薛定谔方程2

ON NONLINEAR SCHRODINGER EQUATIONS, II. HS-SOLUTIONS AND UNCONDIT IONAL WELL-POSEDNESS TOSIO KATO Abstract . We consider the nonlinear Schr6dinger equation (NLS) (see below) with a general "potential" F(u), for which there are in general no conservation laws. The main assumption on F(u) is a growth rate O(lu] k) for large lu], in addition to some smoothness depending on the problem considered. A uniqueness theorem is proved with minimal smoothness assumption on F and u, which is useful in eliminating the "auxiliary conditions" in many cases. A new local existence theorem for HS-solutions is proved using an auxiliary space of Lebesgue type (rather than Besov type); here the main assumption is that k <_ 1 + 4/(m - 2s) if s < m/2, k < oo if s = m/2 (no assumption if s > m/2). Moreover, a general existence theorem is proved for global HS-sohitions with small initial data, under the main additional condition that F(u) = O (] u] 1 +4/m) for small ]u i); in particular F(u) need not be (quasi-) homogeneous or in the critical case. The results are valid for all s >_ 0 if m < 6; there are some restrictions if m _> 7 and if F(u) is not a polynomial in u and ft. 1. Introduct ion This paper is intended to be a continuation of [ 11]; it is concerned with the initial value problem for nonlinear Schr6dinger equations (NLS) Otu = i (Au - F(u)), t ~ O, X E ~m m E 1~. Throughout the paper, we make the following assumptions on the "potential" F(u): (1.1) FEC1(C;C) , F (0 )=0, (1.2) DF(() = O(1r k-~) for some k _> 1, as Ir --* ~ , where the derivative DF is taken in the real sense; thus it may be identified with a real 2 x 2 matrix. Sometimes (1.2) is redundant; in such a case we may set k = c~. Note that (1.2) does not determine k uniquely. There is in general no conservation law. It is our plan to glean some loosely connected results on the well-posedness of the initial value problem for (NLS), which do not seem to have been duly noticed so far. 281 JOURNAL D'ANALYSE MATHEMATIQUE, Vol. 67 (1995) 282 T. KATO We begin with a review of the notion of well-posedness. According to Hadamard's principle, we say that the problem is (locally)~well posed in a function space X defined over •m, if for each 4~ E X there is T > 0 and a unique solu- tion u E C([0, T);X) to (NLS) with u(0) = q$. (In this definition we can include continuous dependence of u on q$, but for simplicity we shall disregard this point.) Actually, however, it often happens that some auxiliary condition [space] is needed to secure well-posedness. This is best illustrated by examples. Recall the following theorems (see e.g. [11, 12, 14]). Theorem A In (1.2) assume thatk < 1 +4/ (m - 2) (no assumption i fm = 1). (i) Given any ~b E H 1 = HI(Rm), there is T > 0 and a unique solution u E C([0, T) ;H l) to (NLS) with u(O) = (9. (ii) u has the additional properties Ou E Lr([O,T);L q) for l /q+ 2 /mr= 1, 1 /2 - I /m < 1/q < 1/2. Theorem B Let k < 1 + 4/m. Given any cb E L 2, there is T > 0 and a unique solution u to (NLS) with the following properties: (i) u E C([0, T);L 2) with u(O) = (9, (ii) u E t r ( (O,T) ; tq) for 1 /q+2/mr = 1, 1 /2 - 1/m < 1/q < 1/2. For uniqueness, conditions (i) and (ii) with a single pair (q, r) suffice. In Theorem A, part (i) constitutes a self-contained theorem by itself. Part (ii) is simply a "bonus", which may or may not appear in the theorem. For this reason we may say that (ii) is a removable auxiliary condition, and that (NLS) is unconditionally well posed in H 1. In most such cases, auxiliary conditions [spaces] originate as tools for constructing the solution. Whether or not to retain such removable spaces in the theorem is largely a matter of taste. In Theorem B, the Lr([0, T); L q) are auxiliary spaces. Unlike in Theorem A, part (ii) is an essential part of the theorem; without such a condition for at least a pair (r, q), uniqueness might not hold or even make sense. In this case we say that (NLS) is conditionally well posed in L 2, with an auxiliary space Lr((o, T); Lq). Practically, conditional well-posedness is not a definitive notion; it may turn out that an auxiliary condition, so far supposed to be necessary, is in fact removable. One of the purposes of this paper is to show that this is indeed the case in many known existence theorems for (NLS). In this connection we find it advisable to prove a uniqueness theorem stronger than the existing ones. Such a theorem is stated in Section 2, together with some ON NONLINEAR SCHRODINGER EQUATIONS 283 of its consequences on unconditional well-posedness. Section 3 contains the proof of the uniqueness theorem. Sections 4 to 6 are devoted to new existence proofs for HS-solutions, with some restrictions on s _> 0 if m > 7 and F(() is not a polynomial in ( and (. In addition to the increased smoothness of F, the main assumption here is that k < 1 + 4/(m - 2s) i fs < m/2, k < oo i fs -- m/2, and k = c~ (no assumption) i fs > m/2. HS-solutions have been studied in detail by Cazenave-Weissler [3], where Besov-type spaces are used as auxiliary spaces. We use Lebesgue-type spaces instead, which appear to be better adapted to the problem. New features in our results include an estimate for the life span T* of the H s- solution u in terms of ]lAau(0)ll2 alone (where A - - (--A) 1/2) for certain values of cr < s, regardless of Ifu(0)llm, excepting the "critical case". These estimates lead naturally to a regularity theorem. Moreover, we show that small global H s- solutions exist under more general conditions than so far considered, the only essential additional restriction being that F( ( ) = O(lffl l+4/m) for small (; F( ( ) need not be homogeneous or of critical power. Here again the smallness of Ilu(0)llHo for certain cr < s suffices in most cases. If F is a polynomial, the range of the admissible ~r is enlarged. In these proofs we use some recent results on the fractional derivatives of nonlinear functionals [4, 5, 9, 13], which are summarized in the Appendix. In most of the proofs, we make use of geometric notation introduced elsewhere; otherwise the proofs would become too long. Remark (a) In a conditionally well-posed case, there is a possibility that there is another solution with a different auxiliary space. If two auxiliary conditions lead to the same u, we say that they are consistent. In Theorem B, there are infinitely many auxiliary spaces Lr(L q) but they are consistent, a fact easily deducible from the proof. See Section 4, Remark (e) regarding the consistency of Besov- and Lebesgue-type auxiliary spaces for HS-solutions. (b) Among other evolution equations, the Navier-Stokes equation (NS) offers interesting examples. (NS) on I~ m is unconditionally well posed in L p with p > m (see eg[6]; ~r indicates the subspace of solenoidal vectors). I fp = m, it is unknown whether or not this is true, but (NS) is at least conditionally well posed. Different auxiliary spaces are known (see [10]). One is the class of functions u such that t~u E C([0, T); L q) for some q > m, with value zero at t = 0, where a = 1/2-m/2q. Another choice is L1/~((O, T); Lq). All these auxiliary spaces are consistent. In this connection we note that a remarkable uniqueness theorem has been proved by H. Brezis [2] for the vorticity equation on 11~ 2. An adaptation of his method to (NS) with p = m would eliminate the restriction near t = 0 involved in the auxiliary spaces just given, so that they may be replaced by a weaker space such as/_a~((0, T);L q) with any q > m. (Of course such a result could not be expected 284 T. KATO of (NLS), which is time-reversible.) The author is greatly indebted to G. Ponce and G. Staffilani for the estimates for fractional derivatives of nonlinear functionals. 2. A uniqueness theorem To show that an auxiliary condition is removable, one needs a strong uniqueness theorem. The purpose of this section is to state a uniqueness theorem for (NLS) that does not assume differentiability of the solution. Using this theorem, many of the existing auxiliary spaces are shown to be removable. The statement of the theorem is expressed in the form "Uniqueness holds in space X." By this we mean the following: (1) it makes sense to say that u E 2" is a solution to (NLS) on t E (0, T); (2) the initial value u(0) = limt~0 u(t) exists as a distribution on Rm; and (3) there is at most one (weak) solution u E A" to (NLS) with a given distribution as the initial value. Notat ions Lr(L q) means Lr((O,T);Lq(~m)), where 1 < q,r < ~ and 0 < T < c~. L~(L ~) should be interpreted as L~(f~r), where f2r = (0, T) • I~ m. a/X b and a V b mean min{a, b} and max{a, b}, respectively. Theorem 2.1 (un iqueness) Assume (1.1 - 2). If m >_ 2, uniqueness holds in L~176 2) fq Lr(L q) whenever (2.1) 1/q < (1/2 + 1/m)/k and l /q+ 2/mr < 1/2 A2/m(k - 1), with the following exceptions. If k = 1, disregard the last factor. If 1 + 4 / (m - 2) < k < c~ and r = oo, replace < in (2.1) by <. I fk = oo (see Introduction), replace zr(z q) by L~176 If m = 1, uniqueness holds in L~176 2) M Lr(L q) whenever (2.2) 1/q<_l/k and 1 /q+2/ r<_ l /2A2/ (k -1 ) , Remark (a) Recall that F does not determine k; if k satisfies (1.2), any larger k will do. This is reflected in the fact that the condition (2.1) becomes stronger with increasing k. (b) The conditions take simpler forms when applied to a specific k. For example, if k < 1 + 2/m, then (q, r) = (2, c~) is allowed; therefore uniqueness holds in L~(L2). Note also that 1/2 A 2/m(k - 1) = 1/2 i fk < 1 +4/m and = 2/m(k - 1) if ON NONLINEAR SCHRODINGER EQUATIONS 285 k > 1 + 4/m. Again, the first condition in (2.1) is redundant i fk > 1 + 4/ (m- 2) and r < ~, since it is implied by the second condition. Theorem 2.1 will be proved in the next section. Here we give some of its direct consequences. Coro l la ry 2.2 I f m > 2, uniqueness holds in L~(L 2 M Lq) whenever (2.3) 1/q< 1 /2A(1 /2+l /m) /kAZ/m(k -1) (q=c~i fk=e~) . I f m = 1, uniqueness holds in L~(L 2 f~ L2Vk). Proof It suffices to set r = ~ in Theorem 2.1. Coro l la ry 2.3 Let s >_ O. Uniqueness holds in C([O,T);H s) in each of the following cases: (i) s > m/2. (ii) m > 2, 0 < s < m/2 and k < 1 + (4 A (2s + 2))/(m - 2s). (iii) m = 1, 0 < s < 1/2 andk < 2/(1 - 2s). Proof We may decrease T if necessary and replace C([0, T); H e) by �9 ~' -- L~176 T); H 2) C L~176 2 fq Lq), where (use the Sobolev imbedding theorem) ?t = 2m/(m- 2s) if s < m/2; any{: /m/2 . Thus we have only to show that uniqueness holds in L~(L 2 n LO). In view of Corollary 2.2, it suffices to verify (2.3) for q = ~/. The case (i) is obvious, since it implies that ?:/= oo or arbitrarily large. In case (ii), setting q = ~/in (2.3) gives the stated sufficient condition. Similarly, in case (iii), ~/> 2 v k is seen to be sufficient. 3. P roo f o f Theorem 2.1 To simplify the arguments, from now on we use the geometric notation intro- duced in [12] and elsewhere. We denote by P= (x,y), 0 < x,y < 1, a point in the unit square [] = [0, 1] • [0, 1] C ]~;~2. We write x = x(P), y = y(P). The segment connecting P, Q 6 [] is denoted by (PQ), [PQ], [PQ), etc. according as it is open or closed at the end point, and its length by IPQI. We also regard P E [] as a 2-vector 286 T. KATO with origin O = (0, 0), so that aP (a > 0) and P + Q make sense (as long as these are in V]). We introduce a certain vertical order \ in []. Q \ P means that Q is below P, i.e. x(Q) = x(P) and y(Q) < y(P). For any segment E in [], R \ E [E \ R] means that there is S E E such that R \ S[S \ R]. There are some important special points in []. We set B-=(1/2,0), B' -- (1/2,1), C=(1 /2 -1 / rn , 1/2), C '=(1 /2+1/m, 1/2) (C- - (0,1/4), C'--- (1,3/4) if m= 1). The segments ~ = [BC) and e' = [B'C') are most important (~ : [BC], g' = [B'C'] if rn = l). These are parts of the lines given by x + 2y/m = 1/2 and x + 2y/m = 1/2 + 2/m, respectively (see Figs. 1 and la). O B t c 13 Fig. 1. 3' rn=l O B Fig. la. C Finally we set L(P) = Lr(L q) if P = (I /q, I /r) ; the time length T involved in Lr(L q) is mostly understood; The norm in L(P) is denoted by II : L(P)II, or simply by II e l i . BC denotes the class of bounded continuous functions, c may denote any positive constant. With these notations, the following rules hold (cf. [7, 8, 11, 12]). (Rule 1) If R E [PQ], then L(P) o L(Q) c L(R) c L(P) + L(Q), with II : RII _< II : Pll~-~ll : al l ~ i f7 = IPRI/IPQI. (Rule 2 ) f E L(P) and g E L(Q) imply that fagb E L(aP + bQ), with II f~gb : ap + bQII <_ II f : ellallg : Qll b, where a, b > 0. (Rule 3) Let T < c~ and Q \ P. Then L(Q) c L(P), with injection bounded by T o where 0 = y(P - Q). ON NONLINEAR SCHRODINGER EQUATIONS 287 (Rule 4) The integral operator G (see (3.6) below) is bounded on any L(P') with P' E g' into any L(P) with P E g, with bound independent of T. Here L(B) may be replaced by BC([0, T); L2). (Rule 5) The operator F (see (3.6)) is bounded on L 2 into L(P) with any P E g, with bound independent of T. Again L(B) may be replaced by BC([0, T); L2); in this case the bound is 1. Theorem 2.1 will now be deduced from the following general lemma. Lemma 3.1 Let P, Q, R E [] be such that (see Fig. 2) (3.1) [BQ] \ P E g, R= P + (k -1 )Q \ g'. Then uniqueness holds in P( = L(B) n L(Q). O B Fig. 2. Proof Step 1. First we note that A' C L(P). Indeed, Rule 1 shows that X c L(Q) for any (~ E [BQ]. Since there is such a Q \ P by hypothesis, it follows that X C L(P) by Rule 3. Next we show that (NLS) makes sense for u E X. To this end we note that F can be written as the sum of two "single power" potentials: (3.2) F = FI + Fk, (3.3) IFl(r ~ Mlff I, IDFI(r < M, (3.4) IFk(C)l ~ NICI k, IOFk(C)l ~ NICI k-~, 288 T. KATO where M, N _> 0 are constants. This can be achieved by multiplying F with cert~n smooth cutoff functions (cf. [11]). Let u E X. Then it is obvious that FI (u) E L(B). Since u E L(P) M L(Q) and R = P+ (k - 1)Q, it follows from Rule 2 that Fk(u) E L(R). Hence F(u) E L(B) +L(R) E L I (L2+L 1 ), so that (NLS) makes sense, with Otu E L f (H -2 +L I +L 2) c L 1 (H -m- 1 ). From this it is easy to conclude that u(0) E H -m-I E St exists. Thus (NLS) is equivalent to the integral equation (3.5) u = F(u(O)) + GF(u), where (3.6) f0 t (rr = U(t)r Gf(t) = - i U ( t - r) f(r)dr, U(t) = exp(itA). Step 2. To show that uniqueness holds in X, suppose that there are two solutions u, v E 2( of (INT) with u(0) -- v(0). By subtraction we then obtain for w = u - v (3.7) w = G(Fl(u) - Fl(v)) + G(Fk(u) - Fk(v)). From this we shall conclude that w = 0 if T is sufficiently small (as usual the size of T is immaterial in a uniqueness theorem). By (3.3-4) we have (use the mean value theorem) (3.8) IFl(u)-Fl(V)l <_Mlwl, IFk(u)--Fk(V)l <_cglwl(lul k-l +[v l* - l ) . Let S E g' such that R \ S (which exists because R \ g'). According to Rule 4, G is bounded from either of L(B') and L(S) into either of L(B) and L(P). According to Rule 3, L(B) c L(B') with injection bounded by T, and L(R) C L(S) with injection bounded by T ~ where 0 = y(S - R) > O. Therefore IIG(F1(u) - Fl(v)) : nil v Ila(Fl(u) - Fx(v)): Pll ~ cl[Fl(u ) - Fl(v): B'II < cTllFl(u) - Fl(V): Bl[ < cTMIIw: BII; note that w E L(B). Again, I I a (Fk (U) - Fk (V) ) : nil V I I a (Fk (U) - Fk (v ) ) : PI[ c l l Fk (u) - Fk(v) : all ~ cZ~ - Fk (V) : R[I cT~ k-~ + Ivlk-~) : e + (k - 1)Q[I <__ cT~ ell(l lu : al l k-~ + IIv: al l*-~); [by (3.1), (3.8)] [by Rule 2] ON NONLINEAR SCHRODINGER EQUATIONS 289 recall that u, v E L(P) n L(Q). If we set p = IIw : nil v IIw : PII, we thus obtain from (3.7) p <<_ c(MT+NT~ K = Ilu: QII k-~ + IIv: QII k-~. The factor of p on the right can be made arbitrarily small by taking T small. This is true even if 0 = 0. Indeed 8 = 0 implies R E g'; since P E g, we then have (k - 1)y(Q) = y(R) - y(P) > 0, hence y(a) > 0. Thus II " all is an integral norm and K becomes arbitrarily small with T. This proves that p = 0, hence w = 0, completing the proof of Lemma 3.1. P roo f o f Theorem 2.1 Set Q = (1/q, 1/r). We have to show that uniqueness holds in L(B) M L(Q). For this we (mostly) verify the assumptions of Lemma 3.1. Since this is an elementary geometry, it will suffice to indicate how to choose P, Q, R, leaving the detail to the reader. First let m >_ 2. We consider several cases separately. (a) 1 _< k < 1 + 2/m. We may assume that Q = B, for if uniqueness holds in L(B) then it holds, afortiori, in L(B) N L(Q) for any Q. Thus set P = Q = B. Then R = kB \ g'. (b) 1 +2/m <_ k < 1 +4/m. Let gbe the maximal extension in [] ofg. (2.1) implies that Q is on or to the left of g with x(Q) < ak =- (1/2 + 1/m)/k _< 1/2. Using Rules 1 and 3, it is not difficult to see that there is Q E g such that L(B) ML(Q) c L(B) NL(O_) and x(Q) = ak -- e, where e > 0 may be taken arbitrarily small. Thus we may assume from the outset that Q satisfies these conditions for Q. Then set P = Q, and show that kP = R \ g'. (c) 1 + 4/m _< k < ~. Let gk be the maximal segment in [] parallel to g and with bottom at (bk,O), where bk = 2/m(k - 1) _< 1/2. The second condition in (2.1) is equivalent to that Q is on or to the left of gk. As in the case (b), it is easy to see that L(B) n L(Q) c L(B) fq L(Q) with a certain Q E gk. Here y(Q) = 0 is excluded by the assumption. Again we may assume that Q itself satisfies these conditions for Q. Then it is easy to see that we can choose P E g and R E g' so that (PR) is parallel to (OQ); note that (PR) may take any slope in the interval (0, c~]. For any such pair P, R, a simple geometry shows that P + (k - 1)Q = R. It remains to show that [BQ] \ P. This is true i lk _> 1 +4/ (m - 2), which implies that bk _< 1/2 - 1/m = x(C). (In this case the first condition of (2.1) is implied by the second.) The case 1 +4/m < k < 1 +4/ (m - 2) is more complicated. But it can be shown that the required conditions can be satisfied by a proper choice of P E g and R E g', provided that x(Q) < ak, which is the first condition in (2.1). 290 T. KATO (d) k = oo. This means that nothing is assumed about the growth rate of F((). In this case we return to the integral equation (3.7). If u, v E L~(f~T), the values of F(() for sufficiently large KI are irrelevant. Hence we may assume that F(() = 0 for large K I, and any of the preceding conditions may be applied to prove that u = v. (Note that L(B) N L~(f~r) c L(Q) for any O with x(Q) < 1/2.) The case m = 1 can be handled in the same way, noting that g, g' are closed. If k < 2, we may assume (as in (a) above) that Q = B. Set P = Q = B, R = kB \ g'. If 2 < k < 5, (2.2) implies that Q is on or to the left of g with x(Q) < 1/k < 1/2. As in (b) above, we may assume that Q is on g with x(Q) = 1/k. Set P = Q; then R = kP \ g'. If k > 5, (2.2) implies that Q is on or to the left of gk, which has bottom (2/(k - 1),0). As in (c) above, we may assume that Q E g~ with x(Q) < 1/k. Then it is not difficult to see that we can choose P E g, R E g' so that (PR) is parallel to (OQ) and [BQ] \ P; note that (PR) can take any slope in the interval [1/2, c~]. The construction also implies P + (k - 1)Q = R. 4. Local existence of H*-solutions We now turn to some existence problems, and give a new proof of the local solvability of (NLS) in H s for real s > 0. HS-solutions were constructed in [3], mainly for single power potentials F(u), using a Besov-type auxiliary space. We consider more general F(u) with minimal decay at u = 0, using the auxiliary space (4.1) s, =/1 - ( A{L/p); e e})