scholarly journals Deterministic ill-posedness and probabilistic well-posedness of the viscous nonlinear wave equation describing fluid-structure interaction

2021 ◽  
pp. 1
Author(s):  
Jeffrey Kuan ◽  
Sunčica Čanić
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Fluid Structure Interaction ◽  
Well Posedness ◽  
Fluid Structure ◽  
Structure Interaction

scholarly journals On the Parabolic and Hyperbolic Liouville Equations

2021 ◽  
Author(s):  
Tadahiro Oh ◽  
Tristan Robert ◽  
Yuzhao Wang
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Gibbs Measures ◽  
Nonlinear Heat Equation ◽  
Well Posedness ◽  
Exponential Nonlinearity ◽  
Definite Structure ◽  
Beta 2 ◽  
Moment Bounds

AbstractWe study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity $$\lambda \beta e^{\beta u }$$ λ β e β u , forced by an additive space-time white noise. (i) We first study SNLH for general $$\lambda \in {\mathbb {R}}$$ λ ∈ R . 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 $$0< \beta ^2 < \frac{8 \pi }{3 + 2 \sqrt{2}} \simeq 1.37 \pi $$ 0 < β 2 < 8 π 3 + 2 2 ≃ 1.37 π . Our argument yields stability under the noise perturbation, thus improving Garban’s local well-posedness result (2020). (ii) In the defocusing case $$\lambda >0$$ λ > 0 , 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: $$0< \beta ^2 < 4\pi $$ 0 < β 2 < 4 π . (iii) As for SdNLW in the defocusing case $$\lambda > 0$$ λ > 0 , 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 $$\Phi ^4_3$$ Φ 3 4 -model) and prove local well-posedness of SdNLW for the range: $$0< \beta ^2 < \frac{32 - 16\sqrt{3}}{5}\pi \simeq 0.86\pi $$ 0 < β 2 < 32 - 16 3 5 π ≃ 0.86 π . 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 $$\lambda > 0$$ λ > 0 , these models formally preserve the associated Gibbs measures with the exponential nonlinearity. Under the same assumption on $$\beta $$ β as 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 general $$\lambda \in {\mathbb {R}}$$ λ ∈ R without using the positivity of the Gaussian multiplicative chaos. This proves local well-posedness of SNLH for the range $$0< \beta ^2 < \frac{4}{3} \pi \simeq 1.33 \pi $$ 0 < β 2 < 4 3 π ≃ 1.33 π , slightly smaller than that in (i), but provides Lipschitz continuity of the solution map in initial data as well as the noise.

scholarly journals Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation

2020 ◽  
Author(s):  
Bjoern Bringmann
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Local Existence ◽  
Approximate Solutions ◽  
Well Posedness ◽  
Dispersive Equations ◽  
Deterministic Theory ◽  
Nonlinear Smoothing ◽  
Iterative Decomposition

Abstract We study the derivative nonlinear wave equation $- \partial _{tt} u + \Delta u = |\nabla u|^2$ on $\mathbb{R}^{1 +3}$. The deterministic theory is determined by the Lorentz-critical regularity $s_L = 2$, and both local well-posedness above $s_L$ as well as ill-posedness below $s_L$ are known. In this paper, we show the local existence of solutions for randomized initial data at the super-critical regularities $s\geqslant 1.984$. In comparison to the previous literature in random dispersive equations, the main difficulty is the absence of a (probabilistic) nonlinear smoothing effect. To overcome this, we introduce an adaptive and iterative decomposition of approximate solutions into rough and smooth components. In addition, our argument relies on refined Strichartz estimates, a paraproduct decomposition, and the truncation method of de Bouard and Debussche.

A Perturbation Method for Low-Frequency Fluid-Structure Interaction Problems

1973 ◽  
Vol 40 (2) ◽  
pp. 388-394 ◽  
Author(s):  
Y. K. Lou
Keyword(s):  
Wave Equation ◽  
Perturbation Method ◽  
Electromagnetic Scattering ◽  
Perturbation Methods ◽  
Separation Of Variables ◽  
Fluid Structure Interaction ◽  
Low Frequency ◽  
Approximate Solutions ◽  
Fluid Structure ◽  
Structure Interaction

Perturbation methods have been used for electromagnetic scattering and diffraction problems in recent years. A similar method suitable for low-frequency fluid-structure interaction problems is presented. The essence of the method lies in the fact that approximate solutions for fluid-structure interaction problems can be obtained from a set of Poisson’s equations, rather than from the reduced wave equation. The method is particularly useful for those problems where the Poisson’s equation may be solved by the method of separation of variables while the reduced wave equation cannot. As an illustrative example, the vibrations of a submerged spherical shell is studied using the perturbation method and the accuracy of the method is demonstrated.

scholarly journals ON THE WAVE EQUATION WITH QUADRATIC NONLINEARITIES IN THREE SPACE DIMENSIONS

2011 ◽  
Vol 08 (01) ◽  
pp. 1-8 ◽  
Author(s):  
AXEL GRÜNROCK
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Parameter Range ◽  
Well Posedness ◽  
Quadratic Nonlinearities ◽  
The Cauchy Problem ◽  
Three Space

The Cauchy problem for the nonlinear wave equation [Formula: see text] in three space dimensions is considered. The data (u0, u1) are assumed to belong to [Formula: see text], where [Formula: see text] is defined by the norm [Formula: see text] Local well-posedness is shown in the parameter range 2 ≥ r > 1, [Formula: see text]. For r = 2 this coincides with the result of Ponce and Sideris, which is optimal on the Hs-scale by Lindblad's counterexamples, but nonetheless leaves a gap of ½ derivative to the scaling prediction. This gap is closed here except for the endpoint case. Corresponding results for □u = ∂u2 are obtained, too.

Global Well-Posedness for the Defocusing, Cubic, Nonlinear Wave Equation in Three Dimensions for Radial Initial Data in .H s × .H s-1, s> 1/2

2018 ◽  
Vol 2019 (21) ◽  
pp. 6797-6817
Author(s):  
Benjamin Dodson
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Three Dimensions ◽  
Well Posedness ◽  
Critical Space ◽  
Initial Value ◽  
Long Time ◽  
Well Posed

Abstract In this paper we study the defocusing, cubic nonlinear wave equation in three dimensions with radial initial data. The critical space is $\dot{H}^{1/2} \times \dot{H}^{-1/2}$. We show that if the initial data is radial and lies in $\left (\dot{H}^{s} \times \dot{H}^{s - 1}\right ) \cap \left (\dot{H}^{1/2} \times \dot{H}^{-1/2}\right )$ for some $s&gt; \frac{1}{2}$, then the cubic initial value problem is globally well-posed. The proof utilizes the I-method, long time Strichartz estimates, and local energy decay. This method is quite similar to the method used in [11].

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