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 $s> \frac{1}{2}$, 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].

scholarly journals The Focusing Energy-Critical Nonlinear Wave Equation With Random Initial Data

2019 ◽  
Author(s):  
Carlos Kenig ◽  
Dana Mendelson
Keyword(s):  
Wave Equation ◽  
Euclidean Space ◽  
Initial Data ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Stationary Solutions ◽  
Small Data ◽  
Random Initial Data ◽  
Long Time ◽  
Linearized Operator

Abstract We consider the focusing energy-critical quintic nonlinear wave equation in 3D Euclidean space. It is known that this equation admits a one-parameter family of radial stationary solutions, called solitons, which can be viewed as a curve in $ \dot H^s_x({{\mathbb{R}}}^3) \times H^{s-1}_x({{\mathbb{R}}}^3)$, for any $s> 1/2$. By randomizing radial initial data in $ \dot H^s_x({{\mathbb{R}}}^3) \times H^{s-1}_x({{\mathbb{R}}}^3)$ for $s> 5/6$, which also satisfy a certain weighted Sobolev condition, we produce with high probability a family of radial perturbations of the soliton that give rise to global forward-in-time solutions of the focusing nonlinear wave equation that scatter after subtracting a dynamically modulated soliton. Our proof relies on a new randomization procedure using distorted Fourier projections associated to the linearized operator around a fixed soliton. To our knowledge, this is the 1st long-time random data existence result for a focusing wave or dispersive equation on Euclidean space outside the small data regime.

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 Global well-posedness for the nonlinear wave equation in analytic Gevrey spaces

2021 ◽  
Vol 275 ◽  
pp. 234-249
Author(s):  
Daniel Oliveira da Silva ◽  
Alejandro J. Castro
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Well Posedness ◽  
Gevrey Spaces

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.

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.

scholarly journals Long time stability of KAM tori for nonlinear wave equation

2015 ◽  
Vol 258 (8) ◽  
pp. 2823-2846 ◽  
Author(s):  
Hongzi Cong ◽  
Meina Gao ◽  
Jianjun Liu
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Kam Tori ◽  
Time Stability ◽  
Long Time ◽  
Long Time Stability
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