Global exponential stabilisation for the Burgers equation with localised control

We consider the 1D viscous Burgers equation with a control localised in a finite interval. It is proved that, for any $\varepsilon>0$, one can find a time $T$ of order $\log\varepsilon^{-1}$ such that any initial state can be steered to the $\varepsilon$-neighbourhood of a given trajectory at time $T$. This property combined with an earlier result on local exact controllability shows that the Burgers equation is globally exactly controllable to trajectories in a finite time. We also prove that the approximate controllability to arbitrary targets does not hold even if we allow infinite time of control.


Introduction
Let us consider the controlled Burgers equation on the interval I = (0, 1) with the Dirichlet boundary condition: x u + u∂ x u = h(t, x) + ζ(t, x), (0.1) u(t, 0) = u(t, 1) = 0. (0.2) Here u = u(t, x) is an unknown function, ν > 0 is a parameter, h is a fixed function, and ζ is a control that is assumed to be localised in an interval [a, b] ⊂ I.As is known, the initial-boundary value problem for (0.1) is well posed.Namely, if h ∈ L 2 loc (R + , L 2 (I)) and ζ ≡ 0, then, for any u 0 ∈ L 2 (I), problem (0.1), (0.2) has a unique solution u(t, x) that belongs to the space X = {u ∈ L 2 loc (R + , H1 0 (I)) : ∂ t u ∈ L 2 loc (R + , H −1 (I))} and satisfies the initial condition u(0, x) = u 0 (x); (0.3) see the end of this Introduction for the definition of functional spaces.Let us denote by R t (u 0 , h) the mapping that takes the pair (u 0 , h) to the solution u(t) (with ζ ≡ 0).We wish to study the problem of controllability for (0.1).This question received great deal of attention in the last twenty years, and we now recall some achievements related to our paper.One of the first results was obtained by Fursikov and Imanuvilov [FI95,FI96].They established the following two properties: Local exact controllability.Let û(t, x) be a trajectory of (0.1), (0.2) with ζ ≡ 0 and let T > 0. Then there is ε > 0 such that, for any u 0 ∈ H 1 0 (I) satisfying the inequality u 0 − û(0) H 1 ≤ ε, one can find a control 1 ζ ∈ L 2 (J T × I) supported in J T × [a, b] for which R T (u 0 , h + ζ) = û(T ).Moreover, when T is fixed, the number ε can be chosen to be the same for all û(0) and h varying in bounded subsets of the spaces H 1 0 (I) and L 2 (J T × I), respectively.Absence of approximate controllability.For any u 0 ∈ L 2 (I) and any positive numbers T and R, one can find û ∈ L 2 (I) such that, for any control ζ ∈ L 2 (J T × I) supported by J T × [a, b], we have (0.4) These results were extended and developed in many works.In particular, Glass and Guererro [GG07] and [Léa12] proved global exact boundary controllability to constant states, Coron [Cor07b] and Fernández-Cara-Guererro [FCG07] established some estimates for the time and cost of control, and Chapouly [Cha09] (see also Marbach [Mar14]) proved global exact controllability to trajectories with two boundary and one distributed controls, provided that h ≡ 0. A large number of works were devoted to the investigation of similar question for other, more complicated equations of fluid mechanics; see the references in [Fur00,Cor07a].
In view of the above-mentioned properties, two natural questions arise: • does the exact controllability to trajectories hold for arbitrary initial conditions and nonzero right-hand sides ?
• does the approximate controllability hold if we allow a sufficiently large time of control ?
It turns out that the answer to the first question is positive, provided that the time of control is sufficiently large, whereas the answer to the second question is negative.Namely, the main results of this paper combined with the above-mentioned property of local exact controllability to trajectories imply the following theorem.(a) There exists T > 0 such that, for any u 0 , û0 ∈ L 2 (I) one can find a control For any positive numbers T 0 and R, one can find û ∈ L 2 (I) such that, for any Let us mention that the result about exact controllability to trajectories remain valid for a much larger class of scalar conservation laws in higher dimension.This question will be addressed in a subsequent publication.
The rest of the paper is organised as follows.In Section 1, we formulate a result on exponential stabilisation to trajectories, outline the scheme of its proof, and derive assertion (a) of the Main Theorem.Section 2 is devoted to some preliminaries about the Burgers equation.In Section 3, we present the details of the proof of exponential stabilisation and establish property (b) of the Main Theorem.Finally, the Appendix gathers the proofs of some auxiliary results.
L p (D) and H s (D) are the usual Lebesgue and Sobolev spaces, endowed with natural norms • L p and • s , respectively.In the case p = 2 (or s = 0), we write • and denote by (•, •) the corresponding scalar product.
2 See the Notation below for definition of the spaces used in the statement.
Very often, the context implies the domain on which a functional space is defined, and in this case we omit it from the notation.For instance, we write L 2 , H s , etc. L p (J, X) is the space of Borel-measurable functions f : J → X (where J ⊂ R is a closed interval and X is a separable Banach space) such that In the case p = ∞, this condition should be replaced by is the space of functions whose restriction to any bounded interval J ′ ⊂ J belongs to W k,p (J ′ , X). C(J, X) is the space of continuous functions f : J → X. B X (a, R) denotes the closed ball in X of radius R ≥ 0 centred at a ∈ X.In the case a = 0, we write B X (R).

Exponential stabilisation to trajectories
Let us consider problem (0.1), (0.2), in which ν > 0 is a fixed parameter, h(t, x) is a given function belonging to H 1 ul ∩ L ∞ on the domain I × R + , and ζ is a control taking values in the space of functions in L 2 (I) with support in a given interval [a, b] ⊂ I. Recall that R t (u 0 , h + ζ) stands for the value of the solution for (0.1)-(0.3) at time t.The following theorem is the main result of this paper.
Theorem 1.1.Under the above hypotheses, there exist positive numbers C and γ such that, given arbitrary initial data u 0 , û0 ∈ L 2 (I), one can find a piecewise continuous control Moreover, the control ζ regarded as a function of time may have discontinuities only at positive integers.
As was mentioned in the Introduction, this theorem combined with the Fursikov-Imanuvilov result on local exact controllability (see [FI96, Section I.6]) implies that the Burgers equation is exactly controllable to trajectories in a finite time independent of the initial data.Indeed, for any û0 ∈ L 2 (I) the trajectory û(t) = R t (û 0 , h) is bounded in H 1 0 (I) for t ≥ 1.In view of the local exact controllability, one can find , where v(t, x) stands for the solution of (0.1), (0.2) issued from v 0 at time t = T .Due to (1.1), there is T ε > 0 such that, for any u 0 , û0 ∈ L 2 (I), one can find a piecewise continuous control ζ : Applying the above result on local exact controllability to v 0 = R Tε (u 0 , h + ζ), we arrive at assertion (a) of the Main Theorem stated in the Introduction.
We now outline the main steps of the proof of Theorem 1.1, which is given in Section 3. It is based on a comparison principle for nonlinear parabolic equations and the Harnack inequality.
Step 1: Reduction to bounded regular initial data We first prove that it suffices to consider the case of H 2 -smooth initial conditions with norm bounded by a fixed constant.Namely, let V := H 1 0 ∩ H 2 , and given a number T > 0, let us define the functional space We have the following result providing a universal bound for solutions of (0.1), (0.2) at any positive time.
Proposition 1.2.Let h ∈ (H 1 ∩ L ∞ )(J T × I) for some T > 0 and let ν > 0. Then there is R > 0 such that any solution u ∈ X T of (0.1) with ζ ≡ 0 satisfies the inclusion u(t) ∈ V for 0 < t ≤ T and the inequality where R is the constant in Proposition 1.2 with T = 1.Furthermore, in view of the contraction of the L 1 -norm for the difference of two solutions (cf.Proposition 2.5 below), we have Hence, to prove Theorem 1.1, it suffices to establish the inequality in (1.1) for t ≥ 0 and any initial data u 0 , û0 ∈ B V (R).
Step 2: Interpolation where C 1 , C 2 , and α are positive numbers not depending on u 0 , û0 , and t.In this case, using the interpolation inequality (see Section 15.1 in [BIN79]) we can write where γ = 2α 5 , and C 4 > 0 does not depend on u 0 , û0 , and t.This implies the required inequality for the first term on the left-hand side of (1.1).An estimate for the second term will follow from the construction; see relations (1.13) and (1.14) below.
Step 3: Main auxiliary result Let us take two initial data v 0 , û0 ∈ B V (R) and consider the difference w between the corresponding solutions of problem (0.1) where a = 1 2 (v + û).The following proposition is the key point of our construction.Proposition 1.3.Let positive numbers ν, T , ρ, and s < 1 be fixed, and let a(t, x) be a function such that (1.9) Then, for any closed interval I ′ ⊂ I, there are positive numbers ε and q < 1, depending only on ν, T , ρ, s, and I ′ , such that any solution w ∈ X T of Eq. (1.8) satisfies one of the inequalities In other words, for the difference of any two solutions, either the L 1 -norm undergoes a strict contraction or a non-trivial mass is concentrated on I ′ .In both cases, we can modify the difference between the reference and uncontrolled solutions in the neighbourhood of I ′ so that the resulting function is a solution to the controlled problem, and the L 1 -norm of the difference decreases exponentially with time.We now describe this idea in more detail.
Step 4: Description of the controlled solution Let us fix a closed interval I ′ ⊂ (a, b) and choose two functions , we denote by û(t, x) the reference trajectory and define a controlled solution u(t, x) of (0.1) consecutively on intervals [k, k + 1] with k ∈ Z + by the following rules: Moreover, it will follow from Proposition 1.3 that, for any even integer k ≥ 0, we have where θ < 1 does not depend on û0 , u 0 , and k.On the other hand, the contraction of the L 1 -norm between solutions of (0.1) implies that where [t] stands for the largest integer not exceeding t.These two inequalities give (1.5).The uniform bounds (1.4) for the H 2 -norm will follow from regularity of solutions for problem (0.1), (0.2).

Preliminaries on the Burgers equation
In this section, we establish some properties of the Burgers equation.They are well known, and their proofs can be found in the literature in more complicated situations.However, for the reader's convenience, we outline some of those proofs in the Appendix to make the presentation self-contained.In this section, when talking about Eq. (0.1), we always assume that ζ ≡ 0.

Maximum principle and regularity of solutions
In this subsection, we discuss the well-posedness of the initial-boundary value problem for the Burgers equation.This type of results are very well known, and we only outline their proofs in the Appendix.Recall that V = H 1 0 ∩ H 2 , and the space X was defined in the Introduction.
Let us note that, if u 0 is only in the space L 2 (I), then the conclusions about the L ∞ bound and the regularity remain valid on the half-line R τ := [τ, +∞) for any τ > 0. To see this, it suffices to remark that any solution u ∈ X of (0.1), (0.2) satisfies the inclusion u(τ ) ∈ H 1 0 ∩ H 2 for almost every τ > 0. For any such τ > 0, one can apply Proposition 2.1 to the half-line R τ and conclude that the inclusions mentioned there are true with R + replaced by R τ .

Comparison principle
The Burgers equation possesses a very strong dissipation property due to the nonlinear term.To state and prove the corresponding result, we need the concept of sub-and super-solution for Eq.(0.1) with ζ ≡ 0. Let us fix T > 0 and, given an interval ) is an arbitrary non-negative function.The concept of a sub-solution is defined similarly, replacing ≥ by ≤.
A proof of the following result can be found in Section 2.2 of [AL83] for a more general problem; for the reader's convenience, we outline it in the Appendix.
Proposition 2.3.Let h ∈ L 1 (J T , L 2 ), and let functions u + and u − belonging to X T (I ′ ) be, respectively, super-and sub-solutions for (0.1) such that4 where the inequality holds almost everywhere.Then, for any t ∈ J T , we have (2.4) We now derive an a priori estimate for solutions of (0.1), (0.2).
Corollary 2.4.Let u 0 ∈ L ∞ and h ∈ L ∞ (J T × I) for some T > 0. Then the solution of problem (0.1)-(0.3) with ζ ≡ 0 satisfies the inequality where C > 0 is a number continuously depending only on h L ∞ and T .
Proof.We follow the argument used in the proof of Lemma 9 in [Cor07b, Section 2.1].Given ε > 0 and u 0 ∈ L ∞ (I), we set It is a matter of a simple calculation to check that the functions are, respectively, super-and sub-solutions for (0.1) on the interval J T such that x) for t = 0, x ∈ I and t ∈ [0, T ], x = 0 or 1. Applying Proposition 2.3, we conclude that x) for a.e.x ∈ I. Passing to the limit as ε → 0 + , we arrive at (2.5) with C = T −1 B 0 (B 0 + 1).

Contraction of the L 1 -norm of the difference of solutions
It is a well known fact that the resolving operator for (0.1), (0.2) regarded as a nonlinear mapping in the space L 2 (I) is locally Lipschitz.The following result shows that it is a contraction for the norm of L 1 (I).
Proposition 2.5.Let u, v ∈ X be two solutions of Eq. (0.1), in which ζ ≡ 0 and (2.6) Inequality (2.6) follows from the maximum principle for linear parabolic PDE's, and more general results can be found in Sections 3.2 and 3.3 of [Hör97].A simple proof of Proposition 2.5 is given in Section 4.3.

Harnack inequality
Let us consider the linear homogeneous equation (1.8).The following result is a particular case of the Harnack inequality established in [KS80, Theorem 1.1] (see also Section IV.2 in [Kry87]).
Proposition 2.6.Let a closed interval K ⊂ I and positive numbers ν and T be fixed.Then, for any ρ > 0 and T ′ ∈ (0, T ), one can find C > 0 such that the following property holds: if a(t, x) satisfies the inequality (2.7) (2.8)

Proof of the main results
In this section, we give the details of the proof of Theorem 1.1 (Sections 3.1-3.3)and establish assertion (b) of the Main Theorem stated in the Introduction (Section 3.4).

Reduction to smooth initial data
Let us prove Proposition 1.2.Fix arbitrary numbers T 1 < T 2 in the interval (0, T ).By Proposition 2.1 and the remark following it, for any τ > 0 we have where J τ,T = [τ, T ].Applying Corollary 2.4, we see that Furthermore, it follows from (3.1) that u(t) is a continuous function of t ∈ (0, T ] with range in V .Thus, it remains to establish inequality (1.3) with a universal constant R.
The proof of this fact can be carried out by a standard argument based on multipliers technique (e.g., see the proof of Theorem 2 in [BV92, Section I.6] dealing with the 2D Navier-Stokes system).Therefore, we confine ourselves to outlining the main steps.
Until the end of this subsection, we deal with Eq. (0.1) in which ζ ≡ 0 and denote by C i unessential positive numbers not depending u.
Step 1: Mean H 1 -norm.Taking the scalar product of (0.1) with 2u and performing usual transformations, we derive Integrating in time and using (3.2) with t = T 1 , we obtain (3.3) Step 2: H 1 -norm and mean H 2 -norm.Let us take the scalar product of (0.1) with −2(t − T 1 )∂ 2 x u: Integrating in time and using (3.2) and (3.3), we obtain (3.4) Using (0.1), we also derive the following estimate for ∂ t u: Step 3: L 2 -norm of the time derivative.Taking the time derivative of (0.1), we obtain the following equation for v = ∂ t u: Taking the scalar product with 2(t − T 2 )v, we derive Integrating in time and using (3.2) and (3.5), we obtain Step 4: H 2 -norm.We now rewrite (0.1) in the form In view of (3.4) and (3.6), we have f (T ) ≤ C 5 .Combining this with (3.7), we arrive at the required inequality (1.3).
Remark 3.1.The argument given above shows that, under the hypotheses of Proposition 1.2, if u 0 ∈ B V (ρ), then R t (u 0 , h) 2 ≤ R for all t ≥ 0, where R > 0 depends only on h, ν, and ρ.Moreover, similar calculations enable one to prove that, for any t > 0, the resolving operator R t (u 0 , h) regarded as a function of u 0 is uniformly Lipschitz continuous from any ball of L 2 to H 2 , and the corresponding Lipschitz constant can be chosen to be the same for T −1 ≤ t ≤ T , where T > 1 is an arbitrary number.

Proof of the main auxiliary result
In this subsection, we prove Proposition 1.3.In doing so, we fix parameter ν > 0 and do not follow the dependence of various quantities on it.
Step 1.We begin with the case of non-negative solutions.Namely, we prove that, given q ∈ (0, 1), one can find δ = δ(I ′ , T, q, ρ) > 0 such that, if w ∈ X T is a non-negative solution of (1.8), then either the first inequality in (1.10) holds, or inf x∈I ′ w(T, x) ≥ δ w(0) L 1 . (3.8) To this end, we shall need the following lemma, established at the end of this subsection.
Lemma 3.2.For any 0 < τ < T and ρ > 0, there is M > 0 such that, if w ∈ X T is a solution of Eq. (1.8) with a function a(t, x) satisfying (1.9), then In view of linearity, we can assume without loss of generality that w(0 where |Γ| denotes the Lebesgue measure of a set Γ ⊂ R, and M > 0 is the constant in (3.9) with τ = 2T /3.By Proposition 2.1 and the remark following it, the function w satisfies the hypotheses of Proposition 2.6.Therefore, by the Harnack inequality (2.8), we have sup where C > 0 depends only on T , K, and ρ.Let us set δ = q 2C|K| and suppose that (3.8) is not satisfied.In this case, using (3.9)-(3.11)and the contraction of the L 1 -norm of solutions for (1.8) (see Remark 4.2), we derive This is the first inequality in (1.10) with w(0) L 1 = 1.
Step 2. We now consider the case of arbitrary solutions w ∈ X T , assuming again that w(0) L 1 = 1.Let us denote by w + 0 and w − 0 the positive and negative parts of w 0 := w(0), and let w + and w − be the solutions of (1.8) issued from w + 0 and w − 0 , respectively.Thus, we have Let us set r := w + 0 L 1 and assume without loss of generality that r ≥ 1/2.In view of the maximum principle for linear parabolic equations (see Section 3.2 in [Lan98]), the functions w + and w − are non-negative, and therefore the property established in Step 1 is true for them.If w + (T ) L 1 ≤ r/2, then the contraction of the L 1 -norm of solutions of (1.8) implies that This coincides with the first inequality in (1.10) with w(0) L 1 = 1.
Suppose now that w + (T ) L 1 > r/2.Using the property of Step 1 with q = 1 2 , we find δ 1 > 0 such that inf x∈Q w + (T, x) ≥ δ 1 r. (3.12) Set ε = 1 4 δ 1 |I ′ | and assume that w(T ) L 1 (I ′ ) < ε (in the opposite case, the second inequality in (1.10) holds), so that By the L 1 -contraction for w − , we see that Repeating the argument applied above to w + , we can prove that if , so that the first inequality in (1.10) holds with q = 1− ε 2 .Thus, it remains to consider the case when (3.13) does not hold.Applying the property of Step 1 to w − , we find δ 2 > 0 such that (3.14) Since 1 2 ≤ r ≤ 1 − ε, the right-hand sides in (3.12) and (3.14) are minorised by θ = min{ 1 2 δ 1 , εδ 2 }.Denoting by χ I ′ the indicator function of I ′ , we write In view of the L 1 -contraction for w + and w − , the right-hand side of this inequality does not exceed we conclude that one of the inequalities (1.10) holds for w.Thus, to complete the proof of Proposition 1.3, it only remains to establish Lemma 3.2.
Proof of Lemma 3.2.By the maximum principle and regularity of solutions for linear parabolic equations, it suffices to prove that where C 1 > 0 does not depend on w.To this end, along with (1.8), let us consider the dual equation supplemented with the initial condition z(T, x) = z 0 (x). (3.17) Let us denote by G(t, x, y) the Green function of the Dirichlet problem for (3.16), (3.17).By Theorem 16.3 in [LSU68, Chapter IV], one can find positive numbers C 2 and C 3 depending only on ρ, s, and T such that It follows that, for z 0 ∈ L 2 (I), the solution z ∈ X T of problem (3.16), (3.17) satisfies the inequality where C 4 > 0 does not depend on z 0 .Now let u ∈ X T be a solution of (1.8).Taking any z 0 ∈ L 2 (I) and denoting by z ∈ X T the solution of (3.16), (3.17), we write where (•, •) denotes the scalar product in L 2 (I).Integrating in time and using (3.18), we obtain Taking the supremum over all z 0 ∈ L 2 with z 0 L 1 ≤ 1, we arrive at the required inequality (3.15).

Completion of the proof
We need to prove inequalities (1.4) and (1.5), as well as the piecewise continuity of ζ : R + → H 1 (I) and the estimate Proof of (1.4).The estimate for û(t) = R t (û 0 , h) follows from Remark 3.1.Setting t k = 2k, we now use induction on k ≥ 0 to prove that u(t) = R t (u 0 , h + ζ) is bounded on [t k , t k+1 ] by a universal constant and that u(t k+1 ) ∈ B V (R), provided that u(t k ) ∈ B V (R).Indeed, it follows from (1.14) that sup where s k = 2k + 1.In view of Remark 3.1, the right-hand side of this inequality does not exceed a constant C 3 (R).Furthermore, recalling (1.13) and using Remark 3.1 and inequality (1.3) with T = 1, we see that sup This completes the induction step.
Proof of (1.5).In view of (1.16), it suffices to establish (1.15) for any even integer k ≥ 0. It follows from (1.14), (1.12), and the definition of χ that (3.21) We know that the norms of the functions v and û are bounded in L ∞ ([k, k + 1], H 2 ) by a constant depending only on R. Since they satisfy Eq. (0.1) with ζ ≡ 0, we see that ) by a number depending on R. By interpolation and the continuous embedding H 1 (I) ⊂ C 1/2 (I), we see that Since the difference w = v − û satisfies Eq. (1.8) with a = 1 2 (v + û), we conclude that Proposition 1.3 is applicable to w.Thus, we have one of the inequalities (1.10).If the first of them is true, then it follows from (3.21) that (1.15) holds with θ = q.If the second inequality is true, then using (3.21), the contraction of the L 1 -norm for w, and relations (1.11), we derive and, hence, we obtain (1.15) with θ = 1 − ε.
Proof of the properties of ζ.In view of (1.13), on any interval [k, k + 1] with odd k ≥ 0, the function u satisfies (0.1) with ζ ≡ 0, and the required properties of ζ are trivial.Let us consider the case of an even k ≥ 0. A direct calculation show that . By Proposition 2.1, v and û are V -valued continuous functions, whence we conclude that ζ is continuous in time with range in H 1 0 .Moreover, since the H 2 -norms of v and û are bounded by a number depending only on R, for t ∈ [k, k + 1] we have where

Absence of global approximate controllability
We shall prove that if u(t, x) is a solution of (0.1), (0.2) on the interval [0, T ] with some control ζ ∈ L 2 (J T × I) supported by J T × [a, b], then the restriction of u(T, •) to any closed interval included in [0, a) satisfies an a priori estimate in the L ∞ norm independent of u 0 and ζ.Namely, we claim that, for any positive numbers T 0 and δ < a, there is ρ > 0 such that, if T ≥ T 0 , u 0 ∈ L 2 (I), and where If this is proved, then for any R > 0 we can take û ∈ L 2 (I) such that û(x) ≥ ρ + δ 1/2 R for x ∈ (0, δ), and it is straightforward to check that We now prove (3.23).In view of the regularising property of the resolving operator (see Proposition 2.1 and the remark following it), there is no loss of generality in assuming that u 0 ∈ V .In this case, if ζ ∈ L 2 (J T × I), then u(t, x) is continuous on J T × Ī.Given ε ∈ (0, 1), we fix a number A ε (which will be chosen below) and define the function We claim that, for an appropriate choice of A ε , the function u ε is a super-solution for (0.1) in the domain J T × K a .Indeed, let where Λ > 0 is a large parameter that will be chosen below.For x ∈ K a and t ∈ J T , we have Furthermore, a simple calculation shows that provided that Λ ≥ 4ν(T + 1) + 2(a + 1) 2 (3.26)It follows from (3.25) that if then u ε is a super-solution for (0.1) on the domain J T × K a .Inequalities (3.26) and (3.27) will be satisfied if we choose Λ = C(T + 1)( h 1/2 L ∞ + 1), where C > 0 is sufficiently large and depends only on a and ν.Recalling the definition of u ε , we see that the function is a super-solution for (0.1) on the domain J T × K a .It follows from (3.24) that Proposition 2.3 is applicable to the pair (u ε , u).In particular, we can conclude that u(T, x) ≤ u ε (T, x) for x ∈ K δ .Passing to the limit as ε → 0, we obtain This implies the required inequality (3.23) in which We have thus established assertion (b) of the Main Theorem of the Introduction.
4 Appendix: proofs of some auxiliary assertions 4.1 Proof of Proposition 2.1 The existence and uniqueness of a solution u ∈ X is well known in more complicated situations; see Chapter 15 in [Tay97].We thus confine ourselves to outlining the proofs of the L ∞ bound and regularity.The solution u(t, x) of (0.1), (0.2) can be regarded as the solution of the linear parabolic equation where b ∈ L 2 loc (R + , H 1 0 ) coincides with u.If b, h, and u 0 were regular functions, then the classical maximum principle would imply that (see Section 3.2 in [Lan98]) To deal with the general case, it suffices to approximate u 0 and h by smooth functions and to pass to the (weak) limit in inequality (4.2) written for approximate solutions.This argument shows that the inequality in (4.2) is valid almost everywhere for any solution u.
We now turn to the regularity of solutions.The function u ∈ X is the solution of the linear equation , where the right-hand side f = h − u∂ x u belongs to L 2 loc (R + , L 2 ).By standard estimates for the heat equation, we see that Differentiating (0.1) with respect to time and setting v = ∂ t u, we see that v satisfies the equations where Taking the scalar product of the first equation in (4.4) and carrying out some simple transformations, we conclude that v ∈ X .On the other hand, it follows from (0.1) that whence we see that u ∈ L 2 loc (R + , H 3 ).Combining this with the inclusion ∂ t u ∈ X , we obtain (2.1).

Proof of Proposition 2.3
Without loss of generality, we can assume that t = T .Define u = u − − u + , ψ δ (z) = 1 ∧ (z/δ) ∨ 0 , where δ > 0 is a small parameter, and a ∧ b (a ∨ b) denotes the minimum (respectively, maximum) of the real numbers a and b.In view of inequality (2.2) and its analogue for sub-solutions, the function u is non-positive almost everywhere for t = 0 and satisfies the inequality where w = (u − ) 2 − (u + ) 2 , and ϕ ∈ L ∞ (J T , L 2 ) ∩ L 2 (J T , H 1 0 ) is an arbitrary nonnegative function.Let us take ϕ(t, x) = ψ δ (u(t, x)) in (4.5).It is easy to check that where we used the fact that 0 ≤ u ≤ δ on the support of ψ ′ δ (u).Passing to the limit as δ → 0 + , we derive I u(T ) ∨ 0 dx ≤ 0. This inequality implies that u(T, x) ≤ 0 for a.e.x ∈ I, which is equivalent to (2.4).

Proof of Proposition 2.5
We apply an argument similar to that used in the proof of Lemma 3.2; see Section 3.2.Let us note that the difference w = u − v ∈ X satisfies the linear equation (1.8), in which a = 1 2 (u + v).Along with (1.8), let us consider the dual equation (3.16).The following result is a particular case of the classical maximum principle.Its proof is given in Section III.2 of [Lan98] for regular functions a(t, x) and can be obtained by a simple approximation argument in the general case.
Lemma 4.1.Let a ∈ L 2 (J T , H 1 ) for some T > 0.Then, for any z 0 ∈ L 2 (I), problem (3.16), (3.17) has a unique solution z ∈ X T .Moreover, if z 0 ∈ L ∞ (I), then z(t) belongs to L ∞ (I) for any t ∈ J T and satisfies the inequality z(t) L ∞ ≤ z 0 L ∞ for t ∈ J T . (4.6) To prove (2.6), we fix t = T and assume without loss of generality that s = 0.By duality, it suffices to show that, for any z 0 ∈ L ∞ (I) with norm z 0 L ∞ ≤ 1, we have I w(T )z 0 dx ≤ w(0) L 1 (4.7) Let z ∈ X T be the solution of (3.16), (3.17).Such solution exists in view of Lemma 4.1 and the inclusion a ∈ L 2 (J T , H 1 0 ), which is ensured by the regularity hypothesis for u Using (4.6) with t = 0, we arrive at the required inequality (4.7).
Remark 4.2.We have proved in fact that if w ∈ X T is a solution of the linear equation (1.8), in which the coefficient a belongs L 2 (J T , H 1 ) , then w(t) L 1 ≤ w(s) L 1 for 0 ≤ s ≤ t ≤ T .
and [a, b] ⊂ I. Then the following two assertions hold.
is the indicator function of the interval [k + 1/2, k], and we used the fact that the resolving operator for the Burgers equation is uniformly Lipschitz continuous from any ball of H 1 0 to H 2 for positive times; see Remark 3.1.Since v(k) − û(k) = u(k) − û(k), it follows from (1.5) and (3.22) that (3.20) holds.This completes the proof of Theorem 1.1.