Wave operators with non-lipschitz coefficients: energy and observability estimates Institut de Mathématiques de Jussieu-Paris Rive Gauche UNIVERSITÉ PARIS DIDEROT PARIS 7 JOURNÉE JEUNES CONTRÔLEURS 2014 Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie Paris 6 Paris February 13, 2014
Contents of the talk Energy estimates Wave operators with non-regular coefficients The Cauchy problem: energy estimates (i Operators with non-lipschitz coefficients (ii Zygmund condition: a well-posedness result (iii The control problem: observability estimates (i Classical observability estimates (ii Estimates with loss (iii Remarks and ideas of the proof
THE CAUCHY PROBLEM: ENERGY ESTIMATES
General setting Energy estimates L u := 2 t u N j,k=1 j (a jk (t, x k u on a strip [0, T] R N, with N 0 < λ 0 ξ 2 a jk (t, x ξ j ξ k Λ 0 ξ 2 ξ R N \{0} j,k=1 Aim: studying the Cauchy problem (CP { Lu = f u t=0 = u 0, t u t=0 = u 1 in the Sobolev spaces framework
Classical result Energy estimates Hurd & Sattinger (1968 a jk (t, x { Lipschitz continuous in t only bounded with respect to x = well-posedness of (CP in H 1 L 2 More regularity in x = H s H s 1 Key: energy estimate with no loss of derivatives ( u(t, H s + t u(t, H s 1 sup 0 t T C s ( u(0, H s + t u(0, H s 1 + T 0 Lu(t, H s 1 dt
De Simon & Torelli (1974: a jk BV t Counterexamples: Hurd & Sattinger (1968: discontinuous coefficients Colombini, De Giorgi & Spagnolo (1979: Hölder coefficients General idea: lower regularity assumptions with respect to t suitable hypothesis on x to compensate it = H well-posedness, but eventually with a loss of derivatives in the energy estimates
Coefficients depending only on time Integral log-lipschitz condition 0 Colombini, De Giorgi & Spagnolo (1979 T τ a jk (t + τ a jk (t ( dt C 0 τ log 1 + 1 τ Integral log-zygmund condition Tarama (2007 T τ ajk (t + τ + a jk (t τ 2a jk (t ( dt C0 τ log 1 + 1 τ τ Theorem sup 0 t T ( u(t, H s δ + t u(t, H s 1 δ C s ( u(0, H s + t u(0, H s 1 + T 0 Lu(t, H s 1 δ dt
Remarks Energy estimates Proof: Approximation of the coefficients Fourier transform Linking dual variable and approximation parameter Hölder coefficients = solutions in Gevrey classes ( Colombini, De Giorgi & Spagnolo 1979 Counterexample to distributional solutions ( Colombini, De Giorgi & Spagnolo 1979
Coefficients depending on (t, x Pointwise log-lipschitz condition in all the variables Colombini & Lerner (1995 a jk (t +τ, x+y a jk (t, x ( C (τ + y log sup (t,x 1 + 1 τ + y Log-Zygmund in time & log-lipschitz in space condition (LZ t sup ajk (t + τ, x + a jk (t τ, x 2a jk (t, x C0 τ log ( 1 + 1 τ (LL x sup a jk (t, x + y a jk (t, x ( C 0 y log 1 + 1 y scalar case x R: Colombini-Del Santo (2009 general case x R N : Colombini-Del Santo-F.-Métivier (2013
Theorem ( u(t, H s βt + t u(t, H s 1 βt sup 0 t T C s ( u(0, H s + t u(0, H s 1 + T 0 Lu(t, H s 1 βt dt loss of derivatives linearly increasing in time s ]0, 1[ Local in time estimates: T T β and C s depend only on L In particular a jk C b (RN x = H well-posedness, globally in time
Energy estimates with no loss of derivatives Lu(t, x := 2 t u N j,k=1 j (a jk (t, x k u Pointwise Zygmund condition in all the variables: a jk (t + τ, x + y + a jk (t τ, x y 2a jk (t, x C (τ + y sup (t,x Theorem (Colombini, Del Santo, F. & Métivier 2013 ( u(t H 1/2 + t u(t H 1/2 sup 0 t T C e λt ( u(0 H 1/2 + t u(0 H 1/2 + T 0 e λt Lu(t H 1/2dt
Remarks Energy estimates Original statement for a complete operator Pu = 2 t u N j,k=1 j (a jk (t, x k u + B(t, x (t,x u + c(t, xu B L ([0, T] ; C θ (R N ( θ > 1/2 c L ([0, T] R N Global in time estimate Well-posedness of (CP in H 1/2 H 1/2 Well-posedness in H if a jk Z([0, T] C b (RN
Related results Energy estimates Tarama (2007: a jk = a jk (t Z([0, T] = no loss of derivatives in any H s H s 1 Cicognani & Colombini (2006 Modulus of continuity Loss of derivatives Lipschitz no loss intermediate arbitrarly small loss log-lipschitz finite loss β t
Zygmund functions Energy estimates Definition: f Z(R n if f L (R n and sup f (z + ζ + f (z ζ 2 f (z K 0 ζ z R n Basic properties Lip(R n Z(R n loglip(r n Condition on second derivatives: if f C 2 (R, then f (z + ζ + f (z ζ 2 f (z = ζ 2 f (φ z,ζ Z(R n B 1, (R n, f B 1, where ( := sup 2 ν ν f L ν N
Regularization in time Energy estimates f Z(R t, 0 < λ 0 f (t Λ 0 Approximation by convolution kernel: f ε (t := (ρ ε f (t = 1 ε Then: 0 < λ 0 f ε Λ 0 R f ε (t f (t C ε t f ε (t C log 2 t f ε (t C ε ( τ ρ f (t τ dτ ε ( 1 + 1 ε
Littlewood-Paley Theory Littlewood-Paley decomposition Dyadic partition of unity in phase-space: χ 1 (ξ + supp χ 1 { ξ 1}, + ν=0 ψ ν (ξ 1 supp ψ ν { 2 ν 1 ξ 2 ν+1} = Operators: 1 := χ 1 (D x, ν := ψ ν (D x, S ν := Sobolev spaces = u S (R N, u = u H s + ν= 1 ν u ν 1 j= 1 ( 2 sν ν u L 2 l 2 ν 1 j
Paradifferential calculus with parameters Bony s paraproduct operator: a, u S (R N x T a u := ν 1 S ν 1 a ν u Regularization in space = well defined also if a(x α(t, x, ξ, rough in x Parameter γ 1 starting from high frequencies = α(t, x, ξ positive symbol = T α positive operator Symbolic calculus for Zygmund continuous symbols
Proof of the energy estimate (i a jk a jk,ε α ε (t, x, ξ, γ := j,k a jk,ε(t, x ξ j ξ k + γ 2 with ε = (γ 2 + ξ 2 1/2 (ii Approximation of the operator Lu = 2 t u + Re T αε u + Ru R : H s H s 1 for any s ]0, 1[ (iii Energy E(t := v(t 2 L 2 + w(t 2 L 2, with v(t, x := T α 1/4 ε w(t, x := T 1/4 α u ε t u T t (α u 1/4 ε
In particular, E(t t u(t 2 + u(t 2 H 1/2 H 1/2 ( if γ 1 large enough (iv Differentiation in time = cancellations Tarama s cancellation ( definition of E(t Paradifferential operator Re T αε (v Gronwall s inequality to conclude Remarks Cancellations only for s = 1/2 s 1/2 not clear
Classical observability estimates Main results Remarks and sketch of the proof THE CONTROL PROBLEM: OBSERVABILITY ESTIMATES
Setting Energy estimates Classical observability estimates Main results Remarks and sketch of the proof N = 1, coefficient just depending on x ω(x t 2 u x 2 u = 0 in [0, 1] [0, T] u(t, 0 = u(t, 1 = 0 in [0, T] u(0, x = u 0 (x, t u(0, x = u 1 (x in [0, 1] 0 < ω ω(x ω T > T (in our case, T ω L 1 x (0,1 Energy: E(t := 1 2 1 = E(t E(0 on [0, T] 0 (ω(x u t (t, x 2 + u x (t, x 2 dx
Classical observability estimates Main results Remarks and sketch of the proof Internal observability: for any Ω := ]l 1, l 2 [ [0, 1], T l2 E(0 C (ω(x u t (t, x 2 + u x (t, x 2 dx dt 0 l 1 Boundary observability: Ω = {0, 1} (or a subset, T E(0 C ( u x (t, 0 2 + u x (t, 1 2 dt 0 Remarks: Observability Geometric Control Condition for Ω ( Bardos, Lebeau & Rauch 1992 ; Burq & Gérard 1997 N 2: (i Microlocal analysis = C 2 regularity (ii Carleman estimates = C 1 regularity ( Duyckaerts, Zhang, Zuazua 2008
Previous results Energy estimates Classical observability estimates Main results Remarks and sketch of the proof ω Lipschiz = observability estimates: E(0 C T Avellaneda, Bardos & Rauch (1992: 0 u x (t, 0 2 dt ω ε (x := ω(x/ε = lim ε 0 C ε = + Fernández-Cara & Zuazua (2002: ω BV(0, 1 = observability estimates Castro & Zuazua (2003: ω C s (0, 1 = NO observability estimates
The Zygmund case Energy estimates Classical observability estimates Main results Remarks and sketch of the proof ω(x 2 t u 2 x u = 0 in [0, 1] [0, T] u(t, 0 = u(t, 1 = 0 in [0, T] u(0, x = u 0 (x, t u(0, x = u 1 (x in [0, 1] 0 < ω ω(x ω, ω Z : ω(x + y + ω(x y 2 ω(x dx K y T > T := 2 ω L 1 x (0,1 Theorem ( F. & Zuazua 2013 T u 0 2 H0 1(Ω + u 1 2 L 2 (Ω C x u(t, 0 2 dt 0
with loss Classical observability estimates Main results Remarks and sketch of the proof (LL Integral log-lipschitz condition: ω(x + y ω(x dx C y log( 1 + 1 y (LZ Integral log-zygmund condition: ω(x + y + ω(x y 2ω(x dx C y log( 1 + 1 y Theorem ( F. & Zuazua 2013 T u 0 2 H0 1(Ω + u 1 2 L 2 (Ω C t m x u(t, 0 2 dt 0 Right-hand side finite for smooth enough data
About the first variation Classical observability estimates Main results Remarks and sketch of the proof Modulus of continuity ω Lipschitz = no loss ω between Lipschitz and log-lipschitz = arbitrarly small loss = m = 1 ω log-lipschitz = finite loss ω worse than log-lipschitz = infinite loss
Proof of observability Energy estimates Classical observability estimates Main results Remarks and sketch of the proof Sidewise energy estimates (i Sidewise energy: F(x := 1 2 T (T /2x (T /2x ( ω(x t u(t, x 2 + x u(t, x 2 dt In particular, F(0 = (ω(0/2 T 0 xu(t, 0 2 dt (ii Zygmund log-zygmund Tarama (2007 log-lipschitz Colombini-De Giorgi-Spagnolo (1979 ( thanks to finite propagation speed (iii integration in space
On the counterexamples Classical observability estimates Main results Remarks and sketch of the proof Castro & Zuazua (2003: ω C s (0, 1 = NO observability estimates Proof: counterexample ( ideas from Colombini & Spagnolo (1989 Fractal partition of [0, 1] Construction of the oscillating coefficient: more and more oscillations for x 0 energy decreasing for x 0 energy exponentially concentrated inside the subintervals = energy too small at x = 0 Construction ok also for x 1 and for internal observability
Remarks Energy estimates Classical observability estimates Main results Remarks and sketch of the proof Z B 1 1, = W1,1 Z Example by Tarama (2007 = BV Z Controllability results ω(x t 2 y x 2 y = 0 in [0, 1] [0, T] y(t, 0 = f (t, y(t, 1 = 0 in [0, T] y(0, x = y 0 (x, t y(0, x = y 1 (x in [0, 1] (i ω Z = f L 2 (0, T (ii ω LZ LL = f H m (0, T
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