Global weak solution of 3-D focusing energy-critical nonlinear Schrödinger equation
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917231479226368 |
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| author | Cheng, Xing Guo, Chang-Yu Zheng, Yunrui |
| author_facet | Cheng, Xing Guo, Chang-Yu Zheng, Yunrui |
| contents | In this article, we prove the existence of global weak solutions to the three-dimensional focusing energy-critical nonlinear Schrödinger (NLS) equation in the non-radial case. Furthermore, we prove the weak-strong uniqueness for some class of initial data. The main ingredient of our new approach is to use solutions of an energy-critical Ginzburg-Landau equation as approximations for the corresponding nonlinear Schördinger equation.
In our proofs, we first show the dichotomy of global well-posedness versus finite time blow-up of energy-critical Ginzburg-Landau equation in $\dot{H}^1( \mathbb{R}^d)$ for $d = 3,4 $ when the energy is less than the energy of the stationary solution $W$. We follow the strategy of C. E. Kenig and F. Merle [25,26], using a concentration-compactness/rigidity argument to reduce the global well-posedness to the exclusion of a critical element. The critical element is ruled out by dissipation of the Ginzburg-Landau equation, including local smoothness, backwards uniqueness and unique continuation. The existence of global weak solution of the three dimensional focusing energy-critical nonlinear Schrödinger equation in the non-radial case then follows from the global well-posedness of the energy-critical Ginzburg-Landau equation via a limitation argument. We also adapt the arguments of M. Struwe [37,38] to prove the weak-strong uniqueness when the $\dot{H}^1$-norm of the initial data is bounded by a constant depending on the stationary solution $W$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_01226 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Global weak solution of 3-D focusing energy-critical nonlinear Schrödinger equation Cheng, Xing Guo, Chang-Yu Zheng, Yunrui Analysis of PDEs Classical Analysis and ODEs 35L05, 35L71 In this article, we prove the existence of global weak solutions to the three-dimensional focusing energy-critical nonlinear Schrödinger (NLS) equation in the non-radial case. Furthermore, we prove the weak-strong uniqueness for some class of initial data. The main ingredient of our new approach is to use solutions of an energy-critical Ginzburg-Landau equation as approximations for the corresponding nonlinear Schördinger equation. In our proofs, we first show the dichotomy of global well-posedness versus finite time blow-up of energy-critical Ginzburg-Landau equation in $\dot{H}^1( \mathbb{R}^d)$ for $d = 3,4 $ when the energy is less than the energy of the stationary solution $W$. We follow the strategy of C. E. Kenig and F. Merle [25,26], using a concentration-compactness/rigidity argument to reduce the global well-posedness to the exclusion of a critical element. The critical element is ruled out by dissipation of the Ginzburg-Landau equation, including local smoothness, backwards uniqueness and unique continuation. The existence of global weak solution of the three dimensional focusing energy-critical nonlinear Schrödinger equation in the non-radial case then follows from the global well-posedness of the energy-critical Ginzburg-Landau equation via a limitation argument. We also adapt the arguments of M. Struwe [37,38] to prove the weak-strong uniqueness when the $\dot{H}^1$-norm of the initial data is bounded by a constant depending on the stationary solution $W$. |
| title | Global weak solution of 3-D focusing energy-critical nonlinear Schrödinger equation |
| topic | Analysis of PDEs Classical Analysis and ODEs 35L05, 35L71 |
| url | https://arxiv.org/abs/2308.01226 |