On the Parabolic and Hyperbolic Liouville Equations
On the Parabolic and Hyperbolic Liouville Equations
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DOI：
10.1007/s00220-021-04125-8
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发表时间：
2019-08
影响因子：
2.4
通讯作者：
Tadahiro Oh;T. Robert;Yuzhao Wang
中科院分区：
文献类型：
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作者：
Tadahiro Oh;T. Robert;Yuzhao Wang
研究了在时空白色噪声作用下二维随机非线性热方程和具有指数非线性项的随机阻尼非线性波动方程的解. (i)我们首先研究SNLH的一般性。通过建立相应的高斯乘性混沌的高阶矩上界和利用高斯乘性混沌的正性，我们证明了SNLH的局部适定性。我们的论点在噪声扰动下产生稳定性，从而改进了Garban的局部适定性结果（2020）。(ii)在离焦情形下，我们利用了方程的符号确定性和高斯乘性混沌的正性。这使我们能够证明SNLH在以下范围内的全局适定性：(iii)至于SdNLW在散焦的情况下，我们超越了Da Prato-Debussche的论点，并引入了非线性分量的分解，使我们能够恢复一个符号明确的结构的一个粗略的未知部分，而其他部分享有更强的平滑属性。因此，我们将SdNLW简化为一个方程组（如动力学模型的paracontrolled方法），并证明了SdNLW的局部适定性：。这一结果（转化为具有指数非线性的确定性非线性波动方程的随机数据适定性的背景）解决了Sun和Tzvetkov（2020）提出的一个悬而未决的问题。(iv)当，这些模型形式上保持相关的吉布斯措施与指数非线性。在上述（ii）和（iii）中相同的假设下，我们证明了几乎肯定的全局适定性（特别是对于SdNLW）和Gibbs测度在抛物和双曲设置中的不变性。(v)在附录中，我们给出了一个不利用高斯乘性混沌的正性证明一般情形下的SNLH的局部适定性的论证。这证明了局部适定性的SNLH的范围，略小于（i），但提供了初始数据以及噪声的解映射的Lipschitz连续性。
We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity, forced by an additive space-time white noise. (i) We first study SNLH for general. By establishing higher moment bounds of the relevant Gaussian multiplicative chaos and exploiting the positivity of the Gaussian multiplicative chaos, we prove local well-posedness of SNLH for the range. Our argument yields stability under the noise perturbation, thus improving Garban’s local well-posedness result (2020). (ii) In the defocusing case, we exploit a certain sign-definite structure in the equation and the positivity of the Gaussian multiplicative chaos. This allows us to prove global well-posedness of SNLH for the range:. (iii) As for SdNLW in the defocusing case, we go beyond the Da Prato-Debussche argument and introduce a decomposition of the nonlinear component, allowing us to recover a sign-definite structure for a rough part of the unknown, while the other part enjoys a stronger smoothing property. As a result, we reduce SdNLW into a system of equations (as in the paracontrolled approach for the dynamical-model) and prove local well-posedness of SdNLW for the range:. This result (translated to the context of random data well-posedness for the deterministic nonlinear wave equation with an exponential nonlinearity) solves an open question posed by Sun and Tzvetkov (2020). (iv) When, these models formally preserve the associated Gibbs measures with the exponential nonlinearity. Under the same assumption onas in (ii) and (iii) above, we prove almost sure global well-posedness (in particular for SdNLW) and invariance of the Gibbs measures in both the parabolic and hyperbolic settings. (v) In Appendix, we present an argument for proving local well-posedness of SNLH for generalwithoutusing the positivity of the Gaussian multiplicative chaos. This proves local well-posedness of SNLH for the range, slightly smaller than that in (i), but provides Lipschitz continuity of the solution map in initial data as well as the noise.