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1. Energy Harvesting Performance of a Flapping Foil with Leading Edge Motion

   Prier, M. (April 2019, Dissertation)

   F or c e d at a f or a fl a p pi n g f oil e n er g y h ar v e st er wit h a cti v e l e a di n g e d g e m oti o n o p er ati n g i n t h e l o w r e d u c e d fr e q u e n c y r a n g e i s c oll e ct e d t o d et er mi n e h o w l e a di n g e d g e m oti o n aff e ct s e n er g y h ar v e sti n g p erf or m a n c e. T h e f oil pi v ot s a b o ut t h e mi dc h or d a n d o p er at e s i n t h e l o w r e d u c e d fr e q u e n c y r a n g e of 𝑓𝑓 𝑓𝑓 / 𝑈𝑈 ∞ = 0. 0 6 , 0. 0 8, a n d 0. 1 0 wit h 𝑅𝑅 𝑅𝑅 = 2 0 ,0 0 0 − 3 0 ,0 0 0 , wit h a pit c hi n g a m plit u d e of 𝜃𝜃 0 = 7 0 ∘ , a n d a h e a vi n g a m plit u d e of ℎ 0 = 0. 5 𝑓𝑓 . It i s f o u n d t h at l e a di n g e d g e m oti o n s t h at r e d u c e t h e eff e cti v e a n gl e of att a c k e arl y t h e str o k e w or k t o b ot h i n cr e a s e t h e lift f or c e s a s w ell a s s hift t h e p e a k lift f or c e l at er i n t h e fl a p pi n g str o k e. L e a di n g e d g e m oti o n s i n w hi c h t h e eff e cti v e a n gl e of att a c k i s i n cr e a s e d e arl y i n t h e str o k e s h o w d e cr e a s e d p erf or m a n c e. I n a d diti o n a di s cr et e v ort e x m o d el wit h v ort e x s h e d di n g at t h e l e a di n g e d g e i s i m pl e m e nt f or t h e m oti o n s st u di e d; it i s f o u n d t h at t h e m e c h a ni s m f or s h e d di n g at t h e l e a di n g e d g e i s n ot a d e q u at e f or t hi s p ar a m et er r a n g e a n d t h e m o d el c o n si st e ntl y o v er pr e di ct s t h e a er o d y n a mi c f or c e s. 

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2. Degenerate linear parabolic equations in divergence form on the upper half space

   https://doi.org/10.1090/tran/8892  

   Dong, Hongjie; Phan, Tuoc; Tran, Hung (June 2023, Transactions of the American Mathematical Society)

   We study a class of second-order degenerate linear parabolic equations in divergence form in ( − ∞ , T ) × R + d (-\infty , T) \times {\mathbb {R}}^d_+ with homogeneous Dirichlet boundary condition on ( − ∞ , T ) × ∂ R + d (-\infty , T) \times \partial {\mathbb {R}}^d_+ , where R + d = { x ∈ R d : x d > 0 } {\mathbb {R}}^d_+ = \{x \in {\mathbb {R}}^d: x_d>0\} and T ∈ ( − ∞ , ∞ ] T\in {(-\infty , \infty ]} is given. The coefficient matrices of the equations are the product of μ ( x d ) \mu (x_d) and bounded uniformly elliptic matrices, where μ ( x d ) \mu (x_d) behaves like x d α x_d^\alpha for some given α ∈ ( 0 , 2 ) \alpha \in (0,2) , which are degenerate on the boundary { x d = 0 } \{x_d=0\} of the domain. Our main motivation comes from the analysis of degenerate viscous Hamilton-Jacobi equations. Under a partially VMO assumption on the coefficients, we obtain the well-posedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems. 

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3. Global Sobolev persistence for the fractional Boussinesq equation with zero diffusivity

   Kukavica, Igor; Wang, Weinan (January 2020, Pure and applied functional analysis)

   We prove the persistence of regularity for the 2D alpha-franctional Boussinesq equations with positive viscosity and zero diffusivity in general Sobolev spaces, i.e. for (u_0,rho_0)\in W^s,q(R^2)\times W^s,q(R^2), where s>1 and 2 

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4. Large-time behavior of solutions of parabolic equations on the real line with convergent initial data III: unstable limit at infinity

   Pauthier, Antoine; Poláčik, Peter (August 2022, Partial Differential Equations and Applications)

   Abstract This is a continuation, and conclusion, of our study of bounded solutions u of the semilinear parabolic equation $$u_t=u_{xx}+f(u)$$ u t = u xx + f ( u ) on the real line whose initial data $$u_0=u(\cdot ,0)$$ u 0 = u ( · , 0 ) have finite limits $$\theta ^\pm $$ θ ± as $$x\rightarrow \pm \infty $$ x → ± ∞ . We assume that f is a locally Lipschitz function on $$\mathbb {R}$$ R satisfying minor nondegeneracy conditions. Our goal is to describe the asymptotic behavior of u ( x ,  t ) as $$t\rightarrow \infty $$ t → ∞ . In the first two parts of this series we mainly considered the cases where either $$\theta ^-\ne \theta ^+$$ θ - ≠ θ + ; or $$\theta ^\pm =\theta _0$$ θ ± = θ 0 and $$f(\theta _0)\ne 0$$ f ( θ 0 ) ≠ 0 ; or else $$\theta ^\pm =\theta _0$$ θ ± = θ 0 , $$f(\theta _0)=0$$ f ( θ 0 ) = 0 , and $$\theta _0$$ θ 0 is a stable equilibrium of the equation $${{\dot{\xi }}}=f(\xi )$$ ξ ˙ = f ( ξ ) . In all these cases we proved that the corresponding solution u is quasiconvergent—if bounded—which is to say that all limit profiles of $$u(\cdot ,t)$$ u ( · , t ) as $$t\rightarrow \infty $$ t → ∞ are steady states. The limit profiles, or accumulation points, are taken in $$L^\infty _{loc}(\mathbb {R})$$ L loc ∞ ( R ) . In the present paper, we take on the case that $$\theta ^\pm =\theta _0$$ θ ± = θ 0 , $$f(\theta _0)=0$$ f ( θ 0 ) = 0 , and $$\theta _0$$ θ 0 is an unstable equilibrium of the equation $${{\dot{\xi }}}=f(\xi )$$ ξ ˙ = f ( ξ ) . Our earlier quasiconvergence theorem in this case involved some restrictive technical conditions on the solution, which we now remove. Our sole condition on $$u(\cdot ,t)$$ u ( · , t ) is that it is nonoscillatory (has only finitely many critical points) at some $$t\ge 0$$ t ≥ 0 . Since it is known that oscillatory bounded solutions are not always quasiconvergent, our result is nearly optimal. 

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5. Examples of non-Dini domains with large singular sets

   https://doi.org/10.1515/ans-2022-0058  

   Kenig, Carlos; Zhao, Zihui (January 2023, Advanced Nonlinear Studies)

   Abstract Let u u be a nontrivial harmonic function in a domain D ⊂ R d D\subset {{\mathbb{R}}}^{d} , which vanishes on an open set of the boundary. In a recent article, we showed that if D D is a C 1 {C}^{1} -Dini domain, then, within the open set, the singular set of u u , defined as { X ∈ D ¯ : u ( X ) = 0 = ∣ ∇ u ( X ) ∣ } \left\{X\in \overline{D}:u\left(X)=0=| \nabla u\left(X)| \right\} , has finite ( d − 2 ) \left(d-2) -dimensional Hausdorff measure. In this article, we show that the assumption of C 1 {C}^{1} -Dini domains is sharp, by constructing a large class of non-Dini (but almost Dini) domains whose singular sets have infinite ℋ d − 2 {{\mathcal{ {\mathcal H} }}}^{d-2} -measures. 

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