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LARGE OUTGOING SOLUTIONS TO SUPERCRITICAL WAVE EQUATIONS 6 1 MARIUSBECEANU ANDAVY SOFFER 0 2 n a J Abstract. We prove the existence of global solutions to the energy- 4 supercritical waveequation in R3+1 2 u −∆u+|u|Nu=0, u(0)=u , u (0)=u ,4<N <∞, tt 0 t 1 ] for a large class of radially symmetric finite-energy initial data. P Functions in this class are characterized as being outgoing under the A linear flow — for a specific meaning of “outgoing” defined below. . In particular, we construct global solutions for initial data with large h (even infinite) critical Sobolev norm and large critical Lebesgue norm. t a m [ Contents 1 1. Introduction 1 v 1.1. Statement of the main results 1 5 1.2. History of the problem 5 3 3 2. Notations 6 6 3. Outgoing and incoming states for the free flow 7 0 4. Standard existence results 15 . 1 5. Proof of the main results 18 0 Appendix A. Positive solutions and a comparison principle 28 6 1 Acknowledgments 29 : References 29 v i X r a 1. Introduction 1.1. Statement of the main results. Considerthe semilinear wave equation in R3+1 u ∆u uNu= 0, u(0) = u , u (0) = u . (1.1) tt 0 t 1 − ±| | The equation is called focusing or defocusing according to whether the sign of the nonlinearity is or +. − For α (0, ), this equation is invariant under the scaling symmetries ∈ ∞ u(x,t) α2/Nu(αx,αt), 7→ 2010 Mathematics Subject Classification. 35L05,35A01, 35B40, 35B33, 35B09, 35B51. 1 2 MARIUSBECEANUANDAVYSOFFER as well as under the Lorentz group of transformations. Restricted to the initial data, the rescaling is (u (x),u (x)) (α2/Nu (αx),α1+2/Nu (αx)). (1.2) 0 1 0 1 7→ For s = 3/2 2/N, the H˙sc H˙sc 1 Sobolev norm is invariant under c − − × the rescaling (1.2), making it the critical Sobolev norm for the equation. Equation (1.1) is locally well-posed in the H˙sc H˙sc 1 norm. Note that the − × corresponding (critical) Lebesgue norm is u Lpc with p = 3N/2. 0 c ∈ All non-critical norms of the solution can be made arbitrarily large or small by rescaling, but critical norms remain constant after rescaling. An important conserved quantity for equation (1.1) is energy, defined as 1 1 1 E[u] := u (x,t)2 + u(x,t)2 u(x,t)N+2dx. ZR3 t 2| t | 2|∇ | ± N +2| | ×{} In case the equation is defocusing, energy controls the H˙1 L2 norm of the × solution, also called the energy norm. Equation (1.1)is energy-supercritical (or, inbrief, supercritical) if N > 4. The difficulty of the initial-value problem in this case lies in the fact that solutions cannot becontrolled in theenergy norm (as s > 1) andnohigher- c level conserved quantities can be used either. By the standard local existence theory, based on Strichartz estimates, any initial data in the critical Sobolev space H˙sc H˙sc 1 produces a solu- − × tion, locally in time. If the initial data is sufficiently small in the critical Sobolev norm, then the corresponding solution exists globally in time and disperses, meaning that, for example, it has finite L2N norm (the endpoints t,x 3N/2 N/2 are L L , which is not dispersive, and L L , which is achieved for ∞t x t ∞x N > 4 or N = 4 and radially symmetric solutions). Inthispaperweonlyconsiderthecaseofradiallysymmetric,i.e.rotation- invariant, solutions. We also assume all solutions are real-valued. For simplicity we suppose that N is an integer in (1.1). This makes little actual difference in the proof. Our first result is an existence result for the class of initial data ((H˙1 L ) L2) +H˙s H˙s 1 := (u ,u )= (v ,v )+(w ,w ) ∞ out − 0 1 0 1 0 1 ∩ × × { | (v ,v ) ((H˙1 L ) L2) , (w ,w ) H˙sc H˙sc 1 , 0 1 ∞ out 0 1 − ∈ ∩ × ∈ × } where((H˙1 L ) L2) meansradialandoutgoingfollowingDefinition3.5. ∞ out ∩ × Note that for data in this class the outgoing component is in a weaker space than H˙s H˙s 1, but the incoming component must still be in H˙s H˙s 1. − − × × Theorem1.1. Assumethat N (4,12], the initialdata (u ,u ) = (v ,v )+ 0 1 0 1 ∈ (w ,w ) decompose into a radial and outgoing component (v ,v ) and a sec- 0 1 0 1 ond radial component (w ,w ) H˙sc H˙sc 1 such that 0 1 − ∈ × 4/N 1 4/N kv0kH˙1 kv0kL−∞ +k(w0,w1)kH˙sc H˙sc−1 << 1 × SUPERCRITICAL WAVE EQUATION 3 is sufficiently small. Then the corresponding solution u to (1.1) exists globally, forward in time, remains small in ((H˙1 L ) L2) +H˙sc H˙sc 1, x∩ ∞x × x out x × x − and disperses: 4/N 1 4/N kukLNt /2L∞x . kv0kH˙1 kv0kL−∞ +k(w0,w1)kH˙sc×H˙sc−1. (1.3) In addition u scatters: there exist (w ,w ) H˙sc H˙sc 1 such that 0+ 1+ − ∈ × lim (u(t),u (t)) Φ(t)(v ,v ) Φ(t)(w ,w ) = 0. (1.4) t k t − 0 1 − 0+ 1+ kH˙sc×H˙sc−1 →∞ Here Φ(t) is the flow induced by the linear wave equation. If (v ,v ) ((H˙1 L ) L2) and (w ,w ) H˙sc H˙sc 1 are not 0 1 ∞ out 0 1 − ∈ ∩ × ∈ × small, then there exist an interval I =[0,T] with T > 0 and a solution u to (1.1) defined on R3 I, with (u ,u ) as initial data, such that (u(t),u (t)) 0 1 t × ∈ ((H˙1 L ) L2) +H˙sc H˙sc 1 for t I and x ∩ ∞x × x out x × x − ∈ u < . k kLNt /2L∞x (R3×I) ∞ ThecaseN =4correspondstos = 1andN = 12correspondstos = 4/3. c c The conclusion is still true, but trivial, when N = 4. Thus, we obtain global dispersive and scattering solutions for initial data of arbitrarily high, even infinite critical Sobolev norms. On the other hand, these initial data are still small in the critical Lpc norm. Dropping the scaling invariance, we can obtain a local existence result for large ((H˙1 L ) L2) initial data in the subcritical sense (i.e. where ∞ out ∩ × the time of existence only depends on the size of the initial data). We can also obtain a global existence result for small initial data, such that the solution remains bounded in ((H˙1 L ) L2) +H˙2 H˙1 H˙1 L2 for ∞ out ∩ × ∩ × ∩ all times, i.e. the incoming component of the solution gains a full derivative. See Proposition 5.2 for both results. As a consequence of Theorem 1.1, outgoing andradial finiteenergy initial dataofanysizeleadtoaglobalsolutionforwardintimeiftheyaresupported sufficiently far away from the origin. Corollary 1.2. Assume that N (4,12], the initial data (u ,u ) are radial, 0 1 ∈ outgoing according to Definition 3.5, supported outside the sphere B(0,R), and u 2 R4/N 1 << 1 (1.5) k 0kH˙1 − is sufficiently small. Then the corresponding solution u to (1.1) exists glob- N/2 ally, forward in time, and disperses, having finite L L norm. t ∞x Again, one can add a small H˙sc H˙sc 1 perturbation to the initial data, − × either outgoing or incoming, without changing the result. In addition, note that the conclusion is still true, but trivial, when N = 4. Remark 1.3. Note that condition (1.5) becomes automatically satisfied (up to a small error) if the equation (1.1) is defocusing and we wait for long enough. This is because the H˙1 L2 energy norm remains bounded by × 4 MARIUSBECEANUANDAVYSOFFER the energy E[u], hence the left-hand side of (1.5) improves with time: the solution becomes more outgoing and gets further removed from the origin, through dispersion. Since 4/N 1 < 0, R4/N 1 0 as R . − − → → ∞ Thus, our results could be part of the the process of showing global in time existence and scattering for any large radial solution to (1.1), after the solution is first shown to exist for a sufficiently long, but finite time. The next result shows that it is not necessary to assume that the initial data have finite energy — bounded and of compact support will suffice. Theorem 1.4. Assume that N [4, ) and (u ,u ) are radial initial data, 0 1 ∈ ∞ outgoing according to Definition 3.5, such that u is bounded and supported 0 on B(0,R). Then, as long as u0 L∞R2/N << 1 k k is sufficiently small, the corresponding solution u to (1.1) exists globally, forward in time, and disperses: kukL3xN/2L∞t ∩L∞x LNt /2 . ku0kL∞R2/N. In particular, these solutions have finite homogenous L2N norm. t,x Note that these initial data can bearbitrarily large in the critical Sobolev norm, but must still be small in the critical Lpc norm. Again, it is possible to add a small H˙sc H˙sc 1 perturbation, either − × incoming or outgoing, to the initial data. It is easy to obtain a local existence result for large radial outgoing initial data (u ,u ) when u L , see Proposition 5.1. 0 1 0 ∞ ∈ Finally, it is possible to construct global solutions to (1.1) for initial data with arbitrarily high critical Lpc norm. For simplicity we assume that N is an even integer. Theorem 1.5 (Main result). Assume that N (2, ) is even. For any L > ∈ ∞ 0 there exist initial data (u ,u ) such that u L, the corresponding 0 1 0 Lpc k k ≥ solution u of (1.1) is global, forward in time, and kukhxi−1L∞t,x∩hxi−1L∞x L1t < ∞. This implies that the solutions have finite critical L2N norm. t,x Note that these initial data necessarily also have arbitrarily high H˙sc norms, but we already had such examples from Theorem 1.1. From our construction it follows that u0 L∞(x 1) >>1, see [KrSc] for a similarresultobtainedbycompletely diffkerekntm|et|≥hods. Ourconstruction is based on taking outgoing initial data concentrated on a thin spherical shell. All these results hold in both the focusing and the defocusing case, re- gardless of the sign of the nonlinearity in (1.1). Also note that if we assume the initial data are smooth then the solution is also smooth. SUPERCRITICAL WAVE EQUATION 5 1.2. History of the problem. The first well-posedness result for large data supercritical problems was obtained by Tao [Tao], for the logarithmically supercritical defocusing wave equation that he introduced u ∆u+u5log(2+u2) = 0, u(0) = u , u (0) = u . (1.6) tt 0 t 1 − [Tao]provedglobalwell-posednessandscatteringforradialinitialdata. The starting point of [Tao] was an observation made in [GSV] for the energy- critical problem. Further results in the supercritical case belong to Roy [Roy1] [Roy2], who proved the scattering of solutions to the log-log-supercritical equation u ∆u+u5logc(log(10+u2)) = 0, u(0) = u , u (0) = u , tt 0 t 1 − 0< c < 8 , without the radial assumption. 225 AnothersupercriticalresultbelongstoStruwe[Str],whoprovedtheglobal well-posedness of the supercritical equation u ∆u+ueu2 = 0, u(0) = u , u (0) = u , tt 0 t 1 − (supercriticalinthecase whenE[u] > 2π) forarbitrary radialsmoothinitial data. A different series of results asserts the conditional well-posedness of supercritical equations. If the critical Sobolev norm (u,u ) of a k t kH˙sc H˙sc−1 solution to (1.1) stays bounded, then the solution exists global×ly and disperses. Such findings belong to Killip–Visan [KiVi1], [KiVi2] and Bulut [Bul1], [Bul2], [Bul3] in the defocusing case and Duyckaerts–Kenig–Merle [DKM] in the focusing case. All these conditional results are based on methods developed by Bour- gain[Bou],Colliander–Keel–Staffilani–Takaoka–Tao [CKSTT],Kenig–Merle [KeMe], and Keraani [Ker] in the energy-critical case. Another recent result belongs to Kriger–Schlag [KrSc], who construct large initial data solutions to the supercritical equation (1.1). The new results presented in this paper do not require the solution to be bounded or even finite in the critical Sobolev norm H˙sc H˙sc 1. Indeed, − this critical normcan bereplaced with theH˙1 L2 energy×norm of outgoing × initialdata,providedtheyaresupportedsufficientlyfarout(ortheL norm ∞ is sufficiently small). Notably, any smooth and radial scattering solution of equation (1.1) can be shown to fulfill the conditions of Corollary 1.2 after sufficient time has elapsed. Theproblem,then,rests incontrolling thesolutiononasufficiently long, butfinitetimeinterval. Thiscanbeachieved numerically, forexample. The energy norm being finite is also not necessary, as we construct solutions for bounded and compactly supported initial data. A variant of our construction (Theorem 1.5) leads to initial data that are large in the sense of [KrSc], i.e. u0 L∞(x 1) >>1. k k | |≥ Ourresultandtheresultof[KrSc]areratherdifferent. Ourresultisbased on multilinear estimates and [KrSc] is based on a nonlinear construction. 6 MARIUSBECEANUANDAVYSOFFER In addition, the solution of [KrSc] only logarithmically fails to be in the critical Sobolev space and in fact is bounded in a critical Lorentz space (i.e. L3N/2, ). By contrast, our solutions can fail to be in the critical Sobolev ∞ space by a wide margin and are large in the critical Lebesgue norm. Asanaside,wealsoproveaglobalwell-posednessresultforawideclassof large initial data for the focusingequation (1.1). Thesolutions we construct satisfy the equation in a rather weak sense, being possibly infinite. The decomposition into outgoing and incoming states which is the basis ofourpaperisnotentirelynew. Infact, theincomingconditioninourpaper resembles a condition from Engquist–Majda [EnMa], see formula (1.27) in that paper. We expect the same method to lead to an improvement in the energy- critical and subcritical cases, by allowing us to prove, for example, global well-posedness for (H˙1/2 L ) H˙ 1/2 outgoing initial data, thus requiring ∞ − ∩ × fewerderivatives thanthecritical Sobolevexponentfortheequation. Itmay also be possible to obtain results in the H˙1/2-subcritical range. All these improvements will be explored in a future paper. Another expected result is the well-posedness of equation (1.1) for arbi- trary large initial data, after projecting the nonlinearity on the outgoing states. This will constitute the subject of another paper. This paper is organized as follows: in Section 3 we state several results about incoming and outgoing states for the linear flow, in Section 4 we enounce some standard existence results, and in Section 5 we prove the theorems stated in the introduction. In the Appendix we state a large data global well-posedness result (which holds in the supercritical case), not related to the rest of the paper. 2. Notations A . B means that A C B for some constant C. We denote various | | ≤ | | constants, not always the same, by C. The Laplacian is the operator on R3 ∆ = ∂2 + ∂2 + ∂2 . ∂2 ∂2 ∂2 x1 x2 x3 We denote by Lp the Lebesgue spaces, by H˙s and W˙ s,p (fractional) homogenous Sobolev spaces, and by Lp,q Lorentz spaces. H˙s are Hilbert spaces and so is H˙1 L2, under the norm × (u ,u ) = ( u 2 + u 2 )1/2. k 0 1 kH˙1 L2 k 0kH˙1 k 1kL2 × ByH˙s ,etc.wedesignatetheradialversionofthesespaces. Foraradially rad symmetric function u(x), we let u(r):= u(x) for x = r. | | By(H˙1 L2) wemeanthespaceofoutgoingradiallysymmetricH˙1 L2 out × × initial data, see Definition 3.5. We define the mixed-norm spaces on R3 [0, ) × ∞ 1/p LptLqx := nf | kfkLptLqx := (cid:16)Z0∞kf(x,t)kpLqxdt(cid:17) < ∞o, SUPERCRITICAL WAVE EQUATION 7 with the standard modification for p = , and likewise for the reversed mixed-norm spaces LqLp. We use a simila∞r definition for LpW˙ s,p. Also, for x t t x I ⊂ [0,∞), letkfkLptLqx(R3×I) := kχI(t)fkLptLqx, whereχI isthecharacteristic function of I. We also denote B(0,R) := x R3 x R . { ∈ | | |≤ } Let D be the Fourier multiplier ξ , δ be Dirac’s delta at zero, and χ 0 | | denote the characteristic function of a set. Let Φ(t) :H˙1 L2 H˙1 L2 be the flow of the linear wave equation in × → × three dimensions: for u ∆u= 0, u(0) = u , u (0) = u , tt 0 t 1 − we set Φ(t)(u ,u ) = (Φ (t)(u ,u ),Φ (t)(u ,u )) := (u(t),u (t)). 0 1 0 0 1 1 0 1 t Alsoletφ(t) :L2 H˙ 1 L2 H˙ 1betheflowofthelinearwaveequation − − × → × in dimension one (on a half-line with Neumann boundary conditions): for v v = 0, v(0) = v , v (0) = v ,v (0,t) = 0, tt rr 0 t 1 r − we set φ(t)(v ,v ):= (v(t),v (t)). 0 1 t In this paper we only consider mild solutions to (1.1), i.e. solutions to the following equivalent integral equation: sin(t√ ∆) t sin((t s)√ ∆) u(t)= cos(t√ ∆)u + − u − − u(s)Nu(s)ds. − 0 √ ∆ 1∓Z √ ∆ | | 0 − − 3. Outgoing and incoming states for the free flow In order to define outgoing and incoming states for the linear wave equation,wereducetheequationtoaone-dimensionalproblem,whereidentifying such states is straightforward. Namely, we write each radial function u(r = x ) on R3 as a superposition | | of identical one-coordinate functions in every possible direction. For each radial function u(x) on R3 there exists a function T(u)(s) defined on R such that u(x) = T(u)(x ω)dω. (3.1) Z · S2 Taking T(u) to be supported on [0, ), this works out to ∞ Lemma 3.1. For radial u(x) = u(x ) and T(u) related by (3.1), such that | | T(u) is supported on [0, ), ∞ 1 2π r T(u)(r) = (ru(r)), u(r)= T(u)(s)ds. (3.2) ′ 2π r Z 0 Proof of Lemma 3.1. With no loss of generality assume that x = (0,0,r) and write ω in polar coordinates as ω = (sinθcosφ,sinθsinφ,cosθ). Then π 2π 2π r 2π r u(r)= T(u)(rcosθ)sinθdφdθ = T(u)(s)ds = T(u)(s)ds. Z Z r Z r Z 0 0 r 0 − (cid:3) 8 MARIUSBECEANUANDAVYSOFFER Conveniently, T is aconstant times aunitarymapfromL2 (thespaceof rad radialL2 functions)toH˙ 1([0, ))andboundedfromH˙1 toL2([0, ))— − ∞ rad ∞ the latter by Hardy’s inequality. The inverse operator T 1 is also bounded − from L2([0, )) to H˙1 . ∞ rad Lemma 3.2. With T defined by (3.1), T(u) = 1 u . k kH˙−1([0,∞)) √πk kL2rad Moreover, T(u) . u and u . T(u) . k kL2([0,∞)) k kH˙r1ad k kH˙r1ad k kL2([0,∞)) Remark 3.3. This shows that u := T(u) = (ru(r)) is L2([0, )) ′ L2([0, )) another norm on H˙1 equivalkentkto tkhe usukal on∞e. k k ∞ rad Proof of Lemma 3.2. Note that u L2 if and only if ∈ rad u 2 = 4π ∞ u(r)2r2dr. k kL2rad Z0 | | In particular, u = 2√π ru(r) = 2√π (ru(r)) . Next, note tkhaktLb2rayd Hardy’skinequkaLli2t(y[0,∞)) k ′kH˙−1([0,∞)) u/r 2 = 4π ∞ u(r)2dr . u 2 = 4π ∞ u (r)2r2dr. k kL2rad Z0 | | k kH˙r1ad Z0 | r | Therefore (ru(r)) u + ru(r) . u . k ′kL2([0,∞)) ≤ k kL2[0,∞) k ′ kL2([0,∞)) k kH˙r1ad Finally, let T(u)= v. Then by (3.2) 1 r 2 u 2 = 4π ∞ u (r)2r2dr . ∞ v(r)2dr+ ∞ v(s)ds dr. k kH˙r1ad Z0 | r | Z0 | | Z0 r2(cid:16)Z0 (cid:17) Then, 1 rv(s)ds = 1v(αr)dα and v(α) = α 1/2 v , so r 0 0 k · kL2 − k kL2 R R 1 1 1 v(αr)dα v(α) dα = v α 1/2dα . v . (cid:13)(cid:13)Z0 (cid:13)(cid:13)L2r ≤ Z0 k · kL2 k kL2Z0 − k kL2 This(cid:13)proves the la(cid:13)st statement of the lemma. (cid:3) For non-radial functions, a similar decomposition into one-coordinate functions can be obtained, but the computation is more complicated. One restricts the three-dimensional Fourier transform along each line through the origin, then takes the inverse one-dimensional Fourier transform. This is related to the Radon transform. So far, the transformation T is completely general; however, the subse- quent computation is not and will be different for, say, Schro¨dinger’s equation. Lemma 3.4. There exist bounded operators P and P on H˙1 L2 , + − rad × rad given by 1 1 r 1 u 0 P (u ,u )= u su (s)ds , (u ) +u (3.3) + 0 1 (cid:16)2(cid:16) 0− r Z0 1 (cid:17) 2(cid:0)− 0 r − r 1(cid:1)(cid:17) and 1 1 r 1 u 0 P (u ,u ) = u + su (s)ds , (u ) + +u , (3.4) − 0 1 (cid:16)2(cid:16) 0 r Z0 1 (cid:17) 2(cid:0) 0 r r 1(cid:1)(cid:17) SUPERCRITICAL WAVE EQUATION 9 such that I = P +P , P2 =P , and P2 = P . + + + If Φ(t) is the flow o−f the linear equation−then−for t 0 P Φ(t)P = 0 and + ≥ − for t 0 P Φ(t)P = 0. In addition, for t 0 Φ(t)P (u ,u ) is supported + + 0 1 on R3≤ B(0,t) and−for t 0 Φ(t)P (u ,u )≥is supported on R3 B(0, t). 0 1 \ ≤ − \ − Definition 3.5. P and P are called the projection on outgoing, respec- + − tively incoming states. We call any radial (u ,u ) such that P (u ,u ) = 0 0 1 0 1 − outgoing; if P (u ,u ) = 0 we call it incoming. + 0 1 Remark 3.6. P and P are self-adjoint on H˙1 L2 with the norm (u ,u ) := ( +(ru (r)) −2 + ru (r) 2 rad ×)1/2,rasdee Remark 3.3. k 0 1 k k 0 ′kL2([0, )) k 1 kL2([0, )) ∞ ∞ Proof. Given a radial solution u of the free wave equation u ∆u= 0, u(0) = u , u (0) = u , (3.5) tt 0 t 1 − each of the essentially one-dimensional functions T(u(t))(x ω) fulfills a one- · dimensional wave equation of the form v v = 0,v(0) = v , v (0) = v , (3.6) tt rr 0 t 1 − where 1 1 v = (ru (r)), v = (ru (r)). 0 0 ′ 1 1 ′ 2π 2π Indeed, in radial coordinates one has that u u (2/r)u = 0 or equiv- tt rr r − − alently that (ru) (ru) = 0. Since v = T(u) = 1 (ru(r)), taking a tt − rr 2π ′ derivative leads to (3.6), which holds in the weak sense. To fix ideas we assume that (u ,u ) H˙1 L2 , so (v ,v ) L2 H˙ 1 0 1 ∈ rad× rad 0 1 ∈ × − by Lemma 3.2. Note that v and v are supported on [0, ). 0 1 ∞ In addition, we consider equation (3.6) on a half-line only and impose the Neumann boundary condition v (0,t) = 0. This is justified because for r a smooth radial solution u one necessarily has u (0,t) = 0 and v (0,t) = r r 1 (ru) (0,t) = 1u (0,t). 2π rr π r The Neumann boundary condition means that the solution is reflected back at the boundary or, in other words, that it could be extended by symmetry to the negative half-axis. Equation (3.6) then has solutions of the form v(r,t) = χ v (r t)+χ v (t r)+χ v (r+t)+χ v ( r t), r t + r t r+t 0 r+t 0 + ≥ − ≤ − − ≥ − ≤ − − where by d’Alembert’s formula 1 1 v (r) = v (r) ∂ 1v (r) , v (r) = v (r)+∂ 1v (r) . (3.7) + 2 0 − r− 1 − 2 0 r− 1 (cid:0) (cid:1) (cid:0) (cid:1) Here ∂ 1 denotes the unique antiderivative of a H˙ 1 distribution that be- r− − longs to L2. Note that since v is supported on [0, ), ∂ 1v is also sup- 1 ∞ r− 1 ported on [0, ) (in fact it is given by 1 ru (r)). ∞ 2π 1 Thus both v and v are supported on [0, ). + − ∞ At time t, the outgoing component of v, which moves in the positive direction with velocity 1, consists of v (r,t) =χ v (r t)+χ v (t r). out r t + r t ≥ − ≤ − − 10 MARIUSBECEANUANDAVYSOFFER The incoming component consists of v (r,t) = χ v (r+t)+χ v ( r t). in r+t 0 r+t 0 + ≥ − ≤ − − In particular, at t = 0 the outgoing component is v and the incoming + component is v . − Note that as tgrows theincomingcomponenthits theorigin andbecomes outgoing. Ifwewaitforlongenough,mostofthesolutionbecomesoutgoing. Conversely, if we reverse time flow, as t all of the solution becomes → −∞ incoming. In order to obtain a general formula for the outgoing projection, without loss of generality we restrict our attention to time 0. Let π (v ,v ) be the + 0 1 initial data of the outgoing component, i.e. π (v ,v )(r) := (v (r t) ,∂ (v (r t)) ) = (v (r), v (r)) + 0 1 + − |t=0 t + − |t=0 + − +′ and likewise π (v ,v )(r):= (v (r+t) ,∂ (v (r+t)) ) = (v (r),v (r)). 0 1 t=0 t t=0 ′ − − | − | − − Note that π (v ,v ) is the initial data for the solution v (r t) of equation + 0 1 + − (3.6) on thetime interval [0, ) (which moves with velocity 1in thepositive ∞ direction) and same for π on ( ,0]. 0 −∞ By d’Alembert’s formula (3.7), π then has the form + 1 1 π (v ,v )(r) := v (r) ∂ 1v (r) , v (r)+v (r) + 0 1 (cid:16)2 0 − r− 1 2 − 0′ 1 (cid:17) (cid:0) (cid:1) (cid:0) (cid:1) and the incoming component π has the form − 1 1 π (v ,v )(r) := v (r)+∂ 1v (r) , v (r)+v (r) . − 0 1 (cid:16)2 0 r− 1 2 0′ 1 (cid:17) (cid:0) (cid:1) (cid:0) (cid:1) Note that π +π = I, π2 = π , π2 = π , and π π = π π = 0. In + + + + + particular, note that−∂ 1v = v , becau−se in a−ny case an a−ntider−ivative of v r− 0′ 0 0′ must be of the form v +c and this is in L2 only if c= 0. 0 Since by definition u,v = u,v , a simple computation shows h iL2 h ′ ′iH˙−1 that π and π are bounded, self-adjoint operators on L2 H˙ 1. + − − × Thus, both π and π are orthogonal projections, in a proper setting (on + L2 H˙ 1 or more gene−rally on H˙s H˙s 1). − − × × Note that a solution of the form v := v (r t) preserves the property + − that ∂ v = ∂ v and hence that π (v(t),v (t)) = 0 for all t 0. In other t r t − − ≥ words, if we denote by φ(t) the flow induced by the linear equation (3.6) on L2 H˙ 1, then for t 0 − × ≥ π φ(t)π (v ,v ) = 0. + 0 1 − Furthermore, in this case φ(t)π (v ,v ) is obviously supported on [t, ). + 0 1 ∞ Likewise,fort 0π φ(t)π (v ,v )= 0andsuppφ(t)π (v ,v ) [ t, ). + 0 1 0 1 ≤ − − ⊂ − ∞ The outgoing component of u corresponds to the outgoing component π (v,v ) of v traveling in the positive direction. Conjugating the projec- + t tions π and π by the transformation T defined by (3.1), we obtain the + corresponding o−perators for radial functions in R3. Letting P := T 1π T, + − +