Local Well-posedness and Blow-up for the Half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential

We study the Cauchy problem for the half Ginzburg-Landau-Kuramoto (hGLK) equation with the second order elliptic operator having rough coefficients and potential type perturbation. The blow-up of solutions for hGLK equation with non-positive nonlinearity is shown by an ODE argument. The key tools in the proof are appropriate commutator estimates and the essential self-adjointness of the symmetric uniformly elliptic operator with rough metric and potential type perturbation.


Introduction
In this paper, we study the Cauchy problem for the focusing half Ginzburg-Landau-Kuramoto (hGLK) type equation iB t u`D A,V u " i|u| p´1 u, p ą 1. (1.1) Here D A,V is the fractional Hamiltonian (see [16] for a more general choice of the fractional powers of the Laplacian) c pR n q "´∆ A,V "´∇¨A∇`V "´n ÿ j,k"1 B j pA j,k pxqB k q`V is a self-adjoint non-negative operator with a real-valued potential, such that the positive Hermite matrix A and the potential V satisfy appropriate assumptions given below. The fractional power of H A,V is defined by spectral analysis. For details, see Definition 6 below. Beside the other ones, it is worth mentioning that A is supposed to ensure that H A,0 is an elliptic second order operator in divergence form. Furthermore, focusing stands for the "`" sign in front of the nonlinearity in (1.1). We recall that the classical Ginzburg-Landau equation is instead typically associated with the standard Laplacian as Hamiltonian (see [24] for a recent review and references on this classical subject).
The idea to replace the Laplace operator in the Hamiltonian of some quantum mechanical models by its fractional powers was initiated in [16] and has been intensively studied in the last decade (see [22], for instance, for motivations to take the square root of the Laplacian and for an overview of the results in this context).
The half Ginzburg-Landau-Kuramoto equation (1.1), which is the main subject of this paper, is closely connected with the Kuramoto model (see [15], [1]) and the idea (proposed in [16] and [22]) to use the square root of the Laplacian in the definition of the Hamiltonian.
In order to define D A,V , we need to prove that´∆ A,V has a selfadjoint extension, where we regard the domain of´∆ A,V as C 8 c pR n q. One can find a self-adjoint extension for´∆ A,V with rough coefficients A and rough potential V by using the Friedrichs type extension under the non-negativity assumption (see [4,Theorem 1.2.7]). Recall that the domain of Friedrichs type extension can be defined as the set of all f P H 1 pR n q, such that there exists g P L 2 pR n q satisfyinǵ ∆ A,V f " g (1.2) in distributional sense. On the other hand, since the argument of Friedrichs type extension does not guarantee the uniqueness of selfadjoint extensions, in order to clarify the definition of fractional power of H A,V , we also need to show the uniqueness of self-adjoint extensions of´∆ A,V . In this case, we say that the operator´∆ A,V is essentially self-adjoint (the problem is referred to as quantum completeness, too). Some sufficient conditions for the essential self-adjointness for general symmetric operators on manifolds have been discussed in [3], for instance. In this paper, we give a detailed proof of the essential self-adjointness of´∆ A,V (see the Subsection 1.2 below for the precise hypothesis).
We started the study of this model in [6], where local and global well-posedness were discussed for the defocusing ("´" sign in front of the nonlinearity) equation iB t u`p´∆q 1{2 u "´i|u| p´1 u in space dimensions n " 1, 2, 3. The blow-up result for the focusing equation is obtained instead in [5] for n " 1. In [5], the proof of the blow-up result uses the following simple commutator estimates: where f is a Lipschitz function with corresponding norm }f } Lip . In order to show the blow-up of solutions to (1.1), we shall prove the following estimates where B s p,q and 9 B s p,q are the standard inhomogeneous and homogeneous Besov spaces on R n , respectively. Since it would be natural to pose the question if the estimates (1.3) and (1.4) are optimal for the case of rough coefficients; but this is not our goal, hence we do not investigate this question, as well as the question if the commutator is a bounded operator in L 2 . However, by replacing xxy by xxy a , our aim shall be to check that the commutator rD A,V , xxy a s is an L 2 -bounded operator for any a P p1{2, 1q and this shall be a sufficient tool to obtain our blow-up result at least for n " 1.
1.1. Notations. We collect here some notations used along the paper. Given two quantities A and B, we denote A À B (A Á B, respectively) if there exists a positive constant C such that A ď CB (A ě CB, respectively). We also denote A " B if A À B À A. Given two operators M and N , the commutator between them is defined as the operator rM, N s " MN´N M. For 1 ď p ď 8, the L p " L p pR n ; Cq are the classical Lebesgue spaces endowed with norm }f } L p "`ş R n |f pxq| p dx˘1 {p if p ‰ 8 or }f } L 8 " ess sup xPR n |f pxq| for p " 8. Given an interval I Ă R, bounded or unbounded, we define by L p pI; Xq the Bochner space of vector-valued functions f : I Ñ X endowed with the norm`ş I }f psq} p X dx˘1 {p for 1 ď p ă 8, with similar modification as above for p " 8. If f : I Ñ X is a continuous function up to the m th -order of derivatives, we write f P C m pI; Xq. For any s P R, we set H s " H s pR n ; Cq :" p1´∆q´s {2 L 2 and its homogeneous version 9 H s " 9 H s pR n ; Cq :" p´∆q´s {2 L 2 . For a pair of functions in L 2 , the inner product xf, gy " xf, gy L 2 is classically defined as xf, gy " ş R n fḡ dx, beingz, the usual complex conjugate to z P C. For x P R n instead, xxy :" a 1`|x| 2 . The space W 1,8 " W 1,8 pR n q is the space of Lipschitz functions. The operator Ff pξq "f pξq is the standard Fourier transform, F´1 being its inverse. For s P R and 0 ă p, q ď 8, 9 B s p,q " 9 B s p,q pR n q is the homogeneous Besov space of functions having finite }¨} 9 B s p,q -norm, the last defined as {q with obvious modifications for p, q " 8. The non-homogeneous version B s p,q " B s p,q pR n q is induced by the norm Here the Littlewood-Paley projectors P j are defined by means of a radial cut-off function χ 0 P C 8 c pR n q and the dyadic functions ϕ j pξq " χ 0 p2´jξq´χ 0 p2´j`1ξq yielding to the partition of the unity χ 0 pξqř jě1 ϕ j pξq " 1, for any ξ P R n . Hence the projectors are given by Qf :" F´1 pχ 0 Ff q and P j f :" F´1 pϕ j F f q. The Lorentz space L β,8 is given by For 1 ď p ď 8, p 1 is the conjugate index defined by 1{p`1{p 1 " 1.

1.2.
Assumptions and the main results. We give now the precise assumptions that we make on the structure of our Hamiltonian´∆ A,V and the main results contained in the paper. We start with the hypotheses on A " Apxq, which is a Hermitian matrix-valued function. We assume: A1. Uniform ellipticity of A: There exist two positive constants C 1 and C 2 satisfying A2. Regularity of the coefficients: A is in the Lipschitz class of matrix-valued functions, namely A j,k P W 1,8 pR n q j, k P t1,¨¨¨, nu; A3. Boundedness: The multiplication operator Let us turn our attention to the potential perturbation V " V pxq. It is a real-valued function satisfying the following conditions: H1. Boundedness of the potential: V P L q,8 pR n q`L 8 pR n q for some q with q ą maxt2, n{2u; H2. Non-negativity of the Hamiltonian´∆ A,V : There exists θ P p0, 1q such that θxA∇f, ∇f y`xV f, f y ě 0, @ f P C 8 0 pR n q. Though for the moment it is not our aim to weaken the non-negativity assumption in H2, it is worth mentioning that this hypothesis could be relaxed, at least in the case n " 1. For example for A " 1, perturbations of the Laplacian which belongs to the Miura class should imply positivity of such Hamiltonians (see [11]). The non-negativity assumption is needed to guarantee that the square root of the operator is well defined.
First we state the result on the self-adjoint extension of the operatoŕ ∆ A,V . This theorem is crucial for the local well-posedness theory below and for the commutator estimates we are going to prove. Theorem 1. Assume the assumptions A1, A2, H1 and H2 are satisfied. Then the operator´∆ A,V is essentially self-adjoint, i.e. there exists a unique self-adjoint extension H A,V of this operator with domain The key point in our blow-up result shall be instead the following commutator estimate. Proposition 2. Assume the conditions A1, A2, A3 and H1, H2 are satisfied. We have the two commutator estimates in two cases below.
Case 2. Suppose n ě 3 and f P 9 (1.8) Next we turn to the local well-posedness of (1.1).  Case 1. Let u 0 P L 2 and w P B 1 8,1 satisfy 1{w P L 8 X L 2 and the following estimate: If there exists a solution u P Cpr0, T max q; L 2 X L p`1 q, then the maximal time of existence is finite: T max ă 8.
Case 2. Suppose that V " 0. Let 1 ă p ă 3 and let u 0 P L 2 zt0u. If there exists a solution u P Cpr0, T max q; L 2 X L p`1 q, then the maximal time of existence is finite: T max ă 8.
Here the condition p " 3 corresponds to the critical exponent p F " 1`2{n defined also in a multidimensional framework. See the results in [5].

Self-Adjointness of´∆ A,V
The proof of Theorem 1 can be reduced to the proof that´∆ A,0 is essentially self-adjoint. Indeed, if´∆ A,0 has unique self-adjoint extension H A,0 with domain DpH A,0 q " H 2 pR n q, then we can use the estimate (A.1) of Lemma A.1 in combination with the KLMN lemma (see [21,Theorem X.17]) and deduce that H A`V is an essentially self-adjoint operator with domain H 2 pR n q.
Therefore, it remains to verify that´∆ A,0 is essentially self-adjoint. This is done below in Proposition 5 and this yields to Theorem 1. Firstly, we recall sufficient equivalent conditions guaranteeing the selfadjointness property of an operator. Ker p´∆ A,V`1 q˚" t0u; iii) the range of p´∆ A,V`1 q is dense in L 2 pR n q: Next, we recall some fundamental operator calculus.
Proof. For completeness, we shall sketch the proof. The relation (2.2) follows directly from the simple commutator rule Proof. For completeness, we shall sketch the proof. Noting the identity and applying the resolvent pλ`Aq´1 from the left, we obtain the assertion.
We can now give the following: Proof. We show that the closure p´∆ A,0 q is self-adjoint. Lemma A.2 in the Appendix A below, implies that Thanks to symmetry and regularity of A, there exists at least one self-adjoint extension of´∆ A,0 . Indeed, since´∆ A,0 is symmetric and A P W 1,2 loc pR n q, the quadratic form We recall that any self-adjoint extension of´∆ A,0 is also an extension of p´∆ A,0 q and so is H A . Now we show that the self-adjointness of H A implies that also p´∆ A,0 q is self-adjoint. For this purpose, we shall check the equivalent assertion namely h is orthogonal to the range of p´∆ A,0`1 q. Our goal is to show that h " 0. We define the Yosida type approximation of Laplacian with j ě 1. We show that We remark that, for any j ě 1 and f P L 2 , ρ j f P H 2 . For any b P L 2 , where R j,k " rρ j , p´∆ A,0 qsρ k . Since for g P H 2 pR n q, p´∆ A,0 qg " ∇Ä ∇g in the distributional sense and ρ j commutes with ∇, by Lemma 2.2, rρ j , p´∆ A,0 qs " j∇¨rpj´∆q´1, As∇ " jpj´∆q´1∇¨r´∆, As∇pj´∆q´1, where r∆, As j,k " p∇A j,k q¨∇`∇¨p∇A j,k q. Therefore Recall that for g P L 2 Moreover, as j Ñ 8, Hence, if h satisfies (2.3), then for any f P DpH A q, Therefore, h K RanpH A`1 q and the self-adjointness of H A implies h " 0.
3. Local well-posedness of (1.1) This section is devoted to the proof of the local well-posedness for the Cauchy problem associated with the model (1.1), where u 0 pxq " up0, xq is considered as initial datum. More precisely, we give now a proof of Theorem 3.
At first, we give the definition of D A,V . We use a functional calculus for the fractional powers of self-adjoint operators based on the integral representation below (see (4.7) in [7], for example).
Definition 6. Let A be a non-negative self-adjoint operator. For 0 ă s ă 2, We remark that here the formula plays a critical role. Now we can conclude this section by proving Theorem 3.

Proof of Theorem 3. We rewrite (1.1) in the integral form by means of its Duhamel's formulation
uptq " e itD A,V u 0`ż t 0 e ipt´τ qD A,V |upτ q| p´1 upτ q dτ. Here e itD A,V stands for the propagator associated with linear hGLK equation, namely (1.1) with trivial RHS. Briefly speaking, e itD A,V f solves the linear hGLK with f as initial datum. By Lemma A.2, e itD A,V is a uniformly bounded operator on H s for n " 1 and s " 1 or for n " 2, 3 and s " 2. A standard fixed point argument implies that (3.3) has a solutions in Cpr0, T q; H 1 q if n " 1 and in Cpr0, T q; H 2 q if n " 2, 3.

4.1.
Preliminary. The following representation is essential for our approach to study commutator estimates.
Proof. Using the spectral measures E µ for A (see [20,Theorem VII.7], for instance), we can write In the next lemma, we recall the well-known result that the function t Ñ t s , s P r0, 1s, is operator monotone on the set of bounded operators in a Hilbert space. One can see [18] for the original matrix-valued version of the statement, [10, Proposition 4.2.8] for the case s " 1{2 and [19] for a short proof of the general case. See also [9,12].  for 0 ă s ď 1. Moreover, if A 1 is invertible, so is A 2 , and xf, A´1 2 f y ď xf, A´1 1 f y.

Fractional Leibniz Rules.
Here we collect some useful Leibniz rules for fractional power of the classical Laplace operator.
Remark 4.1. In [17], one can find the refined estimate and more general estimates, but for simplicity, we use only Lemma 4.5.
In the sequel, we shall also need a generalization, obtained in [8], of the classical Kato-Ponce estimate, introduced in the seminal and well celebrated work [13]. We recall it.
For s ą maxt0, n{r´nu or s P 2N (the set of positive even integers), there exists C ą 0 such that

4.3.
Key estimate for Proposition 2. The purpose of this subsection is to show that the commutator between D A,V and a localized weight function is realized as a bounded operator in L 2 under the following assumptions: for some 1{4 ă σ ď 1; for any g, h P H 1{2 xg, ∇pAp∇f qhqy À }p´∆q 1{4 ∇f } L 8 }p´∆q 1{4 g} L 2 }h} L 2 }∇f } L 8 }p´∆q 1{4 g} L 2 }p´∆q 1{4 h} L 2 . . Then for any j P Z, Proof. We first prove (4.6). The same relation given at the beginning of the proof of Lemma 2.2 and the triangular inequality gives where we have used the fact that By (4.3), the second term on the R.H.S. of (4.8) is estimated as We next prove (4.7). By Lemma 4.1, it is enough to showˇˇˇż By (4.4) and (4.5), the L.H.S. of (4.9) is estimated by Then, by Lemma 4.2, the first integral on the R.H.S. of the last inequality is estimated by 2 3j{2 ď 1 satisfying (4.2). The second integral is also estimated by

Proof of Proposition 2.
We are now in a position to prove Proposition 2. We treat separately the two cases.
(4.5) may be obtained by decomposing B j pA j,k pB k f qhq as follows: where is, up to a complex constant, the standard Riesz transform, and The first term on the R.H.S. of (4.10) is easily estimated by the Hölder inequality and Lemma 4.6. Here we recall that (1.5) implies }A j,k } L 8 ă 8. The other terms are estimated similarly, since by Lemma 4.5 and (1.6), we have We now show Proposition 2 with V " 0. Since xg, rD A,0 , f shy " xP ďj g, rD A,0 , f shy`xg, rD A,0 , f sP ďj hy`xP ąj g, rD A,0 , f sP ąj hy "´xh, rD A,0 , f sP ďj gy`xg, rD A,0 , f sP ďj hy`xP ąj g, rD A,0 , f sP ąj hy, Lemma 4.2 implies the estimate. We next show (1.7) with V ı 0. (1.7) follows from the fact that for any g P H 1 , }pD A,0´DA,V qg} L 2 ď }V } L q,8 }g} L 2 . Indeed, by Lemma 4.1, The L 2 -norm of the first integral on the R.H.S. of the last equality is shown to be bounded by the fact that for any non-negative self-adjoint operator A }pλ`Aq´1g} L 2 ď λ´1}g} L 2 . By (1.5) and Lemma 4.3, Then, the L 2 -norm of the second integral is shown to be bounded by

The finite time blow-up result
Theorem 4 may be concluded be means of the following ODE argument.
Lemma 5.1. Let A, B ą 0 and q ą 1. If f P C 1 pr0, T q; R`q satisfies f p0q ą 0 and Proof. For completeness, we sketch the proof. Let f " e´A t g. Then g 1 " Be´A pq´1qt g q and therefore, pe´A pq´1qt´1 q.
The conclusion follows straightforward.
We exploit Lemma 5.1 in the proof of Theorem 4.
Proof of Theorem 4. Case 1. Let w P B 1 8,1 pRq be a non-negative function satisfying 1{w P L 8 X L 2 . We put u " vw. Then v satisfies By multiplying v on the both hand sides of (5.1), integrating the resulting equation, and taking the real part, where we have used that By (5.2), we apply Lemma 5.1 with Then (1.9) implies that }vptq} L 2 is not uniformly controlled. Case 2. We rescale w P 9 B 1 8,1 as w R " wp¨{Rq with R ą 0. Then by (5.2), We apply Lemma 5.1 with which means AB´1 " R´1`p p´1q{2 . Therefore, if 1 ă p ă 3, AB´1 Ñ 0 as R Ñ 8 and this shows Theorem 4.

Appendix A. Equivalence of Sobolev norms
We show the equivalence of the standard H s -norms (for s " 1, 2) and the ones induced by the Hamiltonian H A,V . We begin with simple a priori estimates that imply the equivalence of H 1 norms.
Lemma A.1. Assume H2. If V P L q,8 pR n q`L 8 pR n q with q ą maxt1, n{2u, then for any α P p0, 1q there exists C ą 0, so that for any f P C 8 c pR n q, xp´α∇¨A∇´|V |qf, f y ě´C}f } 2 Proof. We know that uniform ellipticity assumption implies xp´∇¨A∇q f, f y " }f } 9 H 1 . We need to prove the inequality with 0 ă s ă 1, since this estimate and the Gagliardo-Nirenberg interpolation inequality so we have (A.1) and (A.2). In order to prove (A.3), we take and then we can writê H s due to Hölder inequality in Lorentz spaces and Sobolev embedding. The requirement 0 ă s ă 1 is fulfilled due to the assumption q ą n{2.
Lemma A.2. Assume A1, A2, H1, and H2. Then one can find positive constants C 1 ă C 2 so that for any f P C 8 c pR n q, Proof. The right inequality of (A.4) follows directly from the representation of ∆ A,V . Indeed Further we can take 1 r " 1 2´1 q , s " n q and then we can write }V f } L 2 À }V } L q,8 }f } L r,2 À }V } L q,8 }f } H s (A.5) with s P p0, 2q so interpolation yields the right-side estimate. Next we show the left inequality of (A.4) with V " 0. By A1, where G A " rp´∆q 1{2 , As∇ so that ∇¨G A f " rp´∆ A,0 q, p´∆q 1{2 sf " ∇rA, p´∆q 1{2 s∇f.
This inequality and (A.5) prove the left estimate in (A.4).

Appendix B. Estimate of the weight function
Our choice of w for the proof of the blow-up result is wpxq " xxy a with a P p1{2, 1q. The lower bound of a is required to guarantee that 1{w P L 2 pRq for Theorem 4. The upper bound of a follows from the following Proposition: Proposition 7. For a ă 1, x¨y a P 9 B 1 8,1 . Proof. We recall that 2´s j P j p´∆q s{2 is a bounded operator on L 8 . Therefore for j ě 0, }P j p´∆q 1{2 xxy a } L 8 À 2´j}2 j P j p´∆q´1 {2 ∆xxy a } L 8 À 2´j}∆xxy a } L 8 which implies P ě0 x¨y a P 9 B 1 8,1 . Moreover, for a ą 0 since }P j f } L 8 À 2 jn{p }f } L p and |∇xxy a | À xxy a´1 , by taking p " 2n 1´a }P j p´∆q 1{2 xxy a } L 8 À 2 jn{p }∇p´∆q´1 {2 ∇xxy a } L p À 2 jp1´aq{2 }xxy´1} 1´a L 2n .
Therefore P ď0 xxy a P 9 B