stochastic model of chemotaxis driven Page 3 of 28 57
future work, to obtain more information on the probability of blow-up in this case.
See Remark 2.9 for a longer discussion of these points.
The study of blow-up of solutions to SPDEs is a large topic of which we only
mention some examples. It was shown by [3] that additive noise can eliminate global
well-posedness for stochastic reaction–diffusion equations, while a similar statement
has been shown for both additive and multiplicative noise in the case of stochastic
nonlinear Schrödinger equations by [9,10]. In addition, non-uniqueness results for
stochastic fluid equations have been studied by [21] and [35].
In the case of SPDE models of chemotaxis, the study of blow-up phenomena has
begun to be considered and we mention here two very recent works, by [16] and [28].
In [16], the authors show that under a particular choice of the vector fields, σ, a similar
model to (1.1)onTdfor d=2,3 enjoys delayed blow-up with 1 −εafter choosing
γand σw.r.t. χand ε∈(0,1).In[28], the authors study global well-posedness and
blow-up of a conservative model similar to (1.1) with a constant family of vector fields
σk(x)=σand a single common Brownian motion. Translating their parameters into
ours, they establish global well-posedness of solutions to (1.1), with σk(x)≡1 and
for χ<8π, as well as finite time blow-up when χ>(1+γ)8π.
The main contribution of this paper is the above-mentioned blow-up criterion for
an SPDE version of the Keller–Segel model in the case of a spatially inhomogeneous
noise term. To the best of our knowledge, this is a new result. An interesting point is
that, unlike the deterministic criterion, it relates the chemotactic sensitivity with the
initial variance, regularity and intensity of the noise term. In addition, we close the
gap in [28], as in the case of constant vector fields we show that finite time blow-up
occurs for χ>8π(see Remark 2.9). In addition, we show that χ>(1+γ)8πcannot
be a sharp blow-up threshold for all sufficiently regular initial data.
Our technique of proof follows the deterministic approach by tracking a priori the
evolution in time of the spatial variance of solutions to (1.1). We derive an SDE
satisfied by this quantity which we analyze both pathwise and probabilistically to
obtain criterion for blow-up.
Notation
•For n≥1 and p∈[1,∞)(resp. p=∞), we write Lp(R2;Rn)for the spaces
of pintegrable (resp. essentially bounded) Rn-valued functions on R2.
For α∈R, we write Hα(R2;Rn)for the inhomogeneous Sobolev spaces of
order α—a full definition and some useful facts are given in Appendix A.
For k≥0 and α∈(0,1), we write Ck(R2;Rn)for the kcontinuously differen-
tiable maps and Ck,α(R2;Rn)for the kcontinuously differentiable maps with α
Hölder continuous kth derivatives.
When the context is clear, we remove notation for the target space, simply writ-
ing Lp(R2),Hα(R2). We equip these spaces with the requisite norms writing
·
Lp,·
Hαremoving the domain as well when it will not cause confusion.
•We write P(R2)for the space of probability measures on R2and for m≥1,
Pm(R2)for the space of probability measures with mfinite moments. By an
57 Page 4 of 28 A. Mayorcas and M. Tomaševi´c J. Evol. Equ.
abuse of notation we write, for example, P(R2)∩Lp(R2)to indicate the space
of probability measures with densities in Lp(R2).
•For μ∈P(R2)and when they are finite, we define the following quantities:
C[μ]:=R2xdμ(x),
V[μ]:=1
2R2|x−C[μ]|2dμ(x)=1
2R2|x|2dμ(x)−1
2|C[μ]|2.
Note that V[μ]is one half the usual variance, we define it in this way for com-
putational ease.
•For T>0, Xa Banach space, and p∈[1,∞)(resp. p=∞), we write
Lp
TX:= Lp([0,T];X)for the space of p-integrable (resp. essentially bounded)
maps f:[0,T]→X. Similarly we write CTX:= C([0,T]; X)for the space
of continuous maps f:[0,T]→X, which we equip with the supremum norm
fCTX:= supt∈[0,T]fX. We define the function space ST:= CTL2(R2)∩
L2
TH1(R2).
•We write ∇for the usual gradient operator on Euclidean space while for k≥2,
∇kdenotes the matrix of k-fold derivatives. We denote the divergence operator
by ∇·and we write := ∇ ·∇ for the Laplace operator.
•If we write ab, we mean that the inequality holds up to a constant which
we do not keep track of. Otherwise we write a≤Cb for some C>0 which is
allowed to vary from line to line.
•Given a,b∈R, we write a∧b:= min{a,b}and a∨b:= max{a,b}.
Plan of the paper In Sect.2, we give the precise assumptions on the noise term and
formulate our main result. Then, in Sect.3we establish some important properties of
weak solutions to (1.1) which are made use of in Sect.4where we prove our main
theorem. Appendix A is devoted to a brief recap of the fractional Sobolev spaces
on R2along with some useful properties. Appendix B gives a sketch proof for the
equivalence between (1.1) and a comparable Itô equation. Finally, in Appendix C, for
the readers convenience, we provide a relatively detailed proof of local existence of
weak solutions in the sense of Definition 2.4.
2. Main result
Before stating our main results, we reformulate (1.1) into a closed form and state
our standing assumptions on the noise.
It is classical that cis uniquely defined up to a harmonic function, hence it can be
written as c=K∗uwith K(x)=−1
2πln (|x|). Therefore, from now on, for t>0,
we work with the expression
∇ct(x):= ∇K∗ut(x)=−1
2πR2
x−y
|x−y|2ut(y)dy.(2.1)
J. Evol. Equ. Blow-up for a stochastic model of chemotaxis driven Page 5 of 28 57
Throughout we fix a complete, filtered, probability space, (, F,(Ft)t≥0,P), satis-
fying the usual assumptions and carrying a family of i.i.d Brownian motions {Wk}k≥1.
Furthermore, we consider a family of vector fields σ:= {σk}k≥1, satisfying the fol-
lowing assumptions.
(H1) For k≥1, σk:R2→R2are measurable and such that ∞
k=1σk2
L∞<∞.
(H2) For every k≥1, σk∈C2(R2;R2)and ∇·σk=0.
(H3) Defining q:R2×R2→R2⊗R2by
qij(x,y)=∞
k=1
σi
k(x)σ j
k(y), ∀i,j=1,...,d,x,y∈R2;
(a) The mapping (x,y)→ q(x,y)=: Q(x−y)∈R2⊗R2depends only on
the difference x−y.
(b) Q(0)=q(x,x)=Id for any x∈R2.
(c) We have Q∈C2(R2;R2⊗R2)and supx∈R2|∇2Q(x)|<∞.
Remark 2.1. For σsatisfying Assumption (H3), it is possible to show that the quantity
Cσ:= sup
x=y∈R2
∞
k=1
|σk(x)−σk(y)|2
|x−y|2(2.2)
is finite. See [7, Rem. 4] for details. Note that due to (H3)-(b) one cannot re-scale
σso as to remove γfrom (1.1).
Remark 2.2. It is important to note that one can instead specify the covariance matrix
Qfirst.Infact,due to [25, Thm. 4.2.5] anymatrix-valuedmap Q:R2×R2→R2⊗R2
satisfying the analogue of (2.2),
sup
x=y∈R2
2
i=1
qii(x,x)−2qii(x,y)+qii(y,y)
|x−y|2<∞
can be expressed as a family of vector fields {σk}k≥1satisfying (H1)–(H3).
Analysis and presentations of vector fields satisfying these assumptions can be
found in [7], [19, Sec. 5] and [11,15]. For the reader’s convenience, we give an explicit
example here in the spirit of Remark 2.2, based on [7, Ex. 5], but adapted to our precise
setting.
Example 2.3. Let f∈L1(R+)besuchthatR+rf(r)dr=π−1andR2|u|2f(|u|)du
<∞. Then, let :R→M2×2(R)be the 2 ×2-matrix-valued map defined by,
(u)=(1−p)Id +(2p−1)u⊗u
|u|2,for p∈[0,1].
Then, we define the covariance function,
Q(z):= R2cos(u·z)−sin(u·z)
sin(u·z)cos(u·z)(u)f(|u|)du.
57 Page 6 of 28 A. Mayorcas and M. Tomaševi´c J. Evol. Equ.
Property (H3) (a) is satisfied by definition, after setting q(x,y):= Q(x−y). Since
Q(0)=R2(u)f(|u|)dy,
property (H3) (b) is easily checked by moving to polar coordinates, making use of ele-
mentary trigonometric identities and the normalization R+rf(r)dr=π−1. Finally,
(H3) (c) can be checked by a straightforward computation using smoothness of the
trigonometric functions and the moment assumption on f.
We now define our notion of weak solutions.
Definition 2.4. Let χ,γ > 0. Then, given u0∈P(R2)∩L2(R2), we say that a weak
solution to (1.1) is a pair (u,¯
T)where
•¯
Tis an {Ft}t≥0stopping time taking values in R+∪{∞},
•For T<¯
T,uis an ST:= CTL2∩L2
TH1-valued random variable such that
Eu2
L∞
TL1+u2
L∞
TL2+u2
L2
TH1<∞.
In addition, for any t∈[0,T],φ∈H1(R2),P-a.s. the following identities hold,
ut,φ=u0,φ−t
0
(∇us,∇φ−χus(∇K∗us), ∇φ)ds
−2γ
k≥1t
0σkus,∇φ◦dWk
s.
(2.3)
In Appendix C, we detail a standard argument to show that there exists a determin-
istic, positive time T>0 such that (u,T)is a weak solution in the above sense. This
is due to the particular structure of the noise and we stress that in general the maximal
time of existence may be random.
Applying the standard Itô-Stratonovich correction, one can prove the following
remark, a sketch is given in Appendix B.
Remark 2.5. Let (u,¯
T)be a solution, in the sense of Definition 2.4,to(1.1). Then it
alsoholdsthat (u,¯
T)isasolution tothefollowingItô equation:Foreveryφ∈H1(R2),
t∈[0,¯
T],P-a.s.
ut,φ=u0,φ−t
0
((1+γ)∇ut,∇φ−χus(∇K∗us), ∇φ)ds
−2γ
k≥1t
0σkus,∇φdWk
s,
(2.4)
Remark 2.6. It follows from Definition 2.4 and the standard chain rule, obeyed by
the Stratonovich integral, that for ua weak, Stratonovich solution to (1.1) and F∈
J. Evol. Equ. Blow-up for a stochastic model of chemotaxis driven Page 7 of 28 57
C3(L2(R2);R),
F[ut]=F[u0]+t
0
DF[us][us−χ∇·(us∇K∗us)]ds
+∞
k=1t
0
DF[us][∇ ·(σkus)]◦dWk
s,
(2.5)
where DF[us][ϕ]denotes the Gateaux derivative of F[us]in the direction ϕ∈
H1(R2). An equivalent Itô formula for nonlinear functional of (2.4) also holds, see
for example [31, Sec. 2].
Remark 2.7. Note that under assumption (H1), for any T>0 and any weak so-
lution on [0,T], the stochastic integral is well defined as an element of L2( ×
[0,T];L2(R2)) ⊂L2( ×[0,T];H−1(R2)), since for any t∈(0,T],wehave
E∞
k=1t
0∇ ·(σk(x)us(x))2
L2ds=E∞
k=1t
0σk(x)·∇us(x)2
L2ds
≤∞
k=1σk2
L∞Et
0∇us2
L2ds<∞.
We are ready to state our main result.
Theorem 2.8. (Blow-up in finite time) Let χ, γ > 0and let u0∈P2(R2)∩L2(R2)
be such that xu0(x)dx=0. Assume σ={σk}k≥1satisfy (H1)-(H3). Let (u,¯
T)be
a weak solution to (1.1). Then
(i) Under the condition
χ>(1+γ)8π, (2.6)
we have
P(¯
T<T∗
1)=1,
for T∗
1:= 4πV[u0]
χ−(1+γ)8π.
(ii) Under the condition
χ>(1+γV[u0]Cσ)8π(2.7)
we have
P(¯
T<T∗
2)>0,
for T∗
2:= log(χ−8π)−log(χ−V[u0]8πγCσ−8π)
2γCσ.
Remark 2.9. •If V[u0]Cσ>1 and χsatisfies (2.7), then χalso satisfies (2.6), in
which case blow-up occurs almost surely before T∗
1. This has relevance to the
setting of [16] in which a model similar to (1.1) is considered on Tdfor d=2,3
where formally Cσcan be taken arbitrarily large.
57 Page 8 of 28 A. Mayorcas and M. Tomaševi´c J. Evol. Equ.
•In the case Cσ=0, which corresponds to noise that is independent of the
spatial variable, criterion (2.7) becomes χ>8πwhich is exactly the criterion
for blow-up of solutions to the deterministic PDE. Applying Theorem 2.8, one
would only recover blow-up with positive probability in this case. However,
using the spatial independence of the noise we can instead implement a change
of variables, setting v(t,x):= u(t,x−√2γσWt). It follows from the Leibniz
rule that vsolves a deterministic version of the PDE with viscosity equal to one.
Hence, it blows up in finite time with probability one for χ>8π. Note that
in [28] a similar model was treated, among others, with spatially homogeneous
noise and positive probability of blow-up was shown only for χ>(1+γ)8π.
•Observe that the second half of Theorem 2.8 demonstrates that (2.6) cannot be a
sharp threshold for almost sure global well-posedness of (1.1) for all initial data
(orallfamiliesofsuitablevectorfields{σk}k≥1). Given any8π<χ<(1+γ)8π,
initial data u0(resp. family of vector fields {σk}k≥1) one can always choose
suitable vector fields (resp. initial data) such that χ>(1+γV[u0]Cσ)8πso
that there is at least a positive probability that solutions cannot live for all time.
However, the results of this paper leave open any quantitative information on
this probability.
Remark 2.10. If we set T∗:= T∗
1∧T∗
2, then it is possible to show that T∗respects
the ordering of V[u0]Cσand 1. That is,
T∗=log(χ−8π)−log(χ−V[u0]8πγCσ−8π)
2γCσ,V[u0]Cσ<1,
4πV[u0]
χ−(1+γ)8π,V[u0]Cσ>1.
Asmentioned before, in thePDEcase blow-up occurs, forχ>8π, and weak solutions
cannot exist beyond T∗=4πV[u0]
χ−8π. It follows that in all parameter regions both the
thresholdforχand definitionof T∗in Theorem2.8agreewiththe equivalentquantities
in the limit γ→0.
Theproof ofTheorem2.8is completedinSect.4afterestablishingsomepreliminary
results in Sect.3. The central point is to analyze an SDE satisfied by t→ V[ut].
3. A priori properties of weak solutions
The following lemma demonstrates that the expression ∇ct:= ∇K∗utis well-
defined Lebesgue almost everywhere.
Lemma 3.1. Let (u,¯
T)be a weak solution to (1.1)in the sense of Definition 2.4.
Then, there exists a C >0such that for all t ∈(0,¯
T],
∇ctL∞≤Cut
1
4
L1ut
1
2
L2ut
1
4
H1.(3.1)
J. Evol. Equ. Blow-up for a stochastic model of chemotaxis driven Page 9 of 28 57
Proof. First, applying [29, Lem. 2.5] with q=3gives,
∇cuL∞u
1
4
L1u
3
4
L3.(3.2)
Interpolation between L2(R2)and L∞(R2)gives,
uL3≤u
2
3
L2u
1
3
L∞.(3.3)
Combined with the embedding H1(R2)→C0,0(R2)(see Lemma A.3), the required
estimate is obtained.
Remark 3.2. Note that the choice of q=3 in the proof of Lemma 3.1 and the resulting
exponents are essentially arbitrary, the only restriction being that a non-zero power of
utLpfor some p∈[1,2)must be included in the right-hand side. The choice of
L1is convenient since we will shortly demonstrate that d
dtutL1=0 for all weak
solutions.
Remark 3.3. Exploiting symmetries of the kernel K,(2.1) and following [36], we can
write the advection term of (2.3) in a different form that will become useful later on.
We note that,
us∇cs,∇φ=R4us(x)∇xK(x−y)·∇φ(x)us(y)dydx.(3.4)
Renaming the dummy variables in the double integral and applying Fubini’s theo-
rem, we also have
us∇cs,∇φ=R4us(y)∇yK(y−x)·∇φ(y)us(x)dydx.(3.5)
Combining (3.4) and (3.5)gives
us∇cs,∇φ
=1
2R4us(x)us(y)(∇xK(x−y)·∇φ(x)+∇yK(y−x)·∇φ(y)) dydx.
Therefore, in view of (2.1) we may re-write us∇cs,∇φas
us∇cs,∇φ=− 1
4πR4
(∇φ(x)−∇φ(y)) ·(x−y)
|x−y|2us(x)us(y)dydx(3.6)
In order to prove our main result, we will need to manipulate the zeroth, first, and
second moments of weak solutions. To do so, we define a family of radial, cut-off
functions, indexed by ε∈(0,1)such that for some C>0
ε(x)=1,for |x|<ε
−1,
0,for |x|>2ε−1,∇εL∞≤Cε, ∇2εL∞≤Cε2.(3.7)
57 Page 10 of 28 A. Mayorcas and M. Tomaševi´c J. Evol. Equ.
For any family of cut-off functions satisfying (3.7), it is straightforward to show that
there exists a C>0 such that
sup
x∈R2
ε∈(0,1)
|∇2(xε(x))|≤C,sup
x∈R2
ε∈(0,1)
|∇2(|x|2ε(x))|≤C.(3.8)
Note also that since supp(∇ε)=supp(ε)=B2ε−1(0)\Bε−1(0), then
∇εL2≤Cε1/2and εL2≤Cε3/2.(3.9)
We start with sign and mass preservation.
Proposition 3.4. Let (u,¯
T)a weak solution to (1.1).Ifu
0≥0, then P-a.s.
(i) ut≥0for all t ∈[0,¯
T),
(ii) utL1=u0L1=1for all t ∈[0,¯
T).
Proof. Let us define
S[ut]=ut(x)u−
t(x)dx=u−
t2
L2,
on L2(R2), where u−
t=ut1{ut<0}. The computations below can be properly justified
by first defining an H1approximation of the indicator function, obtaining uniform
bounds in the approximation parameter using that u∈H1and then passing to the
limit using dominated convergence. For ease of exposition, we work directly with
S[ut]keeping these considerations in mind so that the following calculations should
only be understood formally.
Applying (2.5)gives
S[ut]=S[u0]+2t
0{us<0}∇us(x)·(−∇us(x)+χ∇cs(x)us(x))dxds
+2γ∞
k=1t
0{us<0}
us(x)∇·(σkus(x)) dx◦dWk
s.
(3.10)
Regarding the stochastic integral term, using that ∇·σk=0, us∂{us<0}=0 and
integrating by parts, we have
{us<0}
us(x)∇·(σkus(x)) dx=−1
2{us<0}∇·(u2
s(x)σk)dx=0.
Regarding the finite variation integral,
t
0{us<0}∇us(x)·(−∇us(x)+χ∇cs(x)us(x)) dxds
=−t
0{us<0}|∇us(x)|2dx+χt
0{us<0}∇us(x)·us(x)∇cs(x)dx,
J. Evol. Equ. Blow-up for a stochastic model of chemotaxis driven Page 11 of 28 57
we apply Young’s inequality in the second term, to give
χ{us<0}∇us(x)·us(x)∇cs(x)dx≤χ
2ε{us<0}|∇us(x)|2dx
+∇cs2
L∞
εχ
2{us<0}|us(x)|2dx.
So choosing ε=χ
4,wehave
−{us<0}|∇us(x)|2dx+χ{us<0}∇us(x)·us(x)∇cs(x)dx
≤−
1
2{us<0}|∇us(x)|2dx+∇cs2
L∞χ2
8{us<0}|us(x)|2dx
Putting all this together in (3.10) and using that ∇us∈L2(R2)for almost every
s∈[0,¯
T],
S[ut]≤S[u0]+χ2
4t
0∇cs2
L∞S[us]ds.
So, having in mind Lemma 3.1 and applying Grönwall’s inequality, we almost surely
have
S[ut]≤S[u0]exp Cχ2
4u
1
4
L∞
¯
TL1u
3
4
L2
¯
TH1T,
where Cis the constant from Lemma 3.1. Since S[u0]=0, it follows that P-a.s.
S[ut]=0 for all t∈[0,¯
T)which shows the first claim.
To show the second claim, for ε∈(0,1), we define Mε[ut]:=R2ε(x)ut(x)dx,
where the cut-off functions εare given in (3.7). Using the weak form of the equation
and integrating by parts where necessary we see that
Mε[ut]=Mε[u0]+(1+γ)t
0R2ε(x)us(x)dxds
−χt
0R2∇ε(x)·∇cs(x)us(x)dxds
−2γ∞
k=1t
0R2∇ε(x)·(us(x)σk(x)) dxdWk
s.
(3.11)
Applying the Cauchy–Schwartz inequality, the fact that the Itô integral disappears
under the expectation and in view of (3.9), there exists a C>0 such that
ER2ε(x)ut(x)dx≤R2ε(x)u0(x)dx
+(1+γ)Cε3/2EuL∞
TL2+χCε1/2E∇cL2
TL∞uL∞
TL2.
57 Page 12 of 28 A. Mayorcas and M. Tomaševi´c J. Evol. Equ.
Applying Fatou’s lemma,
ER2ut(x)dx≤lim inf
ε→0ER2ε(x)ut(x)dx≤R2u0(x)dx.
Hence, R2ut(x)dx<∞P-a.s. for every t∈[0,¯
T). We may now apply dominated
convergence to each term in (3.11). In particular, stochastic dominated convergence is
used for the last term on the right-hand side. Thus, to obtain almost sure convergence
all the limits should be taken up to a suitable subsequence. Finally, noting that ε
and ∇εconverge to zero pointwise almost everywhere, we conclude
M[ut]=R2ut(x)dx=R2u0(x)dx.
In combination with the first statement of the lemma, this proves the second claim.
The following corollary to Proposition 3.4 will be crucial to obtaining our central
contradiction in the proof of Theorem 2.8.
Corollary 3.5. Let (u,¯
T)be a weak solution to (1.1). Then for any f :R2→Rsuch
that f >0Lebesgue almost everywhere and any t ∈[0,¯
T),
R2f(x)ut(x)dx>0P-a.s.
Proof. We first show that any weak solution must have positive support. Let us fix
an almost sure realization of the solution, then chose any t∈[0,¯
T)and assume for
a contradiction that, ut(ω) is supported on a set of zero measure. However, since
ut(ω)L1=1, we find that for any p>1,
1=R2ut(x,ω)dx≤R2|ut(x,ω)|pdx1
psupp(ut(ω))
1dxp−1
p
=0,
which is a contradiction. Since fis assumed to be strictly positive, Lebesgue almost
surely, the conclusion follows.
In the following proposition, we derive the evolution for the center of mass and the
variance of a weak solution to (1.1).
Proposition 3.6. Let us assume that u0∈L2(R2)∩P(R2)is such that V[u0]<∞.
Then for any weak solution to (1.1)in the sense of Definition 2.4,P-a.s. for any
t∈[0,¯
T),
C[ut]=−
2γ∞
k=1t
0R2σk(x)us(x)dxdWk
s,(3.12)
V[ut]=V[u0]+2(1+γ)−χ
4πt−1
2|C[ut]|2
−2γ
k≥1t
0R2x·σk(x)us(x)dxdWk
s(3.13)
J. Evol. Equ. Blow-up for a stochastic model of chemotaxis driven Page 13 of 28 57
Proof. Without loss of generality, we may assume that C[u0]=R2xu0(x)dx=
0. Indeed, given a non-centred initial condition ˜u0with C[˜u0]=c= 0 one may
redefine C[ut]:=R2(x−c)ut(x)dxwhose evolution along weak solutions to (1.1)
will again, using the argument given below, satisfy the identity (3.12). The rest of our
analysis therefore holds without further change.
Let p∈{1,2}and we use the convention that for p=2, xp:= |x|2. Since xpε(x)
is an H1(R2)function, we may apply (2.4) along with Remark 3.3 and integrate by
parts where necessary to give that
xpε(x)ut(x)dx
=R2xpε(x)u0(x)dx+(1+γ)t
0R2(xpε(x))us(x)dxds
−χ
4πt
0R4
(∇(xpε(x)) −∇(ypε(y))) ·(x−y)
|x−y|2us(x)us(y)dydxds
(3.14)
−2γ
k≥1t
0R2∇(xpε(x)) ·σk(x)us(x)dxdWk
s.
From (3.8), it follows that uniformly across x∈R2and ε∈(0,1),(xpε(x)) is
bounded and ∇(xpε(x)) is Lipschitz continuous. Hence, using that utL1=1for
all t∈[0,¯
T]there exists a C>0 such that, for all ε∈(0,1),
ER2xpε(x)ut(x)dx≤R2xpε(x)u0(x)dx+(1+γ)+χ
4πtC.
Note that we may directly apply Lebesgue’s dominated convergence to the initial data
term, since |xpε(x)u0(x)|≤|xpu0(x)|where the latter is assumed to be integrable.
Now, let us for the moment take only p=2. Applying Fatou’s lemma,
ER2|x|2ut(x)dx≤lim inf
ε→0ER2|x|2ε(x)ut(x)dx
≤R2|x|2u0(x)dx+(1+γ)+χ
4πtC <∞.
Hence, R2|x|2ut(x)dx<∞P-a.s. From Proposition 3.4,utis a probability measure
on R2, so we have the bound
R2|x|ut(x)dx≤R2|x|2ut(x)dx1/2
.
It follows that for p∈{1,2},R2xput(x)dx<∞P-a.s. Since by definition we also
have,
|xpε(x)ut(x)|≤|xput(x)|,
57 Page 14 of 28 A. Mayorcas and M. Tomaševi´c J. Evol. Equ.
as in the proof of Proposition 3.4, we may apply dominated convergence in each
integral of (3.14). Using that for p∈{1,2}and Lebesgue almost every x,y∈R2
lim
ε→0(xpε(x)) =0,
lim
ε→0∇(xpε(x)−ypε(y)) =0,if p=1,
2(x−y), if p=2,
lim
ε→0∇(xpε(x)) =1,if p=1,
2x,if p=2.
we directly find the claimed identities for C[ut]and 1
2R2|x|2ut(x)dx. To conclude
it only remains to note that
V[ut]=1
2R2|x|2ut(x)dx−1
2|C[ut]|2.
4. Proof of Theorem 2.8
We will prove both statements by demonstrating that in each case the a priori
properties of any weak solution proved in Proposition 3.4 will be violated at some
finite time, either almost surely or with positive probability. Furthermore, we will
make use of the identities shown in Proposition 3.6. Notice that our proofs of both of
these propositions rely heavily on assumptions (H1)–(H3).
To prove (i) let us assume that given u0∈P2(R2)∩L2(R2)and any associated weak
solution (u,¯
T), it holds that P(¯
T=∞)>0 and let us choose any ω∈{¯
T=∞}.We
may in addition assume that ωis a member of the full measure set where the solution
lies in CTL2(R2)∩L2
TH1(R2)for any T>0 and the set where the Itô integral is
well defined. Applying (3.13) of Proposition 3.6, for any 0 <t<∞and the above ω
we have that
V[ut](ω) =V[u0]+2(1+γ)−χ
4πt−1
2|C[ut]|2(ω)
−2γ
k≥1t
0R2x·σk(x)us(x)dxdWk
s(ω).
(4.1)
Now since the stochastic integral is by definition a local martingale, by the Dambis–
Dubin–Schwarz theorem [34, Ch. V. Thm. 1.6], there exists a random time change
t→ q(t), with q(t)being the quadratic variation of the local martingale at time t, and
a real-valued Brownian motion Bsuch that for all tin the range of q,
V[ut](ω) =V[u0]+2(1+γ)−χ
4πt−1
2|C[ut]|2(ω) −2γBq(t)(ω)(ω).
J. Evol. Equ. Blow-up for a stochastic model of chemotaxis driven Page 15 of 28 57
Either the range of qis [0,∞)or qis bounded. If ωis such that the first case holds,
then there exists a T>0 such that √2γBq(T)(ω)(ω) =V[u0], at which point, since
by assumption 2(γ +1)−χ
4πT−1
2|C[ut]|2(ω) < 0 for any T>0, we have
V[uT](ω) < 0.
The latter is in contradiction with Corollary 3.5. Alternatively, if ωis such that t→
q(t)(ω) is bounded, then there exists a B∞(ω) ∈Rsuch that limt∞ Bq(t)(ω)(ω) =
B∞(ω), see cite[Ch. 5, Prop. 1.8]. In which case, for all t>0,
V[ut](ω) ≤V[u0]+2(1+γ)−χ
4πt
−1
2|C[ut]|2(ω) +2γ|B∞|(ω) +|B∞−Bq(t)|(ω).
Since the final term vanishes for large t>0 and using again the fact that 2(1+γ)−
χ
4π<0 there exists a T>0 sufficiently large such that once more
V[uT](ω) < 0.
Again this contradicts Corollary 3.5 and as such our initial assumption that P(¯
T=
∞)>0 must be false. Hence, P(¯
T<∞)=1.
Finally, taking the expectation on both sides of (4.1) we see that for
t≥T∗
1:= 4πV[u0]
χ−(1+γ)8π
one has
E[V[ut]] ≤ 0.
Since V[ut]is non-negative, we must in addition have P(¯
T<T∗
1)=1.
To prove (ii) let us instead assume that given any suitable initial data, for the asso-
ciated weak solution one has P(¯
T=∞)=1. Now taking expectations on both sides
of (3.13), we have
E[V[ut]]=V[u0]+2(1+γ)−χ
4πt−1
2E|C[ut]|2.(4.2)
Using (3.12) and Itô’s isometry, we see that
E|C[ut]|2=2γ∞
k=1t
0
Eσk(x)us(x)dx
2ds
=2γt
0
E ∞
k=1
σk(x)·σk(y)us(y)us(x)dxdyds(4.3)
where we exchanged summation and expectation using dominated convergence and
recalling that us∈P(R2)P-a.s. for all s∈[0,T]and applying (H1).
57 Page 16 of 28 A. Mayorcas and M. Tomaševi´c J. Evol. Equ.
Now the game is to estimate ∞
k=1σk(x)·σk(y)from below, remembering that
ut≥0P-a.s. for all t∈[0,T]. From Remark 2.1,
∞
k=1|σk(x)−σk(y)|2≤Cσ|x−y|2,for all x,y∈R.
As a direct consequence and in view of (H3)-b)
Cσ|x−y|2≥∞
k=1|σk(x)−σk(y)|2=∞
k=1[|σk(x)|2−2σk(x)·σk(y)+|σk(y)|2]
=2Tr(Q(0)) −2∞
k=1
σk(x)·σk(y)
=4−2∞
k=1
σk(x)·σk(y).
Rearranging the above inequality gives
∞
k=1
σk(x)·σk(y)≥2−1
2Cσ|x−y|2.(4.4)
Plugging (4.4)into(4.3) and the fact that ut∈P(R2)P-a.s. for all t∈[0,T], we find
E[|C[ut]|2]≥4γt
0
E us(x)us(y)dxdyds
−Cσγt
0
E |x−y|2us(x)us(y)dxdyds
=4γt−Cσγt
0
E |x−y|2us(x)us(y)dxdyds.
The integrand in the second term can easily be rewritten as
|x−y|2us(y)us(x)dxdy=4V[us].
Thus, we establish the lower bound
E[|C[ut]|2]≥4γt−4γCσt
0
E[V[us]] ds.(4.5)
Therefore, inserting (4.5)into(4.2), we find
E[V[ut]] ≤ V[u0]+2−χ
4πt+2γCσt
0
E[V[us]] ds.
J. Evol. Equ. Blow-up for a stochastic model of chemotaxis driven Page 17 of 28 57
Applying Grönwall lemma,
E[V[ut]] ≤ V[u0]+2−χ
4πt+2γCσt
0V[u0]+2−χ
4πse(t−s)2γCσds.
Evaluating the integrals,
E[V[ut]] ≤ V[u0]− 1
2γCσχ
4π−2e2γCσt+1
2γCσχ
4π−2,
which implies that if χ>8π(1+γCσV[u0])and t≥T∗
2,for
T∗
2:= 1
2γCσ
(log(χ −8π)−log (χ−V[u0]8πγCσ−8π)) ,
then E[V[ut]] ≤ 0.
This is again in contradiction with Corollary 3.5 which shows P-a.s. positivity of V[ut]
for uaweaksolutionto(1.1). Hence, our initial assumption must have been false and
so for any weak solution the probability of global existence must be strictly less than
1. Furthermore, using again the fact that V[ut]is almost surely non-negative, we must
in fact have P(¯
T<T∗
2)>0.
Acknowledgements
The authors warmly thank J. Norris, A. de Bouard, and L. Galeati for helpful dis-
cussions and B. Hambly for useful comments on the manuscript. The authors would
like to express their gratitude to the French Centre National de Recherche Scien-
tifique (CNRS) for the grant (PEPS JCJC) that supported this project. M.T. was partly
supported by Fondation Mathématique Jacques Hadamard. Work on this paper was
undertaken during A.M.’s tenure as INI-Simons Post Doctoral Research Fellow hosted
by the Isaac Newton Institute for Mathematical Sciences (INI) participating in pro-
gramme Frontiers in Kinetic Theory, and by the Department of Pure Mathematics and
Mathematical Statistics (DPMMS) at the University of Cambridge. This author would
like to thank INI and DPMMS for support and hospitality during this fellowship,
which was supported by Simons Foundation (award ID 316017) and by Engineering
and Physical Sciences Research Council (EPSRC) Grant Number EP/R014604/1.
Funding Open Access funding enabled and organized by Projekt DEAL.
Data availability Data sharing is not applicable to this article as no datasets were
generated or analyzed during the current study.
Declarations
Conflict of interest The author’s have no competing financial or non-financial inter-
ests that relate to this work.
57 Page 18 of 28 A. Mayorcas and M. Tomaševi´c J. Evol. Equ.
Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License,
which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source, provide a link to the Creative
Commons licence, and indicate if changes were made. The images or other third party material in this
article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line
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is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission
directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/
by/4.0/.
Publisher’s Note Springer Natureremainsneutralwithregardtojurisdictionalclaims
in published maps and institutional affiliations.
Appendix
A Sobolev spaces on Rd
WeincludesomeusefuldefinitionsandlemmasconcerninginhomogeneousSobolev
spaces on R2.
Definition A.1. Let α∈R. The non-homogeneous Sobolev space Hα(R2)consists
of the tempered distributions u∈S(R2)such that ˆu∈L1
loc(R2)and
uHα:= F−1((1+|·|2)α/2ˆu)L2<∞,
where F−1denotes the inverse Fourier transform.
We recall that Hα(R2)is a Hilbert space for all α∈Rwith inner products,
u,vHα:= R2(1+|ξ|2)αˆu(ξ)ˆv(ξ) dξ.
Whenα=1,wehaveu,vH1=u,vL2+∇u,∇vL2.Itfollowsfromthedefinition
that for α0<α
1,
Hα1(R2)→Hα0(R2). (A.1)
We also recall the following interpolation and embedding results.
Lemma A.2. ([1, Prop. 1.32 & Prop.1.52]) For α0≤α≤α1, it holds that Hα0∩
Hα1→Hα. In particular, for all θ∈[0,1]and α=(1−θ)α0+θα1, one has
uHα≤u1−θ
Hα0uθ
Hα1.(A.2)
Lemma A.3. ([1, Thm. 1.66]) For α∈R, the space Hα(R2)embeds continuously
into
•The Lebesgue space L p(R2),if0<α<d/2and 2≤p≤4
2−2α;
•The Hölder space Ck,ρ(R2),ifα≥1+k+ρfor some k ∈Nand ρ∈[0,1).
J. Evol. Equ. Blow-up for a stochastic model of chemotaxis driven Page 19 of 28 57
B Stratonovich to Itô correction
We briefly detail the necessary calculations to justify Remark 2.5. We refer to [7,
Sec. 2.2], [13, Sec. 2], and [18, Sec. 2.3] for similar arguments. All equalities below
should be interpreted in the weak sense.
Lemma B.1. Let (u,¯
T)be a Stratonovich solution to (1.1), in the sense of Definition
2.4, with σsatisfying Assumptions (H1)–(H3). Then u also solves the Itô SPDE,
⎧
⎪
⎪
⎨
⎪
⎪
⎩
dut=((1+γ)ut+χ∇·(ut∇ct)) dt+√2γ∞
k=1∇·(σkut)dWk
t,
−ct=ut,
u|t=0=u0.
Proof. Repeating the caveat that all equalities should be understood after testing
against suitable test functions, for all k≥1 and making use of (H1) to ensure the
stochastic integrals are well defined,
t
0∇·(σkus)◦dWk
s=t
0∇·(σkus)dWk
s+1
2t
0∇·(σkd[u,Wk]s),
where the process s→[u,Wk]sdenotes the quadratic covariation between uand W.
Using (1.1), we find
[u,Wk]s=2γ∇·(σkus),
so that we have
2γt
0∇·(σkus)◦dWk
s
=2γt
0∇·(σkus)dWk
s+γt
0∇·(σk∇·(σkus)) ds.(B.1)
Summing over k≥1 and applying the Leibniz rule, we see that
∞
k=1∇·(σk(x)∇·(σk(x)us(x))) =
d
i,j=1
∂i∂j(Qij(0)us(x))
−∇·∞
k=1∇σk(x)·σk(x)us(x),
where Qij(0)=∞
k=1σi
k(x)σ j
k(x)for any x∈R2and ∇σk·σkis the vector field
with components,
(∇σk·σk)i=
d
j=1
(∂jσi
k)σ j
k.
57 Page 20 of 28 A. Mayorcas and M. Tomaševi´c J. Evol. Equ.
Applying the Leibniz rule once more, for j=1,...,d, we see that
∞
k=1
d
j=1
(∂jσi
k(x))σ j
k(x)=
d
j=1
∂jqij(x,x)−∞
k=1
σi
k(x)∇·σk(x).
By Assumptions (H2) and (H3), we have that ∇·σk=0 and qij(x,x)=Qij(0)=δij
from which it follows that ∞
k=1(∇σk·σk)i=0 for all i=1,...,dand that
d
i,j=1
∂i∂j(Qij(0)us(x)) =us,
which completes the proof.
C Local existence
In this section, we give a sketched proof of local existence of weak solutions to
(1.1). The method of proof is well known and can be found in a general form in [33].
In the case of (1.1), a similar proof of local existence was exhibited in [16, Prop. 3.6].
For the readers convenience, we supply here a lighter version adapted to our particular
setting.
Theorem C.1. Let u0∈L2(R2). Then there exists a pair (u,¯
T), with ¯
T deterministic,
which is a weak solution to (1.1)in the sense of Definition 2.4. Furthermore, P-a.s.
u∈C([0,¯
T];L2(R2)).
We begin with a local a priori bound on solutions to (1.1).
Lemma C.2. Let u0∈P(R2)∩L2(R2). Then there exists a ¯
T=¯
T(u0L2)>0
and a C >0, such that for any weak solution u to (1.1)on [0,¯
T],
sup
t∈[0,¯
T]ut2
L2+¯
T
0ut2
H1dt<C,P-a.s..(C.1)
Furthermore, it holds that
utL1=u0L1=1for all t ∈[0,¯
T). (C.2)
Remark C.3. Since the constant on the right-hand side of (C.1) is non-random, it
follows immediately that uL∞
¯
TL2+uL2
¯
TH1∈Lp(;R)for any p≥1.
Proof of Lemma C.2.The identity (C.2) is shown by Proposition 3.4 so that we are
only required to obtain (C.1).
By assumption, ut∈H1(R2)for all t∈[0,¯
T]and it satisfies (2.3). In particular, the
Stratonovich integral is well-defined for P-a.e. ω∈. Applying (2.5) to the functional
J. Evol. Equ. Blow-up for a stochastic model of chemotaxis driven Page 21 of 28 57
F[ut]:=ut2
L2, we have the identity,
ut2
L2=u02
L2−2t
0∇us2
L2ds+2χt
0∇us,us∇cs·∇usds
−2γ∞
k=1usσk,∇us◦dWk
t.
(C.3)
For the nonlinear term, integrating by parts and using the equation satisfied by c,
|us∇cs,∇us| = 1
2|∇cs,∇(u2
s)| = 1
2us3
L3.
Then using the Sobolev embedding H1/2(R2)→L3(R2), real interpolation as given
by Lemma A.2 and Young’s inequality, for any ε>0,
us3
H1/2≤us
3
2
H1us
3
2
L2≤3
4εus2
H1+ε
4us6
L2
≤3
4ε∇us2
L2+3
4εus2
L2+ε
4us6
L2.
Regarding the stochastic integral, since each σkis divergence free, it follows that,
|usσk,∇us| = 1
2|σk,∇(u2
s)| = 1
2|1,∇·(σku2
s)| = 0.
So, choosing ε=χ, we find that P-a.s.,
ut2
L2≤u02
L2−5
8t
0∇us2
L2ds+3
4t
0us2
L2ds+χ2
4t
0us6
L2ds.
(C.4)
That is t→utL2satisfies the nonlinear, locally Lipschitz, differential inequality,
d
dtutL2≤3
4utL2+χ2
4ut3
L2,P-a.s.
By standard ODE theory and recalling that u0is non-random, there exists a strictly
positive, but possibly finite time ¯
T(u0L2)and a deterministic constant C>0, such
that,
sup
t∈[0,¯
T]utL2≤C,P-a.s.
Coming back to (C.4) to obtain a bound on t
0∇us2
L2dsfor t≤¯
Tcompletes the
proof of (C.1).
Definition C.4. We say that a mapping A:H1(R2)→H−1(R2)is locally coercive,
locally weakly monotone and hemi-continuous if the following hold:
57 Page 22 of 28 A. Mayorcas and M. Tomaševi´c J. Evol. Equ.
Locally coercive: there exists an α>0 such that if u∈H1(R2)with uH1≤R
for any R>0 there exists a λ>0 for which it holds that
2A(u), u+αu2
H1≤λu2
L2.(C.5)
Locally weakly monotone: for any R>0 there exists a λ>0 such that for all
u,w∈H1(R2)with uH1∨wH1≤R
2A(u)−A(w), u−w≤λu−wL2+u−w2
L2.(C.6)
Hemi-continuous: for any u,w,v∈H1(R2)the mapping,
Rθ→A(u+θw),v∈R,(C.7)
is continuous.
Lemma C.5. The operator A :P(R2)∩H1(R2)→H−1(R2)given by the mapping,
A(u):= u−χ∇·(u∇c),
is locally coercive, locally weakly monotone and hemi-continuous.
Proof. Local Coercivity: Approximating uby smooth compactly supported functions
it follows that,
A(u), u=−∇uL2+χu∇c,∇u.
By Hölder’s inequality, Young’s inequality and Lemma 3.1, for any ε>0
|u∇c,∇u| ≤ uL2∇cL∞∇uL2≤1
2ε∇u2
L2+ε
2u2
L2∇u2
H1
So that under the assumption that uH1≤Rand choosing ε>0 sufficiently small,
there exist α, λ(R)>0 such that
2|A(u), u| ≤ −αu2
H1+λu2
L2.
Local Weak Monotonicity: Let us introduce the notation −cu=u, so that we have
A(u)−A(w), u−w=−∇(u−w)2
L2+χu∇cu−w∇cw,∇(u−w).
Applying Cauchy–Schwarz followed by the triangle inequality, Young’s product in-
equality and Hölder’s inequality give
A(u)−A(w), u−w≤−∇(u−w)2
L2+χ∇(u−w)L2u∇cu−wL2
+∇cw(u−w)L2
≤−1
2∇(u−w)2
L2+χ2u∇cu−w2
L2+χ2
2∇cw(u−w)2
L2
≤χ2u2
L2∇cu−w2
L∞+∇cw2
L∞u−w2
L2.
J. Evol. Equ. Blow-up for a stochastic model of chemotaxis driven Page 23 of 28 57
Making use of Lemma 3.1 and the assumptions that uL1∨wL1=1 and uH1∨
wH1≤R, we find the estimates
∇cu−w2
L∞u−w
1
2
L1u−w
1
2
H1u−wL2R1
2u−wL2
and
∇cw2
L∞w1/2
L1w1/2
H1wL2u−w2
L2≤R3
2.
Hence, again using the assumption uL2≤uH1≤R, we find
A(u)−A(w), u−wχ2R5
2u−wL2+R3
2u−w2
L2,
which proves the claim.
Hemi-continuity: Letting u,v,w∈H1(R2)and θ∈R,wehave
|A(u+θw)−A(u), v| ≤ θ|∇w, ∇v|+χ|(u+θw)∇cu+θw −u∇cu,∇v.
The first term directly converges to 0 as θ→0. For the second term, after applying
Hölder’s inequality we see that we are required to control
(u+θw)∇cu+θw −u∇cu2
L2≤θu∇cw2
L2+w∇cu2
L2+θ2w∇cw2
L2,
which again directly converges to 0 as θ→0.
Lemma C.6. For σ:= {σk}k≥1satisfying (H1) and divergence free, the mapping,
H1(R2)u→
k≥1∇·(σku)∈L2(R2),
is linear and strongly continuous.
Proof. Linearity is clear. Let u,w∈H1(R2), using the divergence free property of
the σk,
k≥1∇·(σk(u−w))
L2≤
k≥1σk·∇(u−w)L2≤σ2L∞u−wH1.
Proof. The strategy of proof is to first define a finite-dimensional approximation to
(1.1) using a Galerkin projection, we project the solution and the nonlinear term to
a finite-dimensional subspace of L2(R2). Using Lemma C.5 and the linearity of the
noise term, it follows that this finite-dimensional system has a global solution and
using the same arguments as in the proof of Lemma C.2, there is a non-trivial interval
[0,¯
T]on which we have uniform control on this solution. By Banach–Alaoglu, we
can extract a convergent subsequence, whose limit, u, will be our putative solution to
57 Page 24 of 28 A. Mayorcas and M. Tomaševi´c J. Evol. Equ.
(1.1). By linearity, the noise term converges so it will remain to show that Aconverges
along this subsequence to A(u)and that uis a solution in the sense of Definition 2.4.
For N≥1, let HN⊂L2(R2)denote the finite-dimensional subspace spanned by
the basis vectors {ek}|k|≤Nand N:L2(R2)→HNbe an orthogonal projection
such that NfL2≤fL2. Then we consider the finite-dimensional system of
Stratonovich SDEs,
duN
t=uN
t+χ∇·(N(uN
t∇cN
t))dt
+2γ∞
k=1
N(σk∇uN
t)◦dWk
t
uN
0=Nu0.
(C.8)
It follows from [33], Thm. 3.1.1 and Lemma C.5 that a unique, global solution exists
for all N≥1. Furthermore, for each N≥1, uNis a smooth solution to a truncated
version of (1.1) with smooth initial data and is such that for all t>0 it holds that
uN
tL1=uN
0L1=1. It is readily shown that
∇uN,N(uN∇cN)=∇uN,uN∇cN.
Hence, using the same arguments as in the proof of Lemma C.2, there exists a ¯
T∈
(0,∞)depending only on uN
0L2≤u0L2such that
sup
N≥1
Esup
t∈[0,¯
T]uN
t2
L2+¯
T
0uN
t2
H1dt<∞.
We can therefore apply the Banach–Alaoglu theorem, [5], Thm. 3.16 & Thm. 3.17, to
seethatthereexistsub-sequences{uk}k≥1,{A(uk)}k≥1,au∈L2(×[0,¯
T];H1(R2))
and a ξ∈L2( ×[0,¯
T];H−1(R2)) such that
uku∈L2( ×[0,¯
T];H1(R2))
A(uk)ξ∈L2( ×[0,¯
T];H−1(R2)).
It follows from the first and Lemma C.6 that the stochastic integrals converge so it
remains to show that ξ=A(u). From the local monotonicity of A, for any t∈(0,¯
T],
v∈L2( ×[0,¯
T];H1(R2)) and N≥1
Et
0A(uN
s)−A(vs), uN
s−vsds
λ
2Et
0uN
s−vsL2+uN
s−vs2
L2ds.(C.9)
Using the identity,
EuN
t2
L2−uN
02
L2=Et
0A(uN
s), uN
sds,
J. Evol. Equ. Blow-up for a stochastic model of chemotaxis driven Page 25 of 28 57
which can be proved directly using the chain rule for Stratonovich integrals and the
arguments of Lemma C.2, it is straightforward to show the inequality,
Et
0ξs,usds≤lim inf
N→∞
Et
0A(uN
s), uN
sds.
It follows, applying (C.9) in the final inequality, that for any v∈L2( ×[0,¯
T];
H1(R2)),
Et
0ξs−A(vs), us−vsds
≤lim inf
N→∞
Et
0A(uN
s)−A(vs), uN
s−vsds
λ
2lim inf
N→∞
Et
0uN
s−vsL2+uN
s−vs2
L2ds
Now, choosing v=u−θw for some θ>0 and w∈L2( ×[0,¯
T];H1(R2)) gives
that
Et
0ξs−A(us−θws), wsds≤θλ
2Et
0ws2
L2ds.
So applying (C.7) and taking θ→0, we finally find that,
Et
0ξs−A(us), wsds≤0,
for all w∈L2( ×[0,¯
T];H1(R2)) from which it follows that ξ=A(us)∈L2( ×
[0,¯
T];H−1(R2)).
It follows that u∈L2(;L∞([0,¯
T];L2(R2))) ∩L2( ×[0,¯
T];H1(R2)) and
satisfies (2.3). We now show that in fact, u∈L2(;C([0,¯
T];L2(R2)). To see this,
we recall that since L2(R2)is a Hilbert space, if utkut∈L2(R2), and utkL2→
utL2∈Rone has
utk−ut2
L2=ut−utk,ut−utk=ut2
L2−2ut,utk+utk2
L2→0.
From (C.3), it follows that given a sequence tk→t,utkL2→utL2. So it suffices
to show that utkut∈L2(R2).Leth∈L2(R2)be arbitrary, {hn}n≥1⊂H1(R2)
be a sequence converging to hstrongly in L2(R2)and ε>0, nε≥1 be large enough
such that,
sup
t∈[0,T]utL2|h−hnL2≤ε
2,for all n≥nε.
Therefore, we have
|ut,h−utk,h|2
L≤|ut,h−hnε|+|ut−utk,hnε|+|utk,h−hnε|
≤ε+ut−utkH−1hnεH1.
57 Page 26 of 28 A. Mayorcas and M. Tomaševi´c J. Evol. Equ.
By definition, for any weak solution utk→utstrongly in H−1(R2)and so conclude
lim sup
n≥nε|ut,h−utk,h| ≤ ε.
Since ε>0 was arbitrary, we may conclude utk→ut∈L2(R2)strongly. Further-
more, inspecting the proof we see that the modulus of continuity is deterministic and
hence u∈Lp(;C([0,¯
T];L2(R2))) for any p≥1.
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Avi Mayorcas
Institute of Mathematics
Technische Universität Berlin
Straße des 17. Juni, 135
10623 Berlin
Germany
E-mail: avimayor[email protected]
Milica Tomaševi´c
CMAP, CNRS, École polytechnique
Institut Polytechnique de Paris
91120 Palaiseau
France
E-mail: milica.tomasevic@polytechnique.edu
Accepted: 2 May 2023
