Randomization and the Gross-Pitaevskii hierarchy. (English) Zbl 1372.35287
The authors study the Gross-Pitaevskii hierarchy on the spatial domain \(\mathbb T^3\).
For a fixed spatial domain \(\Lambda = \mathbb T^d\) or \(\Lambda =\mathbb R^d\), the Gross-Pitaevskii hierarchy on \(\Lambda\) is defined to be a sequence \((\gamma^{(k)}(t))_k\) of functions \(\gamma^k:\mathbb R\times \mathbb \Lambda^k\times \mathbb \Lambda^k\to\mathbb C\), which solve the following infinite system of linear PDEs: \[ \begin{cases} i\partial_t\gamma^{(k)}+(\Delta_{\vec {x}_k}-\Delta_{\vec {x}'_k})\gamma^{(k)}=\sum\limits_{j=1}^{k}B_{j,k+1}(\gamma^{(k+1)}) \\ \gamma^{(k)}\mid_{t=0}=\gamma_{0}^{(k)}. \end{cases} \] Here, \((\gamma_{0}^{(k)})_k\) is a fixed sequence of density matrices \(\gamma_{0}^{(k)}\), and \((\gamma^{(k)})_k=(\gamma^{(k)}(t))_k\) is a sequence of time-dependent density matrices of order \(k\). \(\Delta_{\vec {x}_k}:=\sum\limits_{j=1}^k\Delta_{x_j}\) is the Laplacian operator in the first set of \(k\) spatial variables and \(\Delta_{\vec {x}'_k}:=\sum\limits_{j=1}^k\Delta_{x_j'}\) is the Laplacian operator in the second set of \(k\) spatial variables. The map \(B_{j,k+1}\) is called the collision operator.
The Gross-Pitaevskii hierarchy is related to the nonlinear Schrödinger equation.
Earlier, the conditional uniqueness was obtained as a spacetime estimate for a fixed regularity exponent \(\alpha\) and for a fixed time \(T\in (0,+\infty]\). It was shown that the estimate holds on \(\mathbb T^3\) for \(\alpha>1\).
By using the randomization procedure (multiplying the Fourier coefficients by a sequence of independent identically distributed standard Bernoulli random variables), the authors prove the main randomized spacetime estimate in the strong form:
Theorem 1. Let \(\alpha>\dfrac{3}{4}\) be given. There exists a constant \(C_0\) depending only on \(\alpha\) such that for all \(k\in\mathbb N\) and \(1\leq j\leq k\), the following bound holds. \(\|S^{(k,\alpha)}[B_{j,k+1}]^{\omega}\gamma_0^{(k+1)}\|_{L^2(\Omega\times\mathbb T^{3k}\times\mathbb T^{3k})}\leq C_0\|S^{(k+1,\alpha)}\gamma_0^{(k+1)}\|_{L^2(\mathbb T^{(3k+1)}\times\mathbb T^{(3k+1)})}\)
The operator \([B_{j,k+1}]^{\omega}\) is a randomized collision operator, for a fixed \(\omega\) belonging to the probability space \(\Omega\). It is obtained from the collision operator \(B_{j,k+1}\) by appropriately randomizing the Fourier coefficients by means of standard Bernoulli random variables.
The properties of the randomized Gross-Pitaevskii hierarchy are investigated. It is shown that its factorized solutions are obtained as tensor products of solution to a nonlinear Schrödinger equation with a random nonlinearity. The convergence to zero of a sequence of Duhamel terms in a low-regularity space containing a random component is analyzed.
For a fixed spatial domain \(\Lambda = \mathbb T^d\) or \(\Lambda =\mathbb R^d\), the Gross-Pitaevskii hierarchy on \(\Lambda\) is defined to be a sequence \((\gamma^{(k)}(t))_k\) of functions \(\gamma^k:\mathbb R\times \mathbb \Lambda^k\times \mathbb \Lambda^k\to\mathbb C\), which solve the following infinite system of linear PDEs: \[ \begin{cases} i\partial_t\gamma^{(k)}+(\Delta_{\vec {x}_k}-\Delta_{\vec {x}'_k})\gamma^{(k)}=\sum\limits_{j=1}^{k}B_{j,k+1}(\gamma^{(k+1)}) \\ \gamma^{(k)}\mid_{t=0}=\gamma_{0}^{(k)}. \end{cases} \] Here, \((\gamma_{0}^{(k)})_k\) is a fixed sequence of density matrices \(\gamma_{0}^{(k)}\), and \((\gamma^{(k)})_k=(\gamma^{(k)}(t))_k\) is a sequence of time-dependent density matrices of order \(k\). \(\Delta_{\vec {x}_k}:=\sum\limits_{j=1}^k\Delta_{x_j}\) is the Laplacian operator in the first set of \(k\) spatial variables and \(\Delta_{\vec {x}'_k}:=\sum\limits_{j=1}^k\Delta_{x_j'}\) is the Laplacian operator in the second set of \(k\) spatial variables. The map \(B_{j,k+1}\) is called the collision operator.
The Gross-Pitaevskii hierarchy is related to the nonlinear Schrödinger equation.
Earlier, the conditional uniqueness was obtained as a spacetime estimate for a fixed regularity exponent \(\alpha\) and for a fixed time \(T\in (0,+\infty]\). It was shown that the estimate holds on \(\mathbb T^3\) for \(\alpha>1\).
By using the randomization procedure (multiplying the Fourier coefficients by a sequence of independent identically distributed standard Bernoulli random variables), the authors prove the main randomized spacetime estimate in the strong form:
Theorem 1. Let \(\alpha>\dfrac{3}{4}\) be given. There exists a constant \(C_0\) depending only on \(\alpha\) such that for all \(k\in\mathbb N\) and \(1\leq j\leq k\), the following bound holds. \(\|S^{(k,\alpha)}[B_{j,k+1}]^{\omega}\gamma_0^{(k+1)}\|_{L^2(\Omega\times\mathbb T^{3k}\times\mathbb T^{3k})}\leq C_0\|S^{(k+1,\alpha)}\gamma_0^{(k+1)}\|_{L^2(\mathbb T^{(3k+1)}\times\mathbb T^{(3k+1)})}\)
The operator \([B_{j,k+1}]^{\omega}\) is a randomized collision operator, for a fixed \(\omega\) belonging to the probability space \(\Omega\). It is obtained from the collision operator \(B_{j,k+1}\) by appropriately randomizing the Fourier coefficients by means of standard Bernoulli random variables.
The properties of the randomized Gross-Pitaevskii hierarchy are investigated. It is shown that its factorized solutions are obtained as tensor products of solution to a nonlinear Schrödinger equation with a random nonlinearity. The convergence to zero of a sequence of Duhamel terms in a low-regularity space containing a random component is analyzed.
Reviewer: Artyom Andronov (Saransk)
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35B45 | A priori estimates in context of PDEs |
Keywords:
Gross-Pitaevskii hierarchy; nonlinear Schrödinger equation; randomization; conditional uniqueness; density matrixes; collision operator; Duhamel expansionReferences:
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