Local well-posedness and ill-posedness for the fractal Burgers equation in homogeneous Sobolev spaces

We establish local well-posedness in Sobolev spaces Hs(𝕋), with s ≥ -1/2, for the initial value problem issues of the equation [Formula: see text] where η > 0, (Lu)∧(k) = -Φ(k)û(k), k ∈ ℤ and Φ ∈ ℝ is bounded above. Particular cases of this problem are the Korteweg–de Vries–Burgers equation for Φ(k) = -k2, the derivative Korteweg–de Vries–Kuramoto–Sivashinsky equation for Φ(k) = k2 - k4, and the Ostrovsky–Stepanyams–Tsimring equation for Φ(k) = |k| - |k|3.

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Uniform local well-posedness and inviscid limit for the Benjamin-Ono-Burgers equation

Science China Mathematics ◽

10.1007/s11425-020-1807-4 ◽

2021 ◽

Author(s):

Mingjuan Chen ◽

Boling Guo ◽

Lijia Han

Keyword(s):

Burgers Equation ◽

Inviscid Limit ◽

Well Posedness

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Viscous Flow Around a Rigid Body Performing a Time-periodic Motion

Journal of Mathematical Fluid Mechanics ◽

10.1007/s00021-021-00556-4 ◽

2021 ◽

Vol 23 (1) ◽

Author(s):

Thomas Eiter ◽

Mads Kyed

Keyword(s):

Rigid Body ◽

Sobolev Spaces ◽

Periodic Motion ◽

Viscous Incompressible Fluid ◽

Body Motion ◽

Translational Velocity ◽

Ill Posed ◽

Homogeneous Sobolev Spaces ◽

The Mean ◽

Time Periodic

AbstractThe equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed time-periodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigid-body motion are compatible, and that the mean translational velocity is non-zero, existence of a time-periodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is ill-posed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces.

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Bilinear forms on non-homogeneous Sobolev spaces

Forum Mathematicum ◽

10.1515/forum-2019-0311 ◽

2020 ◽

Vol 32 (4) ◽

pp. 995-1026

Author(s):

Carme Cascante ◽

Joaquín M. Ortega

Keyword(s):

Sobolev Spaces ◽

Bilinear Form ◽

Positive Measure ◽

Bilinear Forms ◽

Homogeneous Sobolev Spaces

AbstractIn this paper, we show that if {b\in L^{2}(\mathbb{R}^{n})}, then the bilinear form defined on the product of the non-homogeneous Sobolev spaces {H_{s}^{2}(\mathbb{R}^{n})\times H_{s}^{2}(\mathbb{R}^{n})}, {0<s<1}, by(f,g)\in H_{s}^{2}(\mathbb{R}^{n})\times H_{s}^{2}(\mathbb{R}^{n})\to\int_{% \mathbb{R}^{n}}(\mathrm{Id}-\Delta)^{\frac{s}{2}}(fg)(\mathbf{x})b(\mathbf{x})% \mathop{}\!d\mathbf{x}is continuous if and only if the positive measure {\lvert b(\mathbf{x})\rvert^{2}\mathop{}\!d\mathbf{x}} is a trace measure for {H_{s}^{2}(\mathbb{R}^{n})}.

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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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