Non-Uniqueness of Weak Solutions of the Quantum-Hydrodynamic System

We investigate the non-uniqueness of weak solutions of the Quantum-Hydrodynamic system. This form of ill-posedness is related to the change of the number of connected components of the support of the position density (called nodal domains) of the weak solution throughout its time evolution. We start by considering a scenario consisting of initial and final time, showing that if there is a decrease in the number of connected components, then we have non-uniqueness. This result relies on the Brouwer invariance of domain theorem. Then we consider the case in which the results involve a time interval and a full trajectory (position-current densities). We introduce the concept of trajectory-uniqueness and its characterization.

1. Introduction. In this paper, we study the non-uniqueness of (so-called Schrödinger-generated) bounded energy weak solutions of the Quantum Hydrodynamic (QHD) system [13] ̺ t + divJ = 0, along with appropriate initial data, as discussed below. We define the quantities ̺, J of the QHD system through their connection with the Schrödinger equation subject to the initial condition Hence, ̺ = ̺ (x, t) := |ψ (x, t)| 2 and J = J (x, t) := Im ψ (x, t) ∇ψ (x, t) . We call ̺ the position density and J the current density generated by the wave function ψ. For a detailed derivation of the QHD system see, e.g., [1,4,5,6,12].
The Schrödinger energy is given by It is a constant of the motion, i.e., E (t) = E (t = 0) ∀t ∈ R (under the assumptions made below). Henceforth we assume that the potential V is in C 1,1 loc R d and bounded from below, such that the Hamiltonian H := − 1 2 ∆ + V is essentially self-adjoint on L 2 R d .
Gasser and Markowich [10] showed that if E (t = 0) < ∞ and ψ 0 ∈ L 2 R d , then the position and current densities corresponding to the uniquely defined energyconserving solution ψ ∈ C R t ; L 2 R d x satisfy the QHD system in the sense of distributions with initial data ̺ 0 (x) = ̺ (x, t = 0) = |ψ 0 (x)| 2 and J 0 (x) = J (x, t = 0) = Im ψ 0 (x) ∇ψ 0 (x) . Clearly we have ̺ ∈ C R t ; L 1 R d Note that the QHD system is formulated above in conservative form, namely, as compressible Euler equations where the enthalpy is given by the sum of the so-called Bohm potential ̺ and the external potential V . In this form, vacuum states ̺ = 0 do not have to be dealt with particularly (see [10]), which is not the case for the associated nonconservative form. The reason for this lies in the fact that the velocity u := J/̺ cannot be reasonably defined for wave-functions ψ which exhibit nodes, i.e., vacuum states where ψ = 0. Contrary to this, the internal energy tensor (J ⊗ J) /̺ makes perfect sense when J and ̺ are defined through any bounded energy wave-function (see [2,3]). However, as we shall point out in the sequel, the conservative form of QHD is prone to an ill-posedness in the form of non-uniqueness of finite energy weak solutions, which is generated precisely by the occurrence of vacuum states.
On the other hand, the QHD equations are the zeroth and first order moment equations of the Bohm equation with the mono kinetic closure. See [9,15,16] for a detailed treatment of this topic.
To motivate our subsequent analysis, take ψ 0 = ψ 0 (x) smooth in L 2 R d and define ̺ 0 = ̺ 0 (x) and J 0 = J 0 (x) as before. Moreover, consider a second L 2 −wave function, ϕ 0 = ϕ 0 (x). We want to determine the conditions under which To this end, define D 0 := x ∈ R d : ̺ 0 = 0 and let Λ 0 1 , Λ 0 2 , . . . be the (countably many) connected components of D 0 . Furthermore, let ψ 0 be given by where ̺ 0 (x) and S 0 (x) are smooth real valued functions. Then, which implies ∇R 0 = ∇S 0 on D 0 . Therefore, we find that due to the connectivity of Λ 0 k , k ∈ N,

Hence, (2) is satisfied if and only if
. . a.e. in R d , for some C k ∈ R. Note that, under appropriate smoothness and geometric assumptions, the energy associated to ϕ 0 is finite for all choices of the constants C k (see Section 2).
We shall argue that, under certain assumptions, the constants C k can be chosen such that the initial wave-function ϕ 0 generates a Schrödinger solution whose position and current densities differ from those generated by the Schrödinger evolution of ψ 0 (the original initial wave-function) at a time T = 0. To be more specific, consider the following scenario. Let d = 1 and ψ 0 = ψ 0 (x) smooth with ψ 0 (0) = 0, ψ 0 (x) = 0 for x = 0, such that at some T > 0 we have ψ (x, T ) = 0 for all x ∈ R (we shall show an example below).
It is straightforward to check that ϕ 0 ∈ H 1 R d . Assume that the position and current densities of ψ (T ) and ϕ (T ) coincide. Since Now solve the Schrödinger equation back to t = 0 and find that ϕ (x, t = 0) = e iβ ψ (x, t = 0) on R, i.e., ϕ 0 (x) = e iβ ψ 0 (x) on R, which is a contradiction. Therefore, we conclude the non-uniqueness of the corresponding initial value problem (IVP) for the QHD system.
As an example, consider the harmonic oscillator in 1D, i.e., equation (1) with V (x) = x 2 /2, x ∈ R. In this case there is a family of solutions (with appropriate initial conditions) for n = 0, 1, 2, . . . and energy levels given by E n = (n + 1/2). The first three members of the family are Due to the linearity of the Schrödinger equation, any linear combination of the previous solutions will also be a solution. Hence, consider Then Thus, we have our required scenario if we choose ψ 0 = ψ(x, t = 0) and any 0 < T < π/2. The primary goal of this paper is to give rather explicit (sufficient) conditions on a bounded energy Schrödinger solution which guarantee that the QHD trajectory (position-current densities) generated by such wave-function is not unique in the sense that a different QHD trajectory intersects it at some t ∈ R. It turns out that we can state such conditions in connection with the topological structure of the so-called nodal domains of the wave function, defined as the connected components of the set where the wave function does not vanish (in other words, the connected domains of the non-vacuum set of the quantum flow).
The rest of the paper is organized as follows. In Section 2 we generalize the previous result to wave function solution of the Schrödinger IVP (QHD-IVP) with less regularity, arbitrary dimension, and an arbitrary number of connected components. In Section 3, we consider the case in which the results involve a time interval and a full QHD trajectory.
Remark 1. Assume that all the connected components Λ α of D 0 are of locally finite perimeter (Λ α are Caccioppoli sets), that ψ is continuous on R d and in H 1 R d .
Then the characteristic function 1 Λα of Λ α has locally bounded total variation, which in turn implies that its distributional gradient is a vector valued (signed) Radon measure supported on the boundary of Λ α . Thus the function ϕ, defined by Proof. To simplify the presentation, assume that N 0 < ∞ . Let Λ 0 1 , . . . , Λ 0 N0 be the connected components of D 0 and let Λ T 1 , . . . , Λ T NT be the connected components of D T .
This map is injective since the backward Schrdinger IVP has the uniqueness property. Continuity of F is a consequence the L 2 R d −continuity of the Schrdinger evolution in the following way. Take a sequence α Therefore, l , x ∈ Λ 0 l , l = 1, . . . , N 0 converges to l , x ∈ Λ 0 l , l = 1, . . . , N 0 in L 2 R d . By L 2 −continuity of the Schrdinger evolution, we have that k , x ∈ Λ T k , k = 1, . . . , N T . By the same computation, we find that k , k = 1, . . . , N T . Hence, F is continuous. Since by assumption N 0 > N T , we have a contradiction due to the fact that, as a consequence of the Brouwer invariance of domain theorem (see, e.g., [7,11,17]), there is no continuous injective mapping from R n to R m when n > m.
It is straightforward to show that the result also holds for N 0 = ∞ > N T .
In simple terms, this means that different trajectories do not intersect, neither forward nor backward in time.

PETER MARKOWICH AND JESÚS SIERRA
Proposition 4. Consider a QHD-trajectory, S = {(̺ (t) , J (t)) : t ∈ R}, with ψ ∈ C R d x × R t and bounded energy. If for some T ∈ R the number N of connected components of R d+1 ⊇ Ω := ̺ (x, t) = 0 : (x, t) ∈ R d+1 is smaller than the number N T of connected components of then S is not trajectory-unique, provided that all connected components have locally finite perimeter.
Let ψ = ψ (x, t) be a smooth solution of the Schrdinger equation, with obviously ψ| Γ = 0. Denote by Υ the unit normal of Γ, pointing (for definiteness sake) into Ω 1 and set Υ = Υ x Υ t according to the coordinate ordering x t .
For the following results, we use the generalized Green's formula by De Giorgi-Federer: where F is any locally Lipschitz function, E ⊆ R n is of finite perimeter, Υ is the normal of ∂E, and ∂ * E is the reduced boundary (see, e.g., [8,14]).
The consequence is: Proposition 8. Let all connected components Ω α of Ω (α ∈ B) be of locally finite perimeter and let ∇ψ ∈ C 0,1 loc R d x × R t . Assume that there is a non-space-like interface segment, Γ, between two connected components such that ∇ψ · Υ x = 0 H d − a.e. on Γ * .
Then the QHD-trajectory generated by ψ = ψ (x, t) is not trajectory-unique on R t .