Introuction Burgers' equation is a natural rst step towars eveloping methos for control of ows. Recent references by Burns an Kang [3], Byrnes, Gillia

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1 Regularity of Solutions of Burgers' Equation with Globally Stabilizing Nonlinear Bounary Feeback Anras Balogh an iroslav Krstic Department of AES University of California at San Diego La Jolla, CA fax: Submitte to the SIA J. Control an Optimization, August, 998 Revise: January, 999. Abstract We consier the viscous Burgers equation uner recently propose nonlinear bounary conitions which guarantee global asymptotic stabilization an semiglobal exponential stabilization in H sense. We show global existence an uniqueness of classical solutions with initial ata which are assume to be only in L. To o this, we establish a priori estimates of up to four spatial an two temporal erivatives, an then employ the Banach xe point theorem to the integral representation with a heat kernel. Our result is global in time an allows arbitrary size of initial ata. It strengthens recent results by Byrnes, Gilliam, an Shubov, Ly, ease, an Titi, an Ito an Yan. We inclue a numerical result which illustrates the performance of the bounary controller. Key wors. Burgers' equation, nonlinear bounary feeback, global stability, regularity AS Subject Classication. 93D, 35B35, 35Q53 This work was supporte by grants from the Air Force Oce of Scientic Research, the National Science Founation an the Oce of Naval Research.

2 Introuction Burgers' equation is a natural rst step towars eveloping methos for control of ows. Recent references by Burns an Kang [3], Byrnes, Gilliam, an Shubov [5, 6], Ly, ease, an Titi [7], an Ito an Yan [] achieve progress in local stabilization an global analysis of attractors. The problem of global exponential stabilization was rst aresse by Krstic [3]. This problem is non-trivial because for large initial conitions the quaratic (convective) term which is negligible in a linear/local analysis ominates the ynamics. Linear bounary conitions o not always ensure global exponential stability [6] or prevent nite blow{up [7] in the case of nonlinear reaction{iusion equations. Nonlinear bounary conitions might cause nite blow{up [6], even for the simple heat equation []. The level of regularity neee from the initial ata is an important question. The smoothing property of the iusion term is well known in several ierent settings. In the case of perioic bounary conitions [4] the solution is C {smooth for t > starting with L initial ata. In the case of non{perioic bounary conitions an/or L initial ata it is not trivial to prove even classical smoothness of solutions for t > (see, e.g [6, ]). With the introuction of cubic Neumann bounary feeback control we obtain a close loop system which is globally asymptotically stable an semi-globally exponentially stable in H norm an, hence in maximum norm whenever the initial ata is in H. The rene a priori estimates of Section 4 with the arguments of Section 5 show moreover, that solutions corresponing to L initial ata are classical an even innitely ierentiable solutions for t >. Also, the L norms of their appropriate erivatives converge to zero exponentially as the time approaches innity. Our analysis in Section 4 benets from, an extens the a priori estimates from [6]. In Section 5 we epart from Galerkin's metho in [6], making use of the heat kernel through an integral representation of solutions. We use the contraction mapping theorem to show the existence an uniqueness of classical solutions. For clarity, our treatment oes not inclue external forcing as in [5, 6,, 7]. External forcing woul preclue equilibrium stability but one coul still establish appropriate forms of isturbance attenuation an regularity of solution. Problem Statement Consier Burgers' equation where > is a constant, with some initial ata Our objective is to achieve set point regulation: W t W xx + W W x = ; (.) W (x; ) = W (x) L (; ): (.) lim W (x; t) = W ; 8x [; ] ; (.3) t! where W is a constant, while keeping W (x; t) boune for all (x; t) [; ][; ). Without loss of generality we assume that W. By ening the regulation error as w(x; t) = W (x; t) W, we get the system with initial ata w t w xx + W w x + ww x = ; (.4) w(x; ) = W (x) W w (x): (.5) We will approach the problem using nonlinear Neumann bounary control propose in [3]: w x (; t) = c + W + w (; t) w(; t); (.6) 9c w x (; t) = c + w (; t) w(; t); (.7) 9c

3 where c ; c >. The choice of w x at the bounary as the control input is motivate by physical consierations. For example in thermal problems one cannot actuate the temperature w, but only the heat ux w x. This makes the stabilization problem nontrivial because, as Byrnes et al [5] argue, homogeneous Neumann bounary conitions make any constant prole an equilibrium solution, thus preventing not only global but even local asymptotic stability. 3 Global Asymptotic Stability Denition. The zero solution of a ynamical system is sai to be globally asymptotically stable in an L spatial norm if kw(t)k L (kw k L ; t) ; 8t : (3.) where (; ) is a class KL function, i.e., a function with the properties that for xe t, (r; t) is a monotonically increasing continuous function of r such that (; t) ; for xe r, (r; t) is a monotonically ecreasing continuous function of t such that lim t! (r; t). The trivial solution is sai to be globally exponentially stable when (r; t) = kre t (3.) for some k; > inepenent of r an t, an it is sai to be semi-globally exponentially stable when where K(r) is a continuous nonecreasing function with K() =. (r; t) = K (r) e t ; (3.3) While irrelevant for nite-imensional systems where all vector norms are equivalent, for PDEs, the question of the type of norm L with respect to which one wants to establish stability is a elicate one. Any meaningful stability claim shoul imply bouneness of solutions. We rst establish global exponential stability in L q for any q [; ), which oes not guarantee bouneness. Then we show global asymptotic (plus local exponential) stability in an H -like sense which, by combining Agmon's an Poincare's inequalities, guarantees bouneness. Consier the Lyapunov function Its time erivative is V (w(t)) = w p x = kw p (t)k = kw(t)k p L p ; p : (3.4) _V = p = p w p (w xx W w x ww x ) x (p ) = p(p )kw p (t)w x (t)k pw p (; t) c + pw p (; t) w p wxx + w p w x W wp p wp+ p + p W w(; t) + p + c + W p + w(; t) + w (; t) p + 9c p(p )kw p (t)w x (t)k p 9c w (; t) w p (; t) c c + w (; t) + wp (; t) c 8 c + w (; t) : (3.5) 8

4 From Poincare's inequality it follows that kw p (t)k w p (; t) + w p (; t) + (p) kw p (t)w x (t)k : (3.6) Thus we get _V p V ; (3.7) p where = minf; c ; c g. It then follows that kw(t)k L p e p (p) t kw k L p : (3.8) Thus the solution w(x; t) is globally exponentially stable in an L q sense for any q [; ). Letting p! in (3.8), we get ess sup x[;] This result is not particularly useful for two reasons: jw(x; t)j ess sup jw(x; )j ; 8t : (3.9) x[;]. without aitional eort to establish continuity, with ess sup we cannot guarantee bouneness for all (but only for almost all) x [; ].. the above estimate oes not guarantee convergence to zero (it guarantees stability but not asymptotic stability). For this reason, we turn our attention to the norm kw(t)k B = p w(; t) + w(; t) + kw x (t)k : (3.) By combining Agmon's an Poincare's inequalities, it is easy to see that max jw(x; t)j p kw(t)k B : (3.) x[;] We will now prove global asymptotic stability in the sense of the B-norm. Let us start by rewriting (3.5) for p = as k t kw(t)k + kw(t)k B ; (3.) where k is a generic positive constant inepenent of initial ata an time, an by writing (3.8) as ultiplying (3.) by e t=(k), we get k t Integrating from to t yiels kw(t)k e t=k kw k : (3.3) e t=(k) kw(t)k + e t=(k) kw(t)k B et=(k) kw(t)k et=(k) kw k : (3.4) where = =(k) >. Now we take the L {inner prouct of (.4) with w xx, w t w xx x + e kw()k B kkw k ; (3.5) w xx x W w x w xx x ww x w xx x = : (3.6) 3

5 The estimation of the various terms follows: w t w xx x = w t w x j + w xt w x x = t kw x(t)k + w t(; t) c w(; t) + w 3 (; t) 9c = c t w (; t) + 8c w4 (; t) + (c + W ) + w t(; t) c w(; t) + W w(; t) + w 3 (; t) 9c w (; t) + 8c w4 (; t) + kw x (t)k ; (3.7) W w x w xx x W kw x (t)kkw xx (t)k W kw x(t)k + 4 kw xx(t)k ; (3.8) Using the notation ww x w xx x kw(t)k L jw x w xx j x kw(t)k L kw x (t)kkw xx (t)k kw(t)k L kw x(t)k + 4 kw xx(t)k kw x(t)k kw(t)k B + 4 kw xx(t)k : (3.9) A(t) = c w (; t) + 8c w4 (; t) + (c + W ) an substituting (3.7){(3.9) into (3.6) we obtain w (; t) + 8c w4 (; t) + kw x (t)k ; (3.) an hence t A(t) + kw xx(t)k W kw(t)k B + kw(t)k4 B ; (3.) A(t) _ kkw(t)k B + kkw(t)k BA(t) : (3.) ultiplying by e t we get t e t A(t) ke t kw(t)k B + e t A + kkw(t)k Be t A(t) ke t kwk B + kkwk Be t A(t) : (3.3) By Gronwall's inequality, we get e t A(t) A() + k Thus e kw()k B e k R t kw( )k B A() + kkw k e kkw k : (3.4) A(t) A() + kkw k e kkw k e t k kw k B + kw k 4 B e kkw k Be t kkw k Be kkw k B e t ; (3.5) which implies kw(t)k B kkw k B e kkw k B e t= : (3.6) This proves global asymptotic stability in the sense of the B-norm with (r; t) = kre kr e t=. It also shows semi-global exponential stability. The last estimate also guarantees that sup t max x[;] jw(x; t)j < (3.7) whenever w(; ), w(; ), an R w x(x; ) x are nite. The above results are summarize in the following theorem. 4

6 Theorem. Consier the system (.4), (.6), (.7). For any q [; ) there exists (q) > such that oreover, there exist k; (; ) such that kw(t)k L q kw k L qe t ; 8t : (3.8) kw(t)k B kkw k B e kkw k Be t ; 8t : (3.9) As we shall see in the next sections, the niteness of kw k B is not necessary for the bouneness an exponential ecay of solutions. Due to the iusion term, solutions which start as only L will \instantly" (for any t > ) have kw(t)k B <. oreover, by embeing theorem [5], the results of the next section imply that all of the terms in (.4) are boune an ecrease exponentially for t > : jw(x; t)j + jw x (x; t)j + jw t (x; t)j + jw xx (x; t)j e t t (3.3) for all t > an x [; ], where > epens only on kw k (continuously) an >, > are constants. 4 A Priori Estimates In this section we establish some a priori estimates on the norms of the solution to problem (.4){(.7). Throughout this presentation we will use the L {norm with respect to the spatial variable x. In orer to simplify notation we are going to use the notation for a generic constant whose size epens continuously on kw k,, c, c an W, but which oes not epen on t. For p =, from (3.5) we get t kwk + kw x k + (c + W )w (; t) + w 4 (; t) + c w (; t) + w 4 (; t) : (4.) 9c 9c We also have, using inequality (4.) an Poincare's inequality (A.3), that where t kw(t)k kw x (t)k c w (; t) Ckw x (t)k Cw (; t) Ckw(t)k ; (4.) C = min f; c g : (4.3) Thus, Returning back to (4.), multiplying it by e Ct= we obtain kw(t)k kw k e Ct : (4.4) e Ct= kw(t)k + e kw Ct= x (t)k + c w (; t) + w 4 (; t) + c w (; t) + w 4 (; t) t 9c 9c Integrating from to t an simplifying we obtain e Ct= kw(t)k + for all t. e C= kw x ()k + + e C= w (; ) + e C= w 4 (; ) + C ect= kw(t)k C kw k e Ct= : (4.5) e C= w (; ) e C= w 4 (; ) kw k (4.6) 5

7 Taking the L {inner prouct of (.4) with w xx we obtain w t w xx x + wxx x W w x w xx x ww x w xx x = : (4.7) While we estimate the rst an thir terms in (4.7) as in (3.7) an (3.8), the estimation of the fourth term is rene as Z ww x w xx x kw(t)k L jw x w xx j x kw(t)k L kw x (t)kkw xx (t)k kw(t)k L kw x(t)k + 4 kw xx(t)k 4kw(t)k kw x (t)k + 3 kw x(t)k kw xx(t)k : (4.8) Here we use Cauchy-Schwarz inequality in the secon step, Young's inequality with p = an = p = in the thir step an (A.4) in the last step. Using notation (3.) we obtain from (4.7) t A(t) + kw xx(t)k kw xx(t)k + W + 4kwk kw x (t)k + 3 kw x(t)k 4 : (4.9) ultiplying through by yiels to t A(t) + kw xx(t)k W + 8kw(t)k kw x (t)k + 6 kw x(t)k 4 kw x (t)k + kw x (t)k kw x (t)k kw x (t)k + kw x (t)k A(t): (4.) ultiplying (4.) by te Ct=4 we get te Ct=4 A(t) + te Ct=4 kw xx k te Ct=4 kw x (t)k + e Ct=4 A(t) + C t 4 tect=4 A(t) + kw x (t)k te Ct=4 A(t): (4.) Omitting the nonnegative secon term on the left an using the notation y(t) = te Ct=4 A(t); an we observe that an hence (4.) gives us a(t) = kw x (t)k b(t) = te Ct=4 kw x (t)k + e Ct=4 A(t) + C 4 tect=4 A(t); b(t) e Ct= kw x (t)k + e Ct= A(t) + C 4 tect= A(t); (4.) Inequality (4.6) implies that lim y(t) =. By Gronwall's inequality t! y(t) With this we arrive at the a priori estimate y(t) a(t)y(t) + b(t): (4.3) t b() exp a() : (4.4) w (; t) + w 4 (; t) + w (; t) + w 4 (; t) + kw x (t)k t e Ct=4 : (4.5) 6

8 Returning back to (4.), integrating it we also obtain that As a consequence of the previous estimates, using (.4) we obtain e C=5 kw ()k e C=4 kw xx (t)k : (4.6) e C=5 kw xx ()k + 3W e C=5 kww x ()k e C=5 kw()k L kw x()k e C=5 kw()k kw x ()k + e C=5 kw x ()k e C=5 kw x ()k 3 kw()k e C=5 3= e C 3=8 e Ct : (4.7) To obtain our next a priori estimate we ierentiate (.4) with respect to t an take the L {inner prouct of the resulting equation with w t. We obtain Here w tt w t x w xxt w t x + W w xt w t x + w t w x x + ww xt w t x = : (4.8) w tt w t x = t kw t(t)k ; (4.9) w xxt w t x = w xt w t j + kw xt(t)k = kw xt (t)k + w t (; t) c + w (; t) t 9c w(; t) + w t (; t) c + W t + w (; t) w(; t) 9c = c w t (; t) + 3c w (; t)w t (; t) + c w t (; t) + W w t (; t) + 3c w (; t)w t (; t) + kw xt (t)k ; (4.) W w xt w t x = W w t w x x kw t (t)k L 4kw x(t)k w t x x = W w t (; t) W w t (; t); (4.) kw t (t)k + kw xt (t)k = kw t (t)k 3= kw x (t)k t = e Ct=8 kw t (t)k + kw xt (t)k = t = e Ct=8 kw t (t)k 3= t = e Ct=8 kw t (t)k + 4 kw xt(t)k + 3t=3 4=3 e Ct=6 4 =3 kw t (t)k : (4.) We use the Cauchy-Schwarz inequality in the rst step, (A.3) with q = 4 in the secon step, (4.5) in the 7

9 thir step an Young's inequality with p = 4 an = =4 in the last step. ww xt w t x kw(t)k L 4kw t (t)k L 4kw tx (t)k kw(t)k + kw x (t)k =4 kw(t)k kw 3=4 t (t)k + kw xt (t)k =4 kw t (t)k 3=4 kw tx (t)k e Ct= kw t (t)k + t =8 e Ct=6 e Ct3=8 kw t (t)k + e Ct= kw xt (t)k =4 kw t (t)k 3=4 + t =8 e Ct=6 e Ct3=8 kw xt (t)k =4 kw t (t)k 3=4 kw tx (t)k = e Ct= kw t (t)kkw xt (t)k + t =8 e Ct7=6 kw t (t)kkw tx (t)k + e Ct= kw xt (t)k 5=4 kw t (t)k 3=4 + t =8 e Ct7=6 kw xt (t)k 5=4 kw t (t)k 3=4 kw xt(t)k + ect kw t (t)k + kw tx(t)k + t=4 e Ct7=8 kw t (t)k + kw xt(t)k + 8=3 ect4=3 kw t (t)k + kw xt(t)k + t=3 e Ct7=6 kw t (t)k ; (4.3) 8=3 where we use the generalize Holer's inequality in the rst step, (A.3) with q = 4 in the secon step, then (4.6) an (4.5), Young's inequality with p = an = = twice an with p = 8=5 an = 5=8 twice.using the notation B(t) = w t (; t) + w (; t)w t (; t) + w t (; t) + w (; t)w t (; t); (4.4) substituting (4.9){(4.3) into (4.8) an choosing appropriate 's we obtain t kw t(t)k + B(t) + kw xt (t)k e Ct + e Ct4=3 + t = e Ct=8 + t =3 e Ct=6 + t =3 e Ct7=6 + t =4 e Ct7=8 kw t (t)k ; (4.5) an after multiplication by t e Ct=6, t e Ct=6 kw t (t)k + t e Ct=6 B(t) + t e Ct=6 kw xt (t)k t t e Ct5=6 + t e Ct=6 + t 3= e Ct=4 + t 4=3 + t 5=3 e Ct + t 7=4 e Ct=48 + te Ct=6 kw t (t)k : (4.6) Now we use (4.7) an the fact that k e l ; 8k ; 8l >. We obtain t e Ct=6 kw t (t)k + Rewriting (.4) as e C=6 w t (; ) + w (; )w t (; ) + w t (; ) + w (; )w t (; ) + e C=6 kw xt ()k : (4.7) w xx = w t + W w x + ww x (4.8) results in the following estimate (with the help of a priori estimates (4.7), (4.5) an (4.6)) kw xx (t)k t e Ct= + t = e Ct=8 + t e Ct9=8 t e Ct= : (4.9) Dierentiate (.4) with respect to t an an take the L {inner prouct of it with w xxt to obtain w tt w xxt x + w xxt w xxt x W w xt w xxt x w t w x w xxt x ww xt w xxt x = : (4.3) 8

10 In (4.3) we estimate w tt w xxt x = w tt w xt j + w xtt w xt x = t kw xt(t)k + w tt(; t) c + w (; t) w(; t) t 9c + w tt(; t) c + W t + w (; t) w(; t) 9c = t kw xt(t)k + w tt(; t) c w t (; t) + w (; t)w t (; t) 3c + w tt(; t) c w t (; t) + W w t(; t) + 3c w (; t)w t (; t) = t kw xt(t)k + c t w t (; t) + w (; t) 6c t w t (; t) + c + W 4 t w t (; t) + w (; t) 6c t w t (; t); (4.3) W w xt w xxt x = W = W t w xt c + 9c w (; t) = W c w t (; t) + 3c w (; t)w t (; t) w(; t) + W = W c w t (; t) + 3 w (; t)w t (; t) + c + W c + W + w (; t) w(; t) 9c t + W c w t (; t) + W w t(; t) + w (; t)w t (; t) 3c 9c w 4 (; t)w t (; t) wt (; t) c + W w (; t)wt (; t) w 4 (; t)w 3c 9c t (; t) # ; (4.3) w t w x w xxt x kw t (t)k L 4kw x (t)k L 4kw xxt (t)k kw t (t)k + kw xt (t)k =4 kw t (t)k kw 3=4 x (t)k + kw xx (t)k =4 kw x (t)k 3=4 kw xxt (t)k kw t (t)kkw x (t)kkw xxt (t)k + kw xt (t)k =4 kw t (t)k 3=4 kw x (t)kkw xxt (t)k + kw t (t)kkw xx (t)k =4 kw x (t)k 3=4 kw xxt (t)k + kw xt (t)k =4 kw t (t)k 3=4 kw xx (t)k =4 kw x (t)k 3=4 kw xxt (t)k kw xxt (t)k + kw t (t)k kw x (t)k + kw xt (t)k = kw t (t)k 3= kw x k +kw t (t)k kw xx (t)k = kw x (t)k 3= + kw xt (t)k = kw t (t)k 3= kw xx (t)k = kw x (t)k 3= kw xxt (t)k + t 3 e Ct5= + t 5= e Ct=4 kw xt (t)k = + t 3=4 e Ct9=48 + t e Ct3=48 kw xt (t)k = kw xxt (t)k + t 3=4 e Ct9=48 + t =4 e Ct=4 kw xt (t)k = kw xxt (t)k + t 3=4 e Ct9=48 + t =3 e Ct=3 + t kw xt (t)k kw xxt (t)k + t =3 e Ct=7 + t kw xt (t)k ; (4.33) where we use Holer's inequality in the rst step, inequality (A.) with q = 4 twice in the secon step, Young's inequality in the fourth step, then estimates (4.5), (4.7) an (4.9) an in the seventh step we 9

11 use Young's inequality again, with p = 4. ww xt w xxt x kw(t)k L kw xt (t)kkw xxt (t)k e Ct= kw xt (t)kkw xxt (t)k + t =4 e Ct5=6 kw xt (t)kkw xxt (t)k kw(t)k + kw x (t)k = kw(t)k = kw xt (t)kkw xxt (t)k kw xxt (t)k + e Ct kw xt (t)k + t = e Ct5=8 kw xt (t)k kw xxt (t)k + t kw xt (t)k : (4.34) Here, after using Cauchy-Schwarz inequality, we use (A.) with q = in the secon step, then estimates (4.4) an (4.5) in the thir step an Young's inequality with p = twice in the fourth step. Appropriate choice of an straightforwar computation results in t kw xt(t)k + kw xxt (t)k + + w (; t) t w t (; t) + + w (; t) wt (; t) + w (; t) + w 4 (; t) + wt (; t) + w (; t) + w 4 (; t) t w t (; t) + t =3 e Ct=7 + t kw xt (t)k : (4.35) ultiply (4.35) by t 3 e Ct=7 an omit the nonnegative secon term kw xxt (t)k to obtain t 3 e Ct=7 kw xt (t)k + t 3 e Ct=7 ( + w (; t))wt (; t) + t 3 e Ct=7 ( + w (; t))wt (; t) t t 3 e Ct=7 w t (; t) + w (; t) + w 4 (; t) + t 3 e Ct=7 w t (; t) + w (; t) + w 4 (; t) + t =3 e Ct=7 + t e Ct=7 kw xt (t)k + t 3 e Ct=7 kw xt (t)k + t e Ct=7 ( + w (; t))w t (; t) + t 3 e Ct=7 ( + w (; t))w t (; t) + t 3 e Ct=7 w(; t)w 3 t (; t) + t e Ct=7 ( + w (; t))w t (; t) + t 3 e Ct=7 ( + w (; t))w t (; t) + t 3 e Ct=7 w(; t)w 3 t (; t): (4.36) After integrating from to t we notice that the right han sie is boune. The bouneness of the rst two terms is an immeiate consequence of (4.5) an (4.7). The bouneness of the thir integral is trivial. The fourth, fth, sixth, seventh, ninth an tenth terms are boune accoring to (4.7). In orer to see that the integral of the eighth an equivalently the eleventh term is nite, let us estimate it. 3 e C=7 w(; )wt 3 (; ) 3 e C=7 jw(; )jkw t ()k 3 L 5= e C=56 kw t ()k 3 + kw t ()k 5= kw xt ()k = + kw t ()k kw xt ()k + kw xt ()k 3= kw t ()k 3= 5= e C=56 3 e C=4 + 5= e C 5=4 kw xt ()k = + e C= kw t ()kkw xt ()k + e C= kw t ()k = kw xt ()k 3= + = e C 3= e C=6 kw xt ()k = 3= e C=6 kw t ()kkw xt ()k + =3 e C 5=8 e C=5 kw t ()k e C=5 kw t ()k 3=4 = =4 3= e C=6 kw t ()k = kw xt ()k 3= e C=6 kw xt ()k e C=6 kw xt ()k =4 = e C=6 kw xt ()k 3=4 : (4.37)

12 Here, in the secon step we use (4.5) an (A.) with q =, in the thir step we use (4.7), in the fth step we use Young's inequality three times an nally we use (4.7) an (4.7). With this we obtain from (4.3){(4.37) that t 3 e Ct=7 kw xt ()k + t 3 e Ct=7 ( + w (; t))w t (; t) + t 3 e Ct=7 ( + w (; t))w t (; t) : (4.38) Returning to (4.35), an using (4.38) we also get 3 e C=7 kw xxt ()k : (4.39) Dierentiate (.4) with respect to x, take its L {norm an multiply it by t 3= e Ct=4 to obtain t 3= e Ct=4 kw xxx (t)k t 3= e Ct=4 kw xt (t)k + W t 3= e Ct=4 kw xx (t)k + t 3= e Ct=4 kw x(t)k + t 3= e Ct=4 kww xx (t)k: (4.4) The rst term on the right han sie is nite accoring to (4.38), an the secon term is nite accoring to (4.9). t 3= e Ct=4 kwx(t)k = t 3= e Ct=4 kw x (t)k L 4 t3= e Ct=4 kw x (t)k + kw xx (t)k = kw x (t)k 3= t 3= e Ct=4 t e Ct=4 + t = e Ct=4 t 3=4 e Ct3=6 ; (4.4) where we use (A.) with q = 4 rst, an then estimates (4.5) an (4.9). t 3= e Ct=4 kww xx (t)k t 3= e Ct=4 kw(t)k L kw xx (t)k t 3= e Ct=4 kw(t)k + kw x (t)k = kw(t)k = kw xx (t)k t 3= e Ct=4 (e Ct + t =4 e Ct=6 e Ct )t e Ct= : (4.4) Here we use (A.) with q = an a priori estimates (4.4), (4.5) an (4.9). Combining (4.4)-(4.4) we arrive at our next a priori estimate kw xxx (t)k t 3= e Ct=4 : (4.43) We obtain an estimate on R t 3 kw xxxx ()k in a similar way. Dierentiating (.4) twice with respect to x, taking its L {norm, multiplying by t 3 e Ct=4 an integrating with respect to t we have 3 e C=4 kw xxxx (t)k + 3 e C=4 kw x (t)w xx (t)k + 3 e C=4 kw xxt (t)k + 3 e C=4 kw xxx (t)k 3 e C=4 kw(t)w xxx (t)k : (4.44) The niteness of the rst two integrals easily follows from (4.39) an (4.43) respectively. 3 e C=4 kw x ()w xx ()k 3 e C=4 kw x ()k L 4kw xx()k L 4 3 e C=4 kw x ()k + kw xx ()k = kw x ()k 3= kw xx ()k + kw xxx ()k = kw xx ()k 3= e C=4 + =4 e C=4 + =4 e C=4 + = e C=4 ; (4.45)

13 where we use Cauchy{Schwarz inequality in the rst step, inequality (A.3) with q = 4 twice in the secon step, an estimates (4.5), (4.9) an (4.43) in the thir step. 3 e C=4 kww xxx ()k 3 e C=4 kw()k L kw xxx()k 3 e C=4 kw()k + kw x ()kkw()k kw xxx ()k 3 e e C=4 C + = e C=8 e C 3 e C=7 e C 5=4 + = e C 5=4 (4.46) using inequality (A.4) an estimates (4.4), (4.5) an (4.43) in the secon an thir step. With (4.44){(4.46) we obtain 3 e C=4 kw xxxx ()k : (4.47) Next we ierentiate (.4) with respect to t, take the L {norms, multiply by t 3 e Ct=4 an integrate the resulting equation with respect to t. 3 e C=4 kw tt ()k 3 e C=4 kw xxt ()k + 3 e C=4 kw xt ()k + 3 e C=4 kw t ()w x ()k + 3 e C=4 kw()w xt ()k : (4.48) The niteness of the rst two terms on the right han sie is an immeiate consequence of (4.39) an (4.38) respectively. 3 e C=4 kw t ()w x ()k 3 e C=4 kw t ()k L 4kw x()k L 4 3 e C=4 kw t ()k + kw xt ()k = kw t ()k 3= kw x ()k + kw xx ()k = kw x ()k 3= e C=4 + t =4 e C=4 + t = e C=4 ; (4.49) where we use (A.) with q = 4 twice an then the previous a priori estimates (4.7), (4.38), (4.5) an (4.9). 3 e C=4 kw()w xt ()k 3 e C=4 kw()k L kw xt()k 3 e C=4 kw()k + kw x ()kkw()k kw xt ()k Here we use (A.) with q = an estimates (4.4), (4.5) an (4.38). With this we arrive at 3 e e C=4 C + t = e C=8 e C t 3 e C=7 e C + t = e C : (4.5) 3 e C=4 kw tt ()k : (4.5)

14 Next we ierentiate (.4) twice with respect to t an take its L {inner prouct with w tt to obtain (w ttt w tt w xxtt w tt + W w xtt w tt + w tt w x w tt + w t w xt w tt + ww xtt w tt ) x = : (4.5) The estimation of the various terms follows. w ttt w tt x = t kw tt(t)k ; (4.53) w xxtt w tt x = w xtt w tt j + kw xtt(t)k = kw xtt (t)k + w tt (; t) t c + + w tt (; ) t c + W 9c w (; t) + w (; t) w(; t) 9c = kw xtt (t)k + wtt(; t) c + w (; t) 3c + wtt(; t) c + W + w (; t) 3c W w xtt w tt x = W w ttw x x kw x (t)k L kw tt (t)k w tt w(; t) + 3c w tt (; t)w t (; t)w(; t) + 3c w tt (; t)w t (; t)w(; t); (4.54) = W w tt(; t) W w tt(; t): (4.55) kw x (t)k + kw xx (t)k = kw x (t)k = kw tt (t)k (t = e Ct=8 + t = e Ct=4 t =4 e Ct=6 )kw tt (t)k = (t = e Ct=8 + t 3=4 e Ct= )kw tt (t)k ; (4.56) where we use (A.) with q = in the secon step, an then estimates (4.5) an (4.9). w t w xt w tt x kw t (t)k L 4kw xt (t)k L 4kw tt (t)k kw t (t)k + kw xt (t)k =4 kw t (t)k kw 3=4 xt (t)k + kw xxt (t)k =4 kw xt (t)k 3=4 kw tt (t)k t e Ct= + t 3=8 e Ct=56 t 3=4 e Ct=6 t 3= e Ct=4 + t 9=8 e Ct3=56 kw xxt (t)k =4 kw tt (t)k t 5= e Ct=7 + t =8 e Ct=7 kw tt (t)k + t 7=8 e C= + t 9=4 e Ct=9 kw xxt (t)k =4 kw tt (t)k: (4.57) Here, after the generalize Holer's inequality in the rst step, we use (A.) with q = 4 twice in the secon step, an the previous estimates (4.7), (4.38) an (4.39) in the thir step. kw(t)k + kw x (t)k = kw(t)k = ww xtt w tt x kw(t)k L kw tt (t)kkw xtt (t)k e Ct= + t =4 e Ct=6 e Ct=4 kw tt (t)kkw xtt (t)k kw tt (t)kkw xtt (t)k kw xtt(t)k + 4 ect=8 kw tt (t)k + 4 t= e Ct=8 kw tt (t)k ; (4.58) 3

15 where the rst step follows from Cauchy-Schwarz inequality. In the secon step we use (A.) with q =, in the thir step we use a priori estimates (4.4) an (4.5). Finally we applie Young's inequality with p = an = =. Substituting (4.53){(4.58) into (4.5) an after a few elementary simplication we obtain t kw tt(t)k + kw xtt (t)k + wtt(; t) + w (; t) + wtt(; t) + w (; t) e Ct=8 + t = e Ct=8 + t 3=4 e Ct= kw tt (t)k + t 5= e Ct=7 + t =8 e Ct=7 kw tt (t)k + t 7=8 e Ct= + t 9=4 e Ct=9 kw xxt (t)k =4 kw tt (t)k + w tt (; t)wt (; t)w(; t) + w tt (; t)w t (; t)w(; t): (4.59) Here the last two terms can be estimate the same way, using (A.3) with q = in the secon step, (4.7), (4.38) an (4.5) in the thir step, an Young's inequality with p = an = = twice in the last step. We obtain, for example for the last term, w tt (; t)w t (; t)w(; t) jw tt (; t)jkw t (t)k L jw(; t)j jw tt(; t)j kw t (t)k + kw xt (t)kkw t (t)k jw(; t)j jw tt (; t)j t e Ct=6 + t 3= e Ct=4 t e Ct= t =4 e Ct=6 jw tt (; t)jt 9=4 e Ct=8 + jw tt (; t)jt =4 e Ct=5 wtt(; t) + t 9= e Ct=4 + t = e Ct=5 : (4.6) Substituting (4.6) into (4.59) after choosing appropriate, we obtain t kw tt(t)k + kw xtt (t)k + wtt(; t) + w (; t) + wtt(; t) + w (; t) e Ct=8 + t = e Ct=8 + t 3=4 e Ct= kw tt (t)k + t 5= e Ct=7 + t =8 e Ct=7 kw tt (t)k + t 7=8 e Ct= + t 9=4 e Ct=9 kw xxt (t)k =4 kw tt (t)k + t 9= e Ct=4 + t = e Ct=5 : (4.6) ultiplying (4.6) by t 5 e Ct=8 results in the inequality t 5 e Ct=8 kw tt (t)k + t 5 e Ct=8 kw xtt (t)k + t 5 e Ct=8 w t tt(; t) + w (; t) + t 5 e Ct=8 wtt(; t) + w (; t) t 5 + t 9= + t 7=4 e Ct=4 kw tt (t)k + t 5= e Ct=56 + t 9=8 e Ct=56 kw tt (t)k + t 3=8 e Ct=4 + t =4 e Ct=7 kw xxt (t)k =4 kw tt (t)k + t = e Ct=8 + t = e Ct=4 : (4.6) Now we integrate (4.6) with respect to t an, after omitting the last three nonnegative terms on the left, we obtain t 5 e Ct=8 kw tt (t)k 5 + 9= + 7=4 e C=4 kw tt ()k Here we use the fact that lim t! = e C=56 + 9=8 e C=56 kw tt ()k 3=8 e C=4 + =4 e C=7 kw xxt ()k =4 kw tt ()k = e C=8 + = e C =4 : (4.63) 3 e C=4 kw tt ()k is nite accoring to (4.5), which implies that (t 5 e Ct=8 kw tt (t)k ) =. The niteness of the rst term on the right han sie of (4.63) follows from 4

16 (4.5) an from the inequality k e l ; 8k ; 8l >. The secon term is nite accoring to (4.5): 5= e C=56 + 9=8 e C=56 kw tt ()k = 3= e C=8 kw tt ()ke C 3= e C=4 kw tt ()k = 3= e C=8 kw tt ()k 7=8 e C 3=56 e C 3=8 = + 7=4 e C 3=8 =! : (4.64) We use the Cauchy-Schwarz inequality twice in the secon step, an then (4.5). The thir integral in (4.63) is estimate as 3=8 e C=4 + t =4 e C=7 kw xxt ()k =4 kw tt ()k = + 3=8 e C=4 kw xxt ()k =4 kw tt ()k + 3 e C=7 kw xxt ()k 3 e C=7 kw xxt ()k =8 =8 =4 e C=7 kw xxt ()k =4 kw tt ()k 3 e C=4 kw tt ()k 3 e C=4 kw tt ()k = = 8=3 e C =63 3=8 7=3 e C =89 3=8 ; (4.65) where the secon step follows from the generalize Holer's inequality. In the nal step we use (4.39) an (4.5). The bouneness of the last integral in (4.63) is trivial. With this we obtain from (4.63) that Returning to (4.6), using (4.66) we obtain after integration that t 5 e Ct=8 kw tt (t)k + + t 5 e Ct=8 kw tt (t)k : (4.66) 5 e C=8 kw xtt ()k 5 e C=8 wtt(; )( + w (; )) + 5 e C=8 wtt(; )( + w (; )) : (4.67) Dierentiate (.4) with respect to t an rearrange it, then take the L -norms of its terms an multiply by t 3 e Ct=7 to obtain t 3 e Ct=7 kw xxt (t)k t 3 e Ct=7 kw tt (t)k + t 3 e Ct=7 W kw xt (t)k + t 3 e Ct=7 kw t (t)w x (t)k + t 3 e Ct=7 kw(t)w xt (t)k: (4.68) We alreay have from (4.66) an (4.38) that the rst two terms on the right han sie are nite. t 3 e Ct=7 kw t (t)w x (t)k t 3 e Ct=7 kw t (t)k L 4kw x (t)k L 4 t 3 e Ct=7 kw t (t)k + kw tx (t)k =4 kw t (t)k 3=4 kw x (t)k + kw xx (t)k =4 kw x (t)k 3=4 an (t 3= e Ct6=48 + t =8 e Ct3=95 + t 9=8 e Ct453=44 ) ; (4.69) t 3 e ct=7 kw(t)w xt (t)k t 3 e ct=7 kw(t)k L kw xt (t)k t 3 e ct=7 kw(t)k + kw x (t)k = kw(t)k = kw xt (t)k t 3 e ct=7 e Ct= + t =4 e Ct=6 e Ct= t 3= e Ct=4 : (4.7) 5

17 With this we obtain kw xxt (t)k t 3 e ct=7 : (4.7) Finally we estimate the L {norms in (.4) after taking its erivative with respect to x twice kw xxxx (t)k kw xxt (t)k + W kw xxx (t)k + 3kw x (t)w xx (t)k + kw(t)w xxx (t)k: (4.7) The rst two terms on the right han sie are boune by t 3 e ct=7 accoring to (4.7) an (4.43). The thir term can be estimate as kw x (t)w xx (t)k kw x (t)k L kw xx (t)k kw x (t)k + kw xx (t)k = kw x (t)k = kw xx (t)k t 3 e Ct=7 ; an the last term as kw(t)w xxx (t)k kw(t)k L kw xxx (t)k Putting together (4.7)-(4.74) we obtain (4.73) kw(t)k + kw x (t)k = kw(t)k = kw xxx (t)k t 3 e Ct=7 : (4.74) kw xxxx (t)k t 3 e Ct=7 : (4.75) Remark. Looking at the previous a priori estimates the question arises whether it is possible to continue the process an obtain similar exponential ecay for all the partial erivatives. The authors believe that this is inee possible, an for this an inuction process containing three or more separate estimates might be use. However these calculations are beyon the scope an length limit of this paper. 5 Global Existence an Uniqueness of Classical Solutions Consier Green's function [, 8] G(x; y; t; ) = + corresponing to the heat operator with Neumann bounary conitions Let us enote = X n= p 4(t ) cos(nx) cos(ny)e n (t ) X N= e (xyn) 4(t) + e (x+yn) 4(t) ; (5.) Lw w t w xx ; x ; t > ; (5.) w x (; t) = w x (; t) = ; t > : (5.3) f(w) W w + w ; (5.4) an g (w) c + W + 9c w w (5.5) g (w) c + w w: (5.6) 9c 6

18 The function w(x; t) is a classical solution of (.4){(.7) with continuous initial ata w (x) if an only if w(x; t) = F w(x; t) + G(x; y; t; )w (y) y G(x; ; t; ) ff (w(; )) + g (w(; )g G y (x; y; t; )f (w(y; )) y G(x; ; t; ) ff (w(; )) + g (w(; ))g : (5.7) The local in time solvability result will follow from contraction mapping argument applie to the iteration w n+ (x; t) = F w n (x; t) with some starting function w (x; t), where max jw (x)j ; an Q T [; ] [; T ]. x The heat kernel G satises the general estimates max (x;t)q T max (x;t)q T w (x; t) jg(x; y; t; )j + p C (xy) C e t ; s G(x; y; t; ) C r;s (t ) r s e C (xy) t (5.9) for all r, s nonnegative integer an for some constants C, C r;s, where C r;s epens on r an s. From these estimates it follows that the function (T ) max jg(x; ; t; )j jg y (x; y; t; )j y ; max jg(x; ; t; )j ; max (x;t)q T (x;t)q T converges to zero monotonically as T!. As a consequence, the same convergence statement hols for K(T ) 3(T ) max jf(w)j + max jwj max jg (w)j ; max jg (w)j jwj jwj (5.) (5.) an for L(T ) 3(T ) max jf (w)j + max jwj max jg(w)j ; max jg(w)j jwj jwj : (5.) In particular L(T ) < an K(T ) = if T > is small enough. As a consequence of the maximum principle We obtain from these remarks that max w n+ (x; t) max (x;t)q T x + max + max (x;t)q T max G (x; y; t; ) w (y) y (x;t)q T max jw (x)j : (5.3) x (x;t)q T jw (x)j + max (x;t)q T jg y (x; y; t; )j y jg(x; ; t; )j max tt ff (wn (; t)) + g (w n (; t))g max (x;t)q T jf (w n (x; t))j jg(x; ; t; )j max tt ff (wn (; t)) + g (w n (; t))g + K(T ) (5.4) 7

19 for all n = ; ; : : : an for suciently small T >. In a similar way we obtain max (x;t)q T w n+ (x; t) w n (x; t) + max (x;t)q T (x;t)q T (x;t)q T (x;t)q T + max + max + max 3(T ) max jf (w)j jwj jg(x; ; t; )j max tt jg(x; ; t; )j max tt jg(x; ; t; )j max tt jg(x; ; t; )j max tt max max (x;t)q T jg y (x; y; t; )j y f (w n (; t)) f w n (; t) g (w n (; t)) g w n (; t) f (w n (; t)) f w n (; t) g (w n (; t)) g w n (; t) (x;t)q T w n (x; t) w n (x; t) + 3(T ) max max f (w n (x; t)) f w n (x; t) (x;t)q T jg (w)j max w n (x; t) w n (x; t) jwj (x;t)q T = L(T ) w n (x; t) w n (x; t) : (5.5) As a result, the sequence fw n (x; t)g converges uniformly to a unique continuous function w(x; t) on Q T for suciently small T >. Once the continuity of the solution is obtaine, the general theory of parabolic equations implies that w(x; t) is innitely ierentiable in (; ) (; T ) (see, e.g [], pg 7, Theorem ). The a priori estimate (4.5) of Section 4 implies that unique continuous solution exists for any time interval (; T ), T >. The general theory tells us again that this solution is classical solution. Consier now the case of L initial ata. Let w (x) L (; ), an choose a sequence of continuous functions fw n (x)g such that w n n!! w in L (; ). Let us enote the classical solution of system (.4){ (.7) corresponing to initial ata w n (x) by w n (x; t). Their global in time existence is guarantee by the previous results. An important consequence of a priori estimate (4.5) is that, for any classical solution w(x; t) of system (.4){(.7), the inequality max jw(x; t)j e Ct=8 t = (kw k) (5.6) x[;] hols for all t (; T ], where the constant epens on kw k continuously. As a result, the sequence fw n (x; t)g converges uniformly to a continuous function w(x; t) on [; ] [t ; T ] for any < t < T. A priori estimate (4.4) shows that the sequence also converges in the space C [; T ]; L (; ) an hence the limiting function w(x; t) satises the initial conition. The equicontinuity follows from the integral representation (5.7) in both cases. Similarly, as a consequence of a priori estimates (4.66), (4.7) an (4.75), an the embeing theorem [5] H 4; ([; ] [t ; T ]) C ; ([; ] [t ; T ]); the appropriate time an spatial erivatives converge as well, showing that w(x; t) is a classical solution on [; ] [t ; T ]. We summarize the above results in the following theorem. Theorem. Given w (x) L (; ),. There exists a unique classical solution w C [; T ]; L (; ) \ C ; ([; ] [t ; T ]) of equation (.4){ (.7) for all < t < T <.. The solution w(x; t) epens continuously on the initial ata w (x) in the L {norm. 3. The trivial solution is stable in the sense that there exists a constant >, such that kw(t)k kw ke t ; (5.7) moreover there is a constant > epening only on kw k,, c, c an W continuously, such that jw(x; t)j + jw x (x; t)j + jw t (x; t)j + jw xx (x; t)j e t t ; (5.8) for all (x; t) [; ] [t ; T ], with some constant >. 8

20 w axis 3 w axis t axis x axis t axis..4.6 x axis Figure : Uncontrolle System, = : Figure : Controlle System, = :, c = c = : 6 Simulation Example It is well known (see, e.g. [9], [4]) that nonlinear problems, especially ui ynamical problems, require extremely careful numerical analysis. Typically there is a trae-o between convergence, accuracy an numerical oscillation. This is the case in particular when the initial ata is large relative to the viscosity coecient in Burgers' equation. Higher orer methos are preferre to lower orer methos only when the time an/or spatial step sizes are suciently small, where the smallness is a elicate question. It is not the purpose of our paper to n the best approximation scheme for our problem, simply to emonstrate our theoretical results. Our numerical simulation is base on an unconitionally stable, fully implicit scheme of secon orer accuracy, using three time level quaratic approximation in time an central ierence scheme in space. The simulations were carrie out on various platforms using several ierent numerical packages (OCTAVE, SCILAB, ATLAB), an they show gri inepenence for suciently small time an spatial gri. We consier rst Burgers' equation (.) with zero Neumann bounary conition (uncontrolle system) an then the regulation error system (.4){(.7) with = : an with initial ata w (x) = W (x) W, where W = 3 an W (x) = (:5 x) 3. The uncontrolle system is shown in Figure. The solution seems to converge to a nonzero \equilibrium" prole, although it eventually approaches zero, which coul be seen only for t (in fact, for some initial ata, the numerical solution gets trappe into this prole an never converges to zero [4]). This unsatisfactory behavior is remeie by applying bounary feeback, as shown in Figure. A Appenix: Auxiliary Lemmas The following inequality is an -imensional extension of a classical inequality (see e.g. [5]). which hols for w W m[a; b], m with w(a) =, where r q, an kwk L q kw x k L mkwk L r ; (A.) = r q = m + r + m m r : 9

21 Lemma. For any w H (; ) an q we have kwk L q kwk + kw x k kwk ; (A.) where = = =q, = + an = 6. We also have kwk L q + kwk kw x k kwk ; (A.3) an for the case of q = a simpler but less sharp inequality kwk L 4kwk + 3kw x k : Proof. As in [5], we consier an arbitrary w H (; ) an its extension ew(x) =( w(x) if x [; ], (x + )w(x) if x [; ]. (A.4) (A.5) Inequality (A.) applies to ew with = = =q an r = q, since ew H [; ] an ew() =. We have kewk L q [;] kew x k L [;]kewk L [;] : We have the following relationships between the norms of ew an w. Z kwk L q [;] kewk L q [;] ; (A.6) (A.7) an kewk L [;] = kew x k L [;] = Z Combining inequalities (A.6)-(A.9) we obtain w (x) x +Z j(x + )w(x)j x kwk wx(x) x +Z jw(x) (x + )w x (x)j x kw(x)k + (x + ) kw x (x)k x kw x k + Z kw x k + kwk + kw x k (A.8) = 3kw x k + kwk : (A.9) kwk L q [;] 3kwx k + kwk kwk 3 kwx k + kwk kwk = + kwk + 6 kw x k kwk ; (A.) which proves the rst part of our lemma. In orer to obtain the secon part we square the rst line of (A.) an use q =, = = to obtain kwk L [;] 3kw x k + kwk kwk 3kw x k + kwk + kwk = 4kwk + 3kw x k : (A.) Lemma (Young's inequality). For a; b, >, an p + q = the following inequality hols: Z Z ab p p ap + q q bq : Lemma 3 (Poincare's Inequality). For any w H (; ), the following inequalities hol: w(x) x w() + Z w(x) x w() + Z w x (x) x; w x (x) x : (A.) (A.3) (A.4)

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