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J > 0 for all n, existence as well as uniqueness are guaranteed, independent of ! 0. 6 Classification Condition. t. 2 ; which means that both spatially ( n large) and temporally (! large) high-frequency excitations are strongly damped. 30). 4. Connection with the Schrödinger Equation. 28). u D 0: p For ! , which we already know from the wave equation. For ! Ä 0, however, we obtain a solution behavior similar to the one for the Poisson equation: The eigenvalues of the homogeneous Dirichlet problem are real and negative, existence and uniqueness are guaranteed independent of !

C. / for the Robin boundary value problem u D f in ; nT ru C ˛u D ˇ on @ . Which condition for ˛ must hold? 5. Let R2 denote a bounded domain with sufficiently smooth boundary. t. v; w/ D vw dx; if one of the following conditions holds on the boundary vnT ru D unT rv or nT ru C ˛u D 0 with ˛ > 0: Obviously, the first one is just a homogeneous Dirichlet or Neumann boundary condition, the second one a Robin boundary condition. 6. 2, derive an eigenmode representation for the solution of the Cauchy problem for the wave equation on the domain D  ; Œ.

Rp C . 13) we obtain, via Newton’s second law, the equation Du D rp C . 13), however, a higher spatial derivative arises, which means that we require more boundary conditions than for the Euler equations. Due to the internal friction between fluid and solid walls we have T uD0 for any tangential vector . 17) together with the constitutive law p D p. /; the Navier–Stokes equations for compressible fluids are now complete: u t C ux u D t p r C . 19a) C div. 19c) In the case of incompressible, homogeneous fluids in the absence of external forces we have div u D 0 and D 0 > 0.

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Mathematics Cambridge Vector Calculus


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