On local well-posedness of logarithmic inviscid regularizations of generalized SQG equations in borderline Sobolev spaces

<p style='text-indent:20px;'>This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar <inline-formula><tex-math id="M1">\begin{document}$ \theta $\end{document}</tex-math></inline-formula> on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these regularized models in borderline Sobolev regularity have previously been studied by D. Chae and J. Wu when the velocity <inline-formula><tex-math id="M2">\begin{document}$ u $\end{document}</tex-math></inline-formula> is of lower singularity, i.e., <inline-formula><tex-math id="M3">\begin{document}$ u = -\nabla^{\perp} \Lambda^{ \beta-2}p( \Lambda) \theta $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M4">\begin{document}$ p $\end{document}</tex-math></inline-formula> is a logarithmic smoothing operator and <inline-formula><tex-math id="M5">\begin{document}$ \beta \in [0, 1] $\end{document}</tex-math></inline-formula>. We complete this study by considering the more singular regime <inline-formula><tex-math id="M6">\begin{document}$ \beta\in(1, 2) $\end{document}</tex-math></inline-formula>. The main tool is the identification of a suitable linearized system that preserves the underlying commutator structure for the original equation. We observe that this structure is ultimately crucial for obtaining continuity of the flow map. In particular, straightforward applications of previous methods for active transport equations fail to capture the more nuanced commutator structure of the equation in this more singular regime. The proposed linearized system nontrivially modifies the flux of the original system in such a way that it coincides with the original flux when evaluated along solutions of the original system. The requisite estimates are developed for this modified linear system to ensure its well-posedness.</p>

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The Muskat problem with surface tension and equal viscosities in subcritical $$L_p$$-Sobolev spaces

Journal of Elliptic and Parabolic Equations ◽

10.1007/s41808-021-00104-1 ◽

2021 ◽

Author(s):

Anca-Voichita Matioc ◽

Bogdan-Vasile Matioc

Keyword(s):

Mathematical Model ◽

Surface Tension ◽

Global Existence ◽

Sobolev Spaces ◽

Existence Of Solutions ◽

Well Posedness ◽

Global Existence Of Solutions ◽

Muskat Problem ◽

The Mathematical Model ◽

Quasilinear Parabolic

AbstractIn this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces $$W^s_p(\mathbb {R})$$ W p s ( R ) , where $${p\in (1,2]}$$ p ∈ ( 1 , 2 ] and $${s\in (1+1/p,2)}$$ s ∈ ( 1 + 1 / p , 2 ) . This is achieved by showing that the mathematical model can be formulated as a quasilinear parabolic evolution problem in $$W^{\overline{s}-2}_p(\mathbb {R})$$ W p s ¯ - 2 ( R ) , where $${\overline{s}\in (1+1/p,s)}$$ s ¯ ∈ ( 1 + 1 / p , s ) . Moreover, we prove that the solutions become instantly smooth and we provide a criterion for the global existence of solutions.

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Well-posedness of the linearized system and general asymptotics

Evolution Equations in Thermoelasticity ◽

10.1201/9781482285789-9 ◽

2000 ◽

pp. 23-44

Keyword(s):

Well Posedness ◽

Linearized System

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On the Parabolic and Hyperbolic Liouville Equations

Communications in Mathematical Physics ◽

10.1007/s00220-021-04125-8 ◽

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.

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Error feedback regulation for 1D anti-stable wave equation

Transactions of the Institute of Measurement and Control ◽

10.1177/01423312211059278 ◽

2021 ◽

pp. 014233122110592

Author(s):

Zhiyuan Li ◽

Feng-Fei Jin

Keyword(s):

Wave Equation ◽

Tracking Error ◽

Feedback Regulation ◽

Original System ◽

Well Posedness ◽

Error Feedback ◽

One Dimensional ◽

Internal Loop ◽

Finite Dimensional ◽

Uniformly Bounded

This paper is concerned with the boundary error feedback regulation for a one-dimensional anti-stable wave equation with distributed disturbance generated by a finite-dimensional exogenous system. Transport equation and regulator equation are introduced first to deal with the anti-damping on boundary and the distributed disturbance of the original system. Then, the tracking error and its derivative are measured to design an observer for both exosystem and auxiliary partial differential equation (PDE) system to recover the state. After proving the well-posedness of the regulator equations, we propose an observer-based controller to regulate the tracking error to zero exponentially and keep the states of all the internal loop uniformly bounded. Finally, some numerical simulations are presented to validate the effectiveness of the proposed controller.

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