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.
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