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AN INTRODUCTION TO DISPERSIVE PARTIAL DIFFERENTIAL
EQUATIONS.
NIKOLAOS TZIRAKIS UNIVERSITY OF ILLINOIS URBANA-CHAMPAIGN
Abstract. Lecture notes concerning basic properties of the solutions to the semi-linear Schr¨odinger and to the KdV equation. Based on these notes a series of lectures were given at the summer school in University of Texas, Austin, July 18-22, 2011.
- Introduction. A partial differential equation (PDE) is called dispersive if, when no boundary conditions are imposed, its wave solutions spread out in space as they evolve in time. As an example consider iut + uxx = 0. If we try a simple wave of the form u(x, t) = Aei(kx−ωt), we see that it satisfies the equation if and only if ω = k^2. This is called the dispersive relation and shows that the frequency is a real valued function of the wave number. If we denote the phase velocity by v = ωk we can write the solution as u(x, t) = Aeik(x−v(k)t)^ and notice that the wave travels with velocity k. Thus the wave propagates in such a way that large wave numbers travel faster than smaller ones. (Trying a wave solution of the same form to the heat equation ut − uxx = 0, we obtain that the ω is complexd valued and the wave solution decays exponential in time. On the other hand the transport equation ut − ux = 0 and the one dimensional wave equation utt = uxx are traveling waves with constant velocity.) If we add nonlinear effects and study iut + uxx = f (u), we will see that even the existence of solutions over small times requires delicate techniques. 1
2 NIKOLAOS TZIRAKIS UNIVERSITY OF ILLINOIS URBANA-CHAMPAIGN Going back to the linear equation, consider u 0 (x) = ∫ R ˆu 0 (k)eikxdk. For each fixed k the wave solution becomes u(x, t) = ˆu 0 (k)eik(x−kt)^ = ˆu 0 (k)eikxe−ik^2 t. Summing over k (integrating) we obtain the solution to our problem
u(x, t) =
Ru^ ˆ^0 (k)e
ikx−ik^2 tdk.
Since |uˆ(k, t)| = |uˆ 0 (k)| we have that ‖u(t)‖L 2 = ‖u 0 ‖L 2. Thus the conservation of the L^2 norm (mass conservation or total probability) and the fact that high frequencies travel faster, leads to the conclusion that not only the solution will disperse into separate waves but that its amplitude will decay over time. This is not anymore the case for solutions over compact domains. The dispersion is limited and for the nonlinear dispersive problems we notice a migration from low to high frequencies. This fact is captured by zooming more closely in the Sobolev norm
‖u‖Hs^ =
|uˆ(k)|^2 (1 + |k|)^2 sdk
and observing that it actually grows over time. To analyze further the properties of dis- persive PDEs and outline some recent developments we start with a concrete example.
- The semi-linear Schr¨odinger equation. Consider the semi-linear Schr¨odinger equation (NLS) in arbitrary dimensions
(1)
iut + ∆u + λ|u|p−^1 u = 0, x ∈ Rn, t ∈ R, u(x, 0) = u 0 (x) ∈ Hs(Rn).
for any 1 < p < ∞. Hs(Rn) (the s Sobolev space) is a Banach space that contains all functions that along with their distributional s−derivatives belong to L^2 (Rn). This norm
4 NIKOLAOS TZIRAKIS UNIVERSITY OF ILLINOIS URBANA-CHAMPAIGN
- Assume u 0 ∈ Hs(Rn) and consider a local solution. If there is a T?^ such that
tlim→T?^ ‖u(t)‖Hs^ =^ ∞,
we say that the solution blows up in finite time. This property is usually proved along with the local theory and is referred to as the blow-up alternative. More precisely one often proves that if (0, T ∗) is the maximum interval of existence, then if T ∗^ < ∞, we have limt→T ∗ ‖u(t)‖Hs = ∞. Analogous statements can be made for (−T ∗, 0). This is the case in the focusing problem when λ = 1.
- As a Corollary to the blow-up alternative one obtains globally defined solutions if there is an a priori bound of the Hs^ norms for all times. This is of the form
sup t∈R ‖u(t)‖Hs < ∞,
and it usually comes from the conservation laws of the equation. For (1) this is usually the case for s = 0, 1. An important comment is in order. Our notion of global solutions in remark 2 does not require that ‖u(t)‖Hs^ remains uniformly bounded in time. As we said unless s = 0, 1, it is not a triviality to obtain such a uniform bound. In case that we have quantum scattering, these uniform bounds are byproducts of the control we obtain on our solutions at infinity.
- If u 0 ∈ Hs(Rn) and we have a well defined local solution, then for each (0, T ) we have that u(t) ∈ Hx(Rn). Regularity refers to the fact that if we consider u 0 ∈ Hs^1 (Rn) with s 1 > s, then u ∈ X ⊂ C t^0 Hxs([0, T 1 ] × Rn), with T 1 = T. Notice that any Hs^1 solution is in particular an Hs^ solution and thus (0, T 1 ) ⊂ (0, T ). Regularity affirms that T 1 = T
AN INTRODUCTION TO DISPERSIVE PARTIAL DIFFERENTIAL EQUATIONS. 5 and thus u cannot blow-up in Hs^1 before it blows-up in Hs^ both backward and forward in time.
- Scattering is the hardest problem of all. Assume that we have a globally defined solution (which is true for large data in the defocusing case). The problem then is divided into an easier (existence of the wave operator) and a harder (asymptotic completeness) problem. We will see shortly that the Lp^ norms of linear solutions decay in time. This time decay is suggestive that for large values of p the nonlinearity can become negligible as t → ±∞. Thus we expect that u can be approximated by the solution of the linear equation. We have to add here that this theory is highly nontrivial for large data. For small data we can have global solutions and scattering even in the focusing problem.
- A solution that will satisfy (at least locally) all these properties will be called a strong solution. We will give a more precise definition later in the notes. Let us only mention here that the equipment of our solutions will all these additional properties is of great importance. For example the fact that local H^1 solutions satisfy the energy conservation law is a byproduct not only of the local-in-time existence but also of the regularity and the continuity with respect to the initial data properties. The arguments are strong enough to prove that most of the times (local-in-time existence after all come from Banach’s fixed point theorems) the data to solution map is uniformly continuous, even analytic in some cases.
The standard treatment of the subject is a the wonderful book of Cazenave: Semi- linear Schr¨odinger equations. We will refer to this book throughout these notes. A
AN INTRODUCTION TO DISPERSIVE PARTIAL DIFFERENTIAL EQUATIONS. 7 and space translation invariance that leads to the conservation of the momentum
(4) ~p(t) = =
Rnu^ ¯∇udx^ =^ ~p(0).
In the case that p = 1 + (^4) n , we also have the pseudo-conformal symmetry where if u is a solution to (1) then for t 6 = 0 1 |t|n^2 u(
x t ,^
t )e
i|x 4 t|^2
is also a solution. This leads to the pseudo-conformal conservation law
K(t) = ‖(x + 2it∇)u‖^2 L 2 − 8 t (^2) λ p + 1
Rn^ |u|
p+1dx = ‖xu 0 ‖ (^2) L 2.
Let’s go back to scaling for a moment. If we compute ‖uλ 0 ‖ (^) H˙s we see that
‖uλ 0 ‖ (^) H˙s = λsc−s‖u 0 ‖ (^) H˙s
where sc = n 2 − (^) p−^21. It is then clear that as λ → ∞:
i) If s > sc (sub-critical case) the norm of the initial data can be made small while at the same time the time interval is made longer: this is the best possible scenario for local well-posedness. Notice that uλ^ lives on [0, λ^2 T ].
ii) If s = sc (critical case) the norm of the initial data is invariant while the time in- terval gets longer: there is still hope but it turns out that to provide globally defined solutions one has to work very hard.
iii )If s < sc (super-critical case) the norms grow as the time interval is made longer: scaling is against us and indeed we cannot expect even locally defined solutions.
8 NIKOLAOS TZIRAKIS UNIVERSITY OF ILLINOIS URBANA-CHAMPAIGN A lot of attention has recently been paid to concrete examples of ill-posed solutions when s < sc. For the focusing problem one can use the specific solution (solitons etc) of the problem and show that the data to solution map is not uniformly continuous (see Kenig, Ponce, Vega). For defocusing equations the problem is a little harder. Christ, Colliander and Tao, among others, have recently demonstrated different modes of ill-posedness for defocusing equations.
- Local Well-Posedness The harder problem to resolve when looking for local solutions is to construct the afore- mentioned Banach space X. This process is delicate (the exception been of course the construction of smooth solutions that is done classically) and is build on certain estimates that the linear solution satisfies. Recall from our undergraduate (or graduate) PDE classes, we can obtain the solution to the linear problem by utilizing the Fourier transform. Then for smooth initial data (say in the Schwartz class S(Rn)) the solution of the linear equation is given as the convolution of the data with the tempered distribution
Kt(x) = (^) (4πit^1 )n 2 ei^ i|x^4 t|^2.
Thus we write for the solution
(5) u(x, t) = U (t)u 0 (x) = eit∆u 0 (x) = Kt? u 0 (x) = (^) (4πit^1 )n 2
Rn^ e
i |x− 4 yt |^2 u 0 (y)dy.
Another fact from our undergraduate (or graduate) machinery is Duhamel’s principle: Let I be any time interval and suppose that u ∈ C^1 t S(I × Rn) and that F ∈ C t^0 S(I × Rn). Then u solves
10 NIKOLAOS TZIRAKIS UNIVERSITY OF ILLINOIS URBANA-CHAMPAIGN Fortunately we can extend these basic dispersive estimates by duality (using a T T?^ ar- gument) and obtain the famous Strichartz estimates. Strichartz, Ginibre and Velo, Keel and Tao are the relevant names here.
Theorem 1. Fix n ≥ 1. We call a pair (q, r) of exponents admissible if 2 ≤ q, r ≤ ∞, (^2) q + nr = n 2 and (q, r, n) 6 = (2, ∞, 2). Then for any admissible exponents (q, r) and (˜q, ˜r) we
have the following estimates: The linear estimate
(8) ‖U (t)u 0 ‖Lqt Lrx(R×Rn). ‖u 0 ‖L^2 ,
and the nonlinear estimate
(9) ‖
∫ (^) t 0 U^ (t^ −^ s)F^ (s)ds‖Lqt^ Lrx(R×Rn)^.^ ‖F^ ‖L˜q^ t′^ Lr^ x˜′^ (R×Rn) where (^1) q˜ + (^) q^1 ˜′ = 1 and (^1) ˜r + (^) ˜r^1 ′ = 1.
Remark: Strichartz estimates actually give you more. In particular the operator eit∆u 0 (x) belongs to C(I, L^2 x) where I is any interval of R and ∫^0 t U (t − s)F (s)ds belongs to C( I, L¯^2 x).
We are now ready for a precise definition of what we mean by local well-posedness of the initial value problem (IVP) (1).
Definition 1. We say that the IVP (1) is locally well-posed (lwp) and admits a strong solution in Hs(Rn) if for any ball B in the space Hs(Rn), there exists a finite time T and a Banach space X ⊂ L∞ t Hxs([0, T ] × Rn) such that for any initial data u 0 ∈ B there exists a unique solution u ∈ X ⊂ C t^0 Hsx([0, T ] × Rn) to the integral equation
u(x, t) = U (t)u 0 − i
∫ (^) t 0 U^ (t^ −^ s)|u|
p− (^1) u(s)ds.
AN INTRODUCTION TO DISPERSIVE PARTIAL DIFFERENTIAL EQUATIONS. 11 Furthermore the map u 0 → u(t) is continuous as a map from Hs(Rn) into C t^0 Hxs([0, T ] × Rn). If uniqueness holds in the whole space C t^0 Hxs([0, T ] × Rn) then we say that the lwp is unconditional.
In this case u satisfies the equation in Hs−^2 (Rn) for almost all t ∈ [0, T ]. The details can be found in Cazenave’s book. In what follows we assume that the nonlinearity is sufficiently smooth (eg p-1=2k).
- We start with the Hs^ well-posedness theory where s > n 2 is an integer.
Theorem 2. Let s > n 2 an integer. For every u 0 ∈ Hs(Rn) there exists T ∗^ > 0 and a unique maximal solution u ∈ C((0, T ∗); Hs(Rn)) that satisfies (1) and in addition satisfies the following properties: i) If T ∗^ < ∞ then ‖u(t)‖Hs → ∞ as t → ∞. Moreover lim sup ‖u(t)‖L∞ = ∞ as t → ∞. ii) u depends continuously on the initial data in the following sense. If un, 0 → u 0 in Hs and if un is the corresponding maximal solution with initial data un, 0 , then un → u in L∞((0, T ); Hs(Rn)) for every interval [0, T ] ⊂ [0, T ∗). iii) In addition u satisfies conservation of mass, (3), and conservation of energy, (2).
Remark. A remark about uniqueness. Suppose that one proves existence and unique- ness in C([−T, T ]; XM ) where XM , M = M (‖u 0 ‖X ), T = T (M ), is a fixed ball in the space X. One can then easily extend the uniqueness to the whole space X by shrinking time by a fixed amount. Indeed, shrinking time to T ′^ we get existence and uniqueness in a larger ball XM ′. Now assume that there are two different solutions one staying in the ball XM and one separating after hitting the boundary at some time |t| < T ′. This is already a contradiction by the uniqueness in XM ′. To prove Theorem 2 we need the following two lemmas:
AN INTRODUCTION TO DISPERSIVE PARTIAL DIFFERENTIAL EQUATIONS. 13 where we used the fact that Hs^ embeds in L∞. To prove (13) notice that the L^2 part of the left hand side follows from (10). For the derivative part consider a multi-index α with |α| = s. Then Dαu is the sum (over k ∈ { 1 , 2 , ..., s}) of terms of the form g(k)(u) ∏kj=1 Dβj^ u where |βj | ≥ 1 and |α| = |β 1 | + ... + |βk|. Now let pj = (^) |^2 βsj | such that ∑kj=1 p^1 j = 12. We have by H¨older’s inequality
‖g(k)(u) ∏^ k j=
Dβj^ u‖L 2. ‖g(k)(u)‖L∞ ∏^ k j=
‖Dβj^ u‖Lpj.
By complex interpolation (or Gagliardo-Nirenberg inequality) we obtain
‖Dβj^ u‖Lpj. ‖u‖ |β sj | Hs^ ‖u‖^1 −^
|β sj | L∞
and thus
‖g(k)(u) ∏^ k j= Dβj^ u‖L^2. ‖g(k)(u)‖L∞^ ‖u‖Hs^ ‖u‖k L−∞^1. ‖u‖^2 Hks+
where in the last inequality we used (11). Thus we obtain
(14) ‖Dαu‖L 2. ‖u‖^2 Hks+.
Again notice that the term Dα(g(u) − g(v)) is the sum of terms of the form
g(k)(u) ∏^ k j=
Dβj^ u − g(k)(v) ∏^ k j=
Dβj^ v = [g(k)(u) − g(k)(v)]^ ∏k j=
Dβj^ u + g(k)(v) ∏^ k j=
Dβj^ wj
where wj ’s are equal to u or v except one that is equal to u − v. The second of the left hand side is estimated as in the proof of (14). For the first the same trick applies but now to estimate ‖g(k)(u) − g(k)(v)‖L∞^ we use (13).
It remains to prove Theorem 2. Existence and Uniqueness. We construct solutions by a fixed point argument. Given M, T > 0 to be chosen later, we set I = (0, T ) and consider
A = {u ∈ L∞(I; Hs(Rn)) : ‖u‖L∞(I;Hs) ≤ M }.
14 NIKOLAOS TZIRAKIS UNIVERSITY OF ILLINOIS URBANA-CHAMPAIGN (E, d) is a complete metric space where the distance is defined by d(u, v) = ‖u − v‖L∞(I;L (^2) ). We now consider (equation (1) with λ = −1)
Φ(u)(t) = eit∆u 0 − i
∫ (^) t 0 e
i(t−s)∆|u| 2 ku(s)ds = eit∆u 0 + H(u)(t).
By the Lemma 2, Minkowski’s inequality and the fact that eit∆^ is an isometry in Hs^ we have that
‖Φ(u)(t)‖Hs^. ‖u 0 ‖Hs^ + T ‖g(u)‖L∞(I;Hs) ≤ T C(M )M.
Furthermore using the Lemma 2 again we have
(15) ‖Φ(u)(t) − Φ(v)(t)‖L 2. T C(M )‖u − v‖L∞(I;L (^2) ).
Therefore we see that if M = 2‖u 0 ‖Hs^ and T C(M ) < 12 , then Φ is a contraction of (E, d) and thus has a unique fixed point. Uniqueness in the full space follows by the remark above. Maximal solutions, blow-up alternative. Let u 0 ∈ Hs^ and define
(16) T ∗^ = sup(T > 0 : there exists a solution on [0, T ]).
Now let T ∗^ < ∞ and assume that there exists a sequence tj → T ∗^ such that ‖u(tj )‖Hs ≤ M. In particular for k such that tk is close to T ∗^ we have that ‖u(tk)‖Hs^ ≤ M. Now we solve our problem with initial data u(tk) and we extend our solution to the interval [tk, tk + T (M )]. But if we pick k such that
tk + T (M ) > T ∗
we then contradict the definition of T ∗. Thus limt→T ∗ ‖u(t)‖Hs = ∞ if T ∗^ < ∞. We now show that if T ∗^ < ∞ then lim sup ‖u(t)‖L∞^ = ∞. Indeed suppose that lim sup ‖u(t)‖L∞^ <
16 NIKOLAOS TZIRAKIS UNIVERSITY OF ILLINOIS URBANA-CHAMPAIGN using again Lemma 2 and the fact that eit∆u 0 (x) ∈ C(R; Hs) we have
‖u(t 1 ) − u 0 ‖Hs^. ‖(eit^1 ∆^ − 1)u 0 ‖Hs^ + |t 1 |‖u‖^2 Lk∞+1((0,t 1 );Hs)
which finishes the proof. Conservation laws: Since we develop the H^1 theory below we implicitly have s ≥ 2. We have at hand a solution that satisfies the equation in the classical sense for high enough s (in general in the Hs−^2 sense with s ≥ 2 and thus in particular u satisfies the equation at least in the L^2 sense. All integrations below then can be justified in the Hilbert space L^2 ). To obtain the conservation of mass we can multiply the equation by ¯u, integrate and then take the real part. To obtain the conservation of energy we multiply the equation by ¯ut, take the real part and then integrate.
- We continue with the H^1 sub-critical theory (Kato, Ginibre and Velo).
Theorem 3. For every u 0 ∈ H^1 (Rn) there exists a unique strong solution of (1) defined on the maximal interval (0, Tmax). Moreover
u ∈ Lγloc((0, Tmax); W (^) x^1 ,ρ(Rn))
for every admissible pair (γ, ρ). In addition
t→^ limTmax^ ‖u(t)‖H^1 =^ ∞
if Tmax < ∞, and u depends continuously on u 0 in the following sense: There exists T > 0 depending on ‖u 0 ‖H 1 such that if (u 0 )n → u 0 in H^1 and un(t) is the corresponding solution of (1), then un(t) is defined on [0, T ] for n sufficiently large and
(17) un(t) → u(t) in C([0, T ]; H^1 )
AN INTRODUCTION TO DISPERSIVE PARTIAL DIFFERENTIAL EQUATIONS. 17 for every compact interval [0, T ] of (0, Tmax). Finally we have that
E(u)(t) =^12
|∇u(t)|^2 dx − (^) p + 1λ
|u(t)|p+1dx = E(u 0 )
and
M (u)(t) = ‖u(t)‖L 2 = ‖u 0 ‖L 2 = M (u 0 ).
We note that W 1 ,ρ^ is the Sobolev space with one weak derivative in Lρ. We first write the solution map using Duhamel’s formula as
Φ(u)(t) = eit∆u 0 − i
∫ (^) t 0 e
i(t−s)∆|u|p− (^1) u(s)ds
Now pick r = p + 1. Fix M, T > 0 to be chosen later and let q be such that the pair (q, r) is admissible (since the admissibility condition reads (^2) q + nr = n 2 we have that q = (^) n4((pp+1)−1) ). We run a contraction argument on the set A = ( u ∈ L∞ t H x^1 ([0, T ] × Rn) ∩ Lq((0, T ); W 1 ,r(Rn)) : ‖u‖L∞ t ((0,T );H^1 ) ≤ M, ‖u‖Lqt W (^) x^1 ,r ≤ M.
Notice that for r = p + 1 we have
‖|u|p−^1 u‖Lr x′. ‖u‖pLrx
and thus
‖|u|p−^1 u‖Lqt Lr x′. ‖u‖p L−∞ t 1 Lrx ‖u‖Lqt Lrx.
The last thing to notice is that since p < 1 + (^) n^4 − 2 we have that q > 2 and thus q > q′. By Sobolev embedding we also have that
‖u‖Lp x+1. ‖u‖H^1
AN INTRODUCTION TO DISPERSIVE PARTIAL DIFFERENTIAL EQUATIONS. 19 provides a unique solution u ∈ A. Notice that by the above estimates and the Strichartz estimates we have that u ∈ C t^0 ((0, T ); H^1 (Rn)). To extend uniqueness in the full space we assume that we have another solution v and consider an interval [0, δ] with δ < T. Then as before
‖u(t) − v(t)‖Lqδ W (^) x 1 ,r+ ‖u(t) − v(t)‖L∞ δ H x 1 ≤ Cδα(‖u‖p L−∞ T 1 H (^1) x + ‖v‖p L−∞ T^1 H x 1 )‖u − v‖Lqδ W (^) x 1 ,r
But if we set
K = max(‖u‖L∞ T H x^1 + ‖v‖L∞ T H x^1 ) < ∞
then for δ small enough we obtain
‖u(t) − v(t)‖Lqδ W (^) x 1 ,r + ‖u(t) − v(t)‖L∞ δ H x 1 ≤ 12 (‖u(t) − v(t)‖Lqδ W (^) x 1 ,r+ ‖u(t) − v(t)‖L∞ δ H (^1) x )
which forces u = v on [0, δ]. To cover the whole [0, T ] we iterate the previous argument Tδ times.
- Continuous dependence is almost identical to the uniqueness argument we outlined and we will skip it.
- For the proofs of the conservation laws we refer to Cazenave’s book.
- The fact that
u ∈ Lγloc((0, T ∗); W (^) x^1 ,ρ(Rn))
for every admissible pair (γ, ρ), follows from the Strichartz estimates on any compact interval inside (0, T ∗).
- What about T ∗^ and maximality. The proof is the same as in the smooth case. Remarks: 1. Notice that when λ = −1 (defocusing case), the mass and energy conservation
20 NIKOLAOS TZIRAKIS UNIVERSITY OF ILLINOIS URBANA-CHAMPAIGN provide a global a priori bound
sup t∈R ‖u(t)‖H 1 ≤ CM (u 0 ),E(u 0 ).
By the blow-up alternative we then have that T ∗^ = ∞ and the problem is globally well- posed (gwp).
- Let I = [0, T ]. An inspection of the proof reveals that we can run the lwp argument in the space S^1 (I × Rn) with the norm
‖u‖S (^1) (I×Rn) = ‖u‖S (^0) (I×Rn) + ‖∇u‖S (^0) (I×Rn)
where
‖u‖S (^0) (I×Rn) = (^) (q,r)−supadmissible ‖u‖Lqt∈I Lrx.
- We now state the lwp theory which is due to Tsutsumi (1987) for the L^2 sub-critical problem.
Theorem 4. Consider 1 < p < 1 + (^4) n , n ≥ 1 and an admissible pair (q, r) with p + 1 < q. Then for every u 0 ∈ L^2 (Rn) there exists a unique strong solution of
(18)
iut + ∆u + λ|u|p−^1 u = 0, u(x, 0) = u 0 (x)
defined on the maximal interval (0, Tmax) such that
u ∈ C t^0 ((0, Tmax); L^2 (Rn)) ∩ Lqloc((0, Tmax); Lr(Rn)).
Moreover
u ∈ Lγloc((0, Tmax); Lρ(Rn))
