Blow-up for a stochastic model of chemotaxis driven by conservative noise on R 2

. We establish criteria on the chemotactic sensitivity χ for the non-existence of global weak solutions (i


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 R 2 . The model we consider, on R + × R 2 , u| t=0 = u 0 ∈ P(R 2 ), on R 2 , 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 {W k } k≥1 is a family of i.i.d. standard Brownian motions on a filtered probability space, ( , F, (F t ) t≥0 , P), satisfying the usual assumptions. Here P(R 2 ) denotes the set of probability measures on R 2 . We will give detailed assumptions on the vector fields σ k : R 2 → R 2 below (see (H1)-(H3)), but for now simply stipulate that they are assumed to be divergence free and such that σ := {σ k } k≥1 ∈ 2 (Z; L ∞ (R 2 )).
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 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) on T d for d = 2, 3 enjoys delayed blow-up with 1 − ε after choosing γ and σ w.r.t. χ and 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 σ k (x) = σ and a single common Brownian motion. Translating their parameters into ours, they establish global well-posedness of solutions to (1.1), with σ k (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 )8π cannot 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 L p (R 2 ; R n ) for the spaces of p integrable (resp. essentially bounded) R n -valued functions on R 2 . For α ∈ R, we write H α (R 2 ; R n ) 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 C k (R 2 ; R n ) for the k continuously differentiable maps and C k,α (R 2 ; R n ) for the k continuously differentiable maps with α Hölder continuous k th derivatives. When the context is clear, we remove notation for the target space, simply writing L p (R 2 ), H α (R 2 ). We equip these spaces with the requisite norms writing · L p , · H α removing the domain as well when it will not cause confusion. • We write P(R 2 ) for the space of probability measures on R 2 and for m ≥ 1, P m (R 2 ) for the space of probability measures with m finite moments. By an abuse of notation we write, for example, P(R 2 ) ∩ L p (R 2 ) to indicate the space of probability measures with densities in L p (R 2 ). • For μ ∈ P(R 2 ) and when they are finite, we define the following quantities: Note that V [μ] is one half the usual variance, we define it in this way for computational ease.
• We write ∇ for the usual gradient operator on Euclidean space while for k ≥ 2, ∇ k denotes the matrix of k-fold derivatives. We denote the divergence operator by ∇· and we write := ∇ · ∇ for the Laplace operator. • If we write a b, 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. 3 we establish some important properties of weak solutions to (1.1) which are made use of in Sect. 4 where we prove our main theorem. Appendix A is devoted to a brief recap of the fractional Sobolev spaces on R 2 along 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.

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 c is uniquely defined up to a harmonic function, hence it can be written as c = K * u with K (x) = − 1 2π ln (|x|). Therefore, from now on, for t > 0, we work with the expression (2.1) Throughout we fix a complete, filtered, probability space, ( , F, (F t ) t≥0 , P), satisfying the usual assumptions and carrying a family of i.i.d Brownian motions {W k } k≥1 . Furthermore, we consider a family of vector fields σ := {σ k } k≥1 , satisfying the following assumptions.
(H1) For k ≥ 1, σ k : R 2 → R 2 are measurable and such that ∞ Remark 2.1. For σ satisfying Assumption (H3), it is possible to show that the quantity is finite. See [7,Rem. 4] for details. Note that due to (H3)-(b) one cannot re-scale σ so as to remove γ from (1.1).

Remark 2.2.
It is important to note that one can instead specify the covariance matrix Q first. In fact, due to [25,Thm. 4.2.5] any matrix-valued map Q : R 2 ×R 2 → R 2 ⊗R 2 satisfying the analogue of (2.2), can be expressed as a family of vector fields {σ k } k≥1 satisfying (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.
Then, we define the covariance function, Property (H3) (a) is satisfied by definition, after setting q(x, y) := Q(x − y). Since property (H3) (b) is easily checked by moving to polar coordinates, making use of elementary trigonometric identities and the normalization R + r f (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 u 0 ∈ P(R 2 ) ∩ L 2 (R 2 ), we say that a weak solution to (1.1) is a pair (u,T ) where In addition, for any t ∈ [0, T ], φ ∈ H 1 (R 2 ), P-a.s. the following identities hold, In Appendix C, we detail a standard argument to show that there exists a deterministic, 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.

4)
Remark 2.6. It follows from Definition 2.4 and the standard chain rule, obeyed by the Stratonovich integral, that for u a weak, Stratonovich solution to (1.1) and F ∈ ). 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 solution on [0, T ], the stochastic integral is well defined as an element of We are ready to state our main result.
we have Remark 2.9.
• If V [u 0 ]C σ > 1 and χ satisfies (2.7), then χ also 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 T d for d = 2, 3 where formally C σ can be taken arbitrarily large.
• In the case C σ = 0, which corresponds to noise that is independent of the spatial variable, criterion (2.7) becomes χ > 8π which 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, It follows from the Leibniz rule that v solves 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 (or all families of suitable vector fields {σ k } k≥1 ). Given any 8π < χ < (1+γ )8π , initial data u 0 (resp. family of vector fields {σ k } k≥1 ) one can always choose suitable vector fields (resp. initial data) such that so 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.
then it is possible to show that T * respects the ordering of V [u 0 ]C σ and 1. That is, As mentioned before, in the PDE case blow-up occurs, for χ > 8π , and weak solutions cannot exist beyond It follows that in all parameter regions both the threshold for χ and definition of T * in Theorem 2.8 agree with the equivalent quantities in the limit γ → 0.
The proof of Theorem 2.8 is completed in Sect. 4 after establishing some preliminary results in Sect. 3. The central point is to analyze an SDE satisfied by t → V [u t ].

A priori properties of weak solutions
The following lemma demonstrates that the expression ∇c t := ∇ K * u t is welldefined Lebesgue almost everywhere.
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 u t L p for some p ∈ [1, 2) must be included in the right-hand side. The choice of L 1 is convenient since we will shortly demonstrate that d dt u t L 1 = 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, Renaming the dummy variables in the double integral and applying Fubini's theorem, we also have Combining (3.4) and (3.5) gives Therefore, in view of (2.1) we may re-write u s ∇c s , ∇φ as 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 For any family of cut-off functions satisfying (3.7), it is straightforward to show that there exists a C > 0 such that We start with sign and mass preservation.
Proof. Let us define The computations below can be properly justified by first defining an H 1 approximation of the indicator function, obtaining uniform bounds in the approximation parameter using that u ∈ H 1 and then passing to the limit using dominated convergence. For ease of exposition, we work directly with S[u t ] keeping these considerations in mind so that the following calculations should only be understood formally.
Applying (2.5) gives (3.10) Regarding the stochastic integral term, using that ∇ · σ k = 0, u s ∂{u s <0} = 0 and integrating by parts, we have Regarding the finite variation integral, we apply Young's inequality in the second term, to give Putting all this together in (3.10) and using that ∇u s ∈ L 2 (R 2 ) for almost every s ∈ [0,T ], So, having in mind Lemma 3.1 and applying Grönwall's inequality, we almost surely have where C is the constant from Lemma 3.1. Since S[u 0 ] = 0, it follows that P-a.s. S[u t ] = 0 for all t ∈ [0,T ) which shows the first claim.
To show the second claim, for ε ∈ (0, 1), we define M ε [u t ] := R 2 ε (x)u t (x) dx, where the cut-off functions ε are given in (3.7). Using the weak form of the equation and integrating by parts where necessary we see that 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 Applying Fatou's lemma, Hence, R 2 u t (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 ∇ ε converge to zero pointwise almost everywhere, we conclude 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. (1.1). Then for any f : R 2 → R such that f > 0 Lebesgue almost everywhere and any t ∈ [0,T ),

Corollary 3.5. Let (u,T ) be a weak solution to
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, u t (ω) is supported on a set of zero measure. However, since u t (ω) L 1 = 1, we find that for any p > 1, which is a contradiction. Since f is 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 u
Then for any weak solution to (1.1) in the sense of Definition 2.4, P-a.s. for any t ∈ [0,T ), Proof. Without loss of generality, we may assume that C[u 0 ] = R 2 xu 0 (x) dx = 0. Indeed, given a non-centred initial conditionũ 0 with C[ũ 0 ] = c = 0 one may redefine C[u t ] := R 2 (x − c)u t (x) dx whose 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, x p := |x| 2 . Since x p ε (x) is an H 1 (R 2 ) function, we may apply (2.4) along with Remark 3.3 and integrate by parts where necessary to give that From (3.8), it follows that uniformly across x ∈ R 2 and ε ∈ (0, 1), (x p ε (x)) is bounded and ∇(x p ε (x)) is Lipschitz continuous. Hence, using that u t L 1 = 1 for all t ∈ [0,T ] there exists a C > 0 such that, for all ε ∈ (0, 1), Note that we may directly apply Lebesgue's dominated convergence to the initial data term, since |x p ε (x)u 0 (x)| ≤ |x p u 0 (x)| where the latter is assumed to be integrable. Now, let us for the moment take only p = 2. Applying Fatou's lemma, Hence, R 2 |x| 2 u t (x) dx < ∞ P-a.s. From Proposition 3.4, u t is a probability measure on R 2 , so we have the bound It follows that for p ∈ {1, 2}, R 2 x p u t (x) dx < ∞ P-a.s. Since by definition we also have, 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 ∈ R 2 lim ε→0 (x p ε (x)) = 0, we directly find the claimed identities for C[u t ] and 1 2 R 2 |x| 2 u t (x) dx. To conclude it only remains to note that

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 u 0 ∈ P 2 (R 2 )∩ L 2 (R 2 ) 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 ω is a member of the full measure set where the solution lies in C T L 2 (R 2 ) ∩ L 2 T H 1 (R 2 ) 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 Either the range of q is [0, ∞) or q is bounded. If ω is such that the first case holds, then there exists a T > 0 such that The latter is in contradiction with Corollary 3.5.
In which case, for all t > 0, 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 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 Since V [u t ] 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 associated weak solution one has P(T = ∞) = 1. Now taking expectations on both sides of (3.13), we have Using (3.12) and Itô's isometry, we see that where we exchanged summation and expectation using dominated convergence and recalling that u s ∈ P(R 2 ) P-a.s. for all s ∈ [0, T ] and applying (H1). Now the game is to estimate ∞ k=1 σ k (x) · σ k (y) from below, remembering that u t ≥ 0 P-a.s. for all t ∈ [0, T ]. From Remark 2.1, As a direct consequence and in view of (H3)-b)

Rearranging the above inequality gives
Plugging (4.4) into (4.3) and the fact that u t ∈ P(R 2 ) P-a.s. for all t ∈ [0, T ], we find The integrand in the second term can easily be rewritten as Thus, we establish the lower bound Therefore, inserting (4.5) into (4.2), we find Applying Grönwall lemma, Evaluating the integrals, This is again in contradiction with Corollary 3.5 which shows P-a.s. positivity of V [u t ] for u a weak solution to (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 [u t ] is almost surely non-negative, we must in fact have P(T < T * 2 ) > 0.

B Stratonovich to Itô correction
We briefly detail the necessary calculations to justify Remark 2.5. We refer to [ Let (u,T ) be a Stratonovich solution to (1.1), in the sense of Definition 2.4, with σ satisfying Assumptions (H1)-(H3). Then u also solves the Itô SPDE, 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, where the process s → [u, W k ] s denotes the quadratic covariation between u and W . Using (1.1), we find Summing over k ≥ 1 and applying the Leibniz rule, we see that for any x ∈ R 2 and ∇σ k · σ k is the vector field with components, Applying the Leibniz rule once more, for j = 1, . . . , d, we see that By Assumptions (H2) and (H3), we have that ∇ ·σ k = 0 and q i j (x, 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. Furthermore, it holds that Since the constant on the right-hand side of (C.1) is non-random, it follows immediately that u L ∞ T L 2 + u L 2T H 1 ∈ L p ( ; 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, u t ∈ H 1 (R 2 ) 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 F[u t ] := u t 2 L 2 , we have the identity, For the nonlinear term, integrating by parts and using the equation satisfied by c, Then using the Sobolev embedding H 1/2 (R 2 ) → L 3 (R 2 ), real interpolation as given by Lemma A.2 and Young's inequality, for any ε > 0, Regarding the stochastic integral, since each σ k is divergence free, it follows that, So, choosing ε = χ , we find that P-a.s., That is t → u t L 2 satisfies the nonlinear, locally Lipschitz, differential inequality, By standard ODE theory and recalling that u 0 is non-random, there exists a strictly positive, but possibly finite timeT ( u 0 L 2 ) and a deterministic constant C > 0, such that, Coming back to (C.4) to obtain a bound on t 0 ∇u s 2 L 2 ds for t ≤T completes the proof of (C.1).
Definition C.4. We say that a mapping A : H 1 (R 2 ) → H −1 (R 2 ) is locally coercive, locally weakly monotone and hemi-continuous if the following hold: Locally coercive: there exists an α > 0 such that if u ∈ H 1 (R 2 ) with u H 1 ≤ R for any R > 0 there exists a λ > 0 for which it holds that Locally weakly monotone: for any R > 0 there exists a λ > 0 such that for all Hemi-continuous: for any u, w, v ∈ H 1 (R 2 ) the mapping, is continuous.
given by the mapping, is locally coercive, locally weakly monotone and hemi-continuous.
Proof. Local Coercivity: Approximating u by smooth compactly supported functions it follows that, By Hölder's inequality, Young's inequality and Lemma 3.1, for any ε > 0 So that under the assumption that u H 1 ≤ R and choosing ε > 0 sufficiently small, there exist α, λ(R) > 0 such that Local Weak Monotonicity: Let us introduce the notation − c u = u, so that we have Applying Cauchy-Schwarz followed by the triangle inequality, Young's product inequality and Hölder's inequality give Making use of Lemma 3.1 and the assumptions that u L 1 ∨ w L 1 = 1 and u H 1 ∨ w H 1 ≤ R, we find the estimates Hence, again using the assumption u L 2 ≤ u H 1 ≤ R, we find which proves the claim. Hemi-continuity: Letting u, v, w ∈ H 1 (R 2 ) and θ ∈ R, we have 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 which again directly converges to 0 as θ → 0.
Proof. Linearity is clear. Let u, w ∈ H 1 (R 2 ), using the divergence free property of the σ k , 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 L 2 (R 2 ). 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 (1.1). By linearity, the noise term converges so it will remain to show that A converges along this subsequence to A(u) and that u is a solution in the sense of Definition 2.4. For N ≥ 1, let H N ⊂ L 2 (R 2 ) denote the finite-dimensional subspace spanned by the basis vectors {e k } |k| ≤ N and N : L 2 (R 2 ) → H N be an orthogonal projection such that N f L 2 ≤ f L 2 . Then we consider the finite-dimensional system of Stratonovich SDEs, (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, u N is 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 u N t L 1 = u N 0 L 1 = 1. It is readily shown that Hence, using the same arguments as in the proof of Lemma C.2, there exists aT ∈ (0, ∞) depending only on u N 0 L 2 ≤ u 0 L 2 such that 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 ∈ L 2 ( × [0,T ]; H 1 (R 2 )) and N ≥ 1 Using the identity, It follows that u ∈ L 2 ( ; L ∞ ([0,T ]; L 2 (R 2 ))) ∩ L 2 ( × [0,T ]; H 1 (R 2 )) and satisfies (2.3). We now show that in fact, u ∈ L 2 ( ; C([0,T ]; L 2 (R 2 )). To see this, we recall that since L 2 (R 2 ) is a Hilbert space, if u t k u t ∈ L 2 (R 2 ), and u t k L 2 → u t L 2 ∈ R one has From (C.3), it follows that given a sequence t k → t, u t k L 2 → u t L 2 . So it suffices to show that u t k u t ∈ L 2 (R 2 ). Let h ∈ L 2 (R 2 ) be arbitrary, {h n } n≥1 ⊂ H 1 (R 2 ) be a sequence converging to h strongly in L 2 (R 2 ) and ε > 0, n ε ≥ 1 be large enough such that, , for all n ≥ n ε .
Therefore, we have By definition, for any weak solution u t k → u t strongly in H −1 (R 2 ) and so conclude lim sup Since ε > 0 was arbitrary, we may conclude u t k → u t ∈ L 2 (R 2 ) strongly. Furthermore, inspecting the proof we see that the modulus of continuity is deterministic and hence u ∈ L p ( ; C([0,T ]; L 2 (R 2 ))) for any p ≥ 1.