scholarly journals Newtonian and Single Layer Potentials for the Stokes System with L∞ Coefficients and the Exterior Dirichlet Problem

2019 ◽  
pp. 237-260
Author(s):  
Mirela Kohr ◽  
Sergey E. Mikhailov ◽  
Wolfgang L. Wendland
Keyword(s):  
Dirichlet Problem ◽  
Stokes System ◽  
Single Layer ◽  
Layer Potentials

ON THE ROBIN PROBLEM FOR STOKES AND NAVIER–STOKES SYSTEMS

2006 ◽  
Vol 16 (05) ◽  
pp. 701-716 ◽  
Author(s):  
REMIGIO RUSSO ◽  
ALFONSINA TARTAGLIONE
Keyword(s):  
Boundary Layer ◽  
Weak Solution ◽  
Lipschitz Domain ◽  
Stokes System ◽  
Navier Stokes ◽  
Robin Problem ◽  
Layer Potentials ◽  
Very Weak Solution ◽  
Stokes Systems ◽  
Compact Boundary

The Robin problem for Stokes and Navier–Stokes systems is considered in a Lipschitz domain with a compact boundary. By making use of the boundary layer potentials approach, it is proved that for Stokes system this problem admits a very weak solution under suitable assumptions on the boundary datum. A similar result is proved for the Navier–Stokes system, provided that the datum is "sufficiently small".

On norm of single layer potentials on segments

2016 ◽  
Vol 22 (1) ◽  
Author(s):  
Seyed M. Zoalroshd
Keyword(s):  
Upper Bound ◽  
Single Layer ◽  
Exact Expression ◽  
Operator Norm ◽  
Equal Length ◽  
Layer Potentials ◽  
Schmidt Norm ◽  
Special Case

AbstractWe show that, for a special case, equality of the spectra of single layer potentials defined on two segments implies that these segments must have equal length. We also provide an upper bound for the operator norm and exact expression for the Hilbert–Schmidt norm of single layer potentials on segments.

Solvability of a Dirichlet problem for a time fractional diffusion-wave equation in Lipschitz domains

2012 ◽  
Vol 15 (2) ◽  
Author(s):  
Jukka Kemppainen
Keyword(s):  
Wave Equation ◽  
Dirichlet Problem ◽  
Original Problem ◽  
Single Layer ◽  
Fractional Diffusion ◽  
Lipschitz Domains ◽  
Single Layer Potential ◽  
Layer Potential ◽  
Diffusion Wave ◽  
Sesquilinear Form

AbstractThis paper investigates a Dirichlet problem for a time fractional diffusion-wave equation (TFDWE) in Lipschitz domains. Since (TFDWE) is a reasonable interpolation of the heat equation and the wave equation, it is natural trying to adopt the techniques developed for solving the aforementioned problems. This paper continues the work done by the author for a time fractional diffusion equation in the subdiffusive case, i.e. the order of the time differentiation is 0 < α < 1. However, when compared to the subdiffusive case, the operator ∂ tα in (TFDWE) is no longer positive. Therefore we follow the approach applied to the hyperbolic counterpart for showing the existence and uniqueness of the solution.We use the Laplace transform to obtain an equivalent problem on the space-Laplace domain. Use of the jump relations for the single layer potential with density in H −1/2(Γ) allows us to define a coercive and bounded sesquilinear form. The obtained variational form of the original problem has a unique solution, which implies that the original problem has a solution as well and the solution can be represented in terms of the single layer potential.

scholarly journals Suppressing blow-up by gradient-dependent flux limitation in a planar Keller–Segel–Navier–Stokes system

2021 ◽  
Vol 72 (2) ◽  
Author(s):  
Michael Winkler
Keyword(s):  
Dirichlet Problem ◽  
Initial Data ◽  
Classical Solution ◽  
Smooth Function ◽  
Stokes System ◽  
Blow Up ◽  
Navier Stokes ◽  
Diffusive Fluxes ◽  
Fluid Interaction ◽  
Flux Limitation

AbstractThe flux-limited Keller–Segel–Navier–Stokes system $$\begin{aligned} \left\{ \begin{array}{lcl} n_t + u\cdot \nabla n &{}=&{} \Delta n - \nabla \cdot \Big ( n f(|\nabla c|^2) \nabla c\Big ), \\ c_t + u\cdot \nabla c &{}=&{} \Delta c - c + n, \\ u_t + (u\cdot \nabla ) u &{}=&{} \Delta u + \nabla P + n\nabla \Phi , \qquad \nabla \cdot u=0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$ n t + u · ∇ n = Δ n - ∇ · ( n f ( | ∇ c | 2 ) ∇ c ) , c t + u · ∇ c = Δ c - c + n , u t + ( u · ∇ ) u = Δ u + ∇ P + n ∇ Φ , ∇ · u = 0 , ( ⋆ ) is considered in a smoothly bounded domain $$\Omega \subset {\mathbb {R}}^2$$ Ω ⊂ R 2 . It is shown that whenever the suitably smooth function f models any asymptotically algebraic-type saturation of cross-diffusive fluxes in the sense that $$\begin{aligned} |f(\xi )| \le K_f\cdot (\xi +1)^{-\frac{\alpha }{2}} \end{aligned}$$ | f ( ξ ) | ≤ K f · ( ξ + 1 ) - α 2 holds for all $$\xi \ge 0$$ ξ ≥ 0 with some $$K_f>0$$ K f > 0 and $$\alpha >0$$ α > 0 , for any all reasonably regular initial data a corresponding no-flux/no-flux/Dirichlet problem admits a globally defined classical solution which is bounded, inter alia, in $$L^\infty (\Omega \times (0,\infty ))$$ L ∞ ( Ω × ( 0 , ∞ ) ) with respect to all its components. By extending a corresponding result known for a fluid-free counterpart of ($$\star $$ ⋆ ), this confirms that with regard to the possible emergence of blow-up phenomena, the choice $$f\equiv const.$$ f ≡ c o n s t . retains some criticality also in the presence of fluid interaction.

True and Spurious Eigensolutions of an Elliptical Membrane by Using the Nondimensional Dynamic Influence Function Method

2014 ◽  
Vol 136 (2) ◽  
Author(s):  
Jeng-Tzong Chen ◽  
Jia-Wei Lee ◽  
Ying-Te Lee ◽  
Wen-Che Lee
Keyword(s):  
Dirichlet Problem ◽  
Influence Function ◽  
Single Layer ◽  
Boundary Integral ◽  
Double Layer Potential ◽  
Single Layer Potential ◽  
Layer Potential ◽  
Potential Formulation ◽  
The Neumann Problem ◽  
Spurious Eigenvalues

In this paper, we employ the nondimensional dynamic influence function (NDIF) method to solve the free vibration problem of an elliptical membrane. It is found that the spurious eigensolutions appear in the Dirichlet problem by using the double-layer potential approach. Besides, the spurious eigensolutions also occur in the Neumann problem if the single-layer potential approach is utilized. Owing to the appearance of spurious eigensolutions accompanied with true eigensolutions, singular value decomposition (SVD) updating techniques are employed to extract out true and spurious eigenvalues. Since the circulant property in the discrete system is broken, the analytical prediction for the spurious solution is achieved by using the indirect boundary integral formulation. To analytically study the eigenproblems containing the elliptical boundaries, the fundamental solution is expanded into a degenerate kernel by using the elliptical coordinates and the unknown coefficients are expanded by using the eigenfunction expansion. True and spurious eigenvalues are simultaneously found to be the zeros of the modified Mathieu functions of the first kind for the Dirichlet problem when using the single-layer potential formulation, while both true and spurious eigenvalues appear to be the zeros of the derivative of modified Mathieu function for the Neumann problem by using the double-layer potential formulation. By choosing only the imaginary-part kernel in the indirect boundary integral equation method (BIEM) to solve the eigenproblem of an elliptical membrane, spurious eigensolutions also appear at the same position with those of NDIF since boundary distribution can be lumped. The NDIF method can be seen as a special case of the indirect BIEM by lumping the boundary distribution. Both the analytical study and the numerical experiments match well with the same true and spurious solutions.

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