Partial Differential Equations

Evans, Lawrence C.

doi:10.1090/gsm/019

Cited by 3,693 publications

(3,557 citation statements)

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“…for 0 s t T , where C ε depends on ε and T but is independent of t, s. It is standard theory (see, for instance, [15] Section 5.9, Theorem 4) that, if u ∈ L 2 (0, which implies (126). These conditions are true for solutions of (1), since, by the gradient flow property,…”

Section: Continuity Argumentsmentioning

confidence: 97%

Vortex Liquids and the Ginzburg–landau Equation

Kurzke

Spirn

2014

Forum math. Sigma

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We establish vortex dynamics for the time-dependent Ginzburg-Landau equation for asymptotically large numbers of vortices for the problem without a gauge field and either Dirichlet or Neumann boundary conditions. As our main tool, we establish quantitative bounds on several fundamental quantities, including the kinetic energy, that lead to explicit convergence rates. For dilute vortex liquids, we prove that sequences of solutions converge to the hydrodynamic limit.

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Section: Continuity Argumentsmentioning

confidence: 97%

Vortex Liquids and the Ginzburg–landau Equation

Kurzke

Spirn

2014

Forum math. Sigma

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“…also Evans [34]), with references back to the fundamental L 2 -theory including boundary points of Ladyshenskaya, Solonnikov and Uraltseva [35]. A fundamental framework of functional analysis for parabolic Cauchy problems was developed by Lions and Magenes [8].…”

Section: Notesmentioning

confidence: 99%

Final Value Problems for Parabolic Differential Equations and Their Well-Posedness

Christensen

Johnsen

2018

Axioms

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This article concerns the basic understanding of parabolic final value problems, and a large class of such problems is proved to be well posed. The clarification is obtained via explicit Hilbert spaces that characterise the possible data, giving existence, uniqueness and stability of the corresponding solutions. The data space is given as the graph normed domain of an unbounded operator occurring naturally in the theory. It induces a new compatibility condition, which relies on the fact, shown here, that analytic semigroups always are invertible in the class of closed operators. The general set-up is evolution equations for Lax-Milgram operators in spaces of vector distributions. As a main example, the final value problem of the heat equation on a smooth open set is treated, and non-zero Dirichlet data are shown to require a non-trivial extension of the compatibility condition by addition of an improper Bochner integral.

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“…The nonzero diagonal entry corresponding to a vertex is 0.5 times the sum of lengths of edges containing that vertex. Then the terms α ij p j on the left hand side of (13) are discretized as * 0 α ij p j + E δ i0 δ j0 α v + δ in δ jn α a p j (15) and f i on right hand side of (13) is discretized as…”

Section: Spatial Discretization Using Exterior Calulusmentioning

confidence: 99%

“…An analytical solution to problem (18) can be constructed using standard techniques in partial differential equations [15]. The solution p is written as the sum of two parts, namely p = v + u, where v accounts for the Dirac source in Ω c and u accounts for the boundary conditions on ∂Ω c .…”

Section: Testcase To Verify the Discretization Of The Boundary Conditmentioning

confidence: 99%

Effect of ocular shape and vascular geometry on retinal hemodynamics: a computational model

Dziubek

Guidoboni

Harris

et al. 2015

Biomech Model Mechanobiol

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Thistleton, W. (2016Effect of ocular shape and vascular geometry on retinal hemodynamics: a computational modelReceived: date / Accepted: date Abstract A computational model for retinal hemodynamics accounting for ocular curvature is presented. The model combines: (i) a hierarchical Darcy model for the flow through small arterioles, capillaries and small venules in the retinal tissue, where blood vessels of different size are comprised in different hierarchical levels of a porous medium; and (ii) a one-dimensional network model for the blood flow through retinal arterioles and venules of larger size. The non-planar ocular shape is included by: (i) defining the hierarchical Darcy flow model on a two-dimensional curved surface embedded in the three-dimensional space; and (ii) mapping the simplified one-dimensional network model onto the curved surface. The model is solved numerically using a finite element method in which spatial domain and hierarchical levels are discretized separately. For the finite element method we use an exterior calculus based implementation which permits an easier treatment of non-planar domains. Numerical solutions are verified against suitably constructed analytical solutions. Numerical experiments are performed to investigate how

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Partial Differential Equations

Cited by 3,693 publications

References 0 publications

Vortex Liquids and the Ginzburg–landau Equation

Vortex Liquids and the Ginzburg–landau Equation

Final Value Problems for Parabolic Differential Equations and Their Well-Posedness

Effect of ocular shape and vascular geometry on retinal hemodynamics: a computational model

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