Well-posedness

On the concept of well-posedness for a difference problem. There is another matter which is one of some interest. In conformity with statements of problems of mathematical physics, it is fairly common to call a problem well-posed if the following conditions are satisfied ... 

The definition of well-posedness and ill-posedness of a scheme is closely connected with the selection rules for norms II j and (-2,)- It niay happen that for some choices of these norms estimate (25) is fulfilled, while... 

Existence results for unsteady flows are important in two ways. First, the (local) well-posedness of the initial and boundary value problem proves the adequacy of a given model to describe (at least locally) dynamical situations. Second, global well-posedness is preliminary to any nonlinear stability study. 

We now turn to local existence of solutions for Maxwell-type models. The situation is much trickier here since these models can display Hadamard instabilities (see Section 2.1), and no general results seem to be known so far. One has, in any case, to restrict initial data to Hadamard stable ones. A possible way to overcome the difficulty is to consider models satisfying an eUipticity condition, which will imply well-posedness. This approach was followed by Renardy [41], whose results are briefly described below. 

The condition number (CN) is used in the numerical examples to control well-posedness of the SLAE. In the case of high CN, the methods for ill-posed problems should be used [8],... 

The well-posedness of the two-fluid model has been a source of controversy reflected by the large number of papers on this issue that can be found in the literature. This issue is linked with analysis of the characteristics, stability and wavelength phenomena in multi -phase flow equation systems. The controversy originates primarily from the fact that with the present level of knowledge, there is no general way to determine whether the 3D multi-fluid model is well posed as an initial-boundary value problem. The mathematical theory of well posedness for systems of partial differential equations describing dispersed chemical reacting flows needs to be examined. 

The principle of well-posedness states that the description of the motion should be such that a solution to the initial boundary value problem exists and depends continuously on the initial and boundary conditions. 

Convergence Problems and Failure of Well-posedness of Granular theory closures... 

Sensitize fhe reader to fhe issues of well-posedness of a mafhemafical problem... 

The definition of well-posedness and ill-posedness of a scheme is closely... 

In practice, the prior distribution may be used as a regularizer [164,169] to improve the well-posedness of the inverse problem ... 

The equation for the gas phase degenerates for = 1, which is reached by both parametrizations, so well-posedness of the system cannot be guaranteed In the case where the gas phase disappears, the system becomes indefinite. Physically, this effect describes the fact that some parts of the modeling domain are fully saturated with water. The same effect could occur - using a different parametrization - with the water phase. 

INTEGRATED STABILITY AND WELL-POSEDNESS CRITERIA, 350 Stability and Well-Posedness Map, 350 Constructing the Stratified Flow Boundaries, 352... 

It was shown by Brauner and Moalem Maron that linear stability analysis is insufficient to predict the stratified flow boundaries [40]. Parallel analyses on the stability as well as on the well-posedness of the (hyperbolic) equations which govern the stratified flow has been invoked. It has been shown that the departure from stratified configuration is associated with a buffer zone confined between the conditions derived from stability analysis (a lowerbound) and those obtained by requiring well-posedness of the transient governing equations (an upper-bound). These two bounds form a basis for the construction of the complete stratified/non-stratified transitional boundary to the various bounding flow patterns. 

The transient continuity equations and the combined momentum equation constitute a set of hyperbolic equations. The formulation is well-posed provided the equations possess real characteristics. The conditions of well-posedness of averaged two-fluid models were extensively discussed in the literature (e.g., Lyczkowski et al. [106], Ramshaw and Trapp [107], Banerjee and Chan [56], Drew [108], Jones and Prosperetti [109], Prosperetti and Jones [110], Moe [111]). The condition under which the characteristic roots of Equations 1, 2, 7 are real reads, (derived in 43 for C, = 0) ... 

The identity between condition (41) and condition (38) for stable dynamic wave indicates that the region of well-posedness coincides with that of stable dynamic waves, c > 0. The region of c < 0, corresponds to unstable waves and their evolution, as formulated by the initial value set of equations, is ill-posed. As the stability condition for inviscid flows (obtained with = 0) is equivalent to that of stable dynamic waves, the well-posedness condition (with = 0, = 1) is... 

Inspection of Equation 41 points out that for fluids of zero surface tension and equal densities (or zero gravity conditions) the formulation is ill-posed for plug flow (y = y =1). However, since the condition for well-posedness is independent of the... 

Thus, while the neutral stability boundary may represent preliminary transition from smooth-stratified flow to a wavy interfacial structure, the well-posedness boundary, which is within the wavy unstable region, represents an upper bound for the existence of a stratified wavy configuration. Beyond the well-posedness boundary transition to a different flow pattern takes place. In the ill-posed region, the model is no longer capable of describing the physical phenomena involved therefore, amplification rates predicted for ill-posed modes or numerical simulation of their growth is actually meaningless. 

It is to be noted that the criterion for ill-posedness is affected only by the terms which are proportional to the gradients of h, u, u (derivatives with respect to time and space) and, therefore, apparently unaffected by the quasi-steady modelling of the shear stresses. However, the test for well-posedness is carried out on a stratified wavy configuration, which is represented by the averaged values of H, U, (obtained from the solution of AF = 0, Equation 11). Obviously, their values depend on the models used for the wall and interfacial shear stresses. In particular, the modelling of x. deserves a special attention since in the wavy regime the augmentation of the interfacial friction factor, due to the interfacial waviness is to be considered. 

INTEGRATED STABILITY AND WELL-POSEDNESS CRITERIA Stability and Well-Posedness Map... 

Given the fluid physical properties and system geometry (tube size and inclination) the stability and well-posedness boundaries can be mapped in the coordinate system of the two-fluids flow rates, U j,. The construction of the stability and well-posedness map (SWP map) is demonstrated in Figure 8 for horizontal air-water flow in a 2.5 cm pipe. 

Figure 8. Stability and well-posedness map for air-water horizontal system, D = 2.53cm.

In parallel to the zero neutral stability boundary defined by = 0, a zero real characteristics boundary (ZRC) is built-up by searching for all combinations of (U j, U j) which yield by (41) real characteristics for long waves, = 0. The ZRC curve confines the region of operational conditions for which well-posedness is ensured for all wave modes. Generally, the ZRC boundary is composed of two branches the left one corresponds to while along the right one... 

As a corollary, it can be stated that the condition of unstable dynamic waves or ill-posedness is sufficient to indicate instability, whereas the condition of c < 0, or well-posedness, is necessary but insufficient to ensure stability. The above ideas and interpretations as detailed above with regards to the horizontal system of Figure 8 also prevail basically in inclined flows, although limiting stability and well-posedness boundaries may demonstrate entirely different structures (Brauner and Moalem Maron [45]). 

The general implication of the stability boundary (ZNS or ZNS ) and the well-posedness boundary (ZRC) is in defining three zones the area within the stability boundary is well-understood to be the stable smooth stratified zone. Beyond the ZRC boundary, the complex characteristics indicate that the governing equations of the stratified flow configuration are ill-posed with respect to long wave modes in the wave spectra. In this sense, the ZRC boundary represents an upper bound... 

The various boundaries which evolve from stability and well-posedness analyses and the associated physical interpretations form a basis for constructing a flow... 

The well-posedness boundary (ZRC) (included in Figures 10, 11, 13) represents the limit of operational conditions (U, U, ) for which the governing set of continuity and momentum equations is still well-posed with respect to all wave modes. Hence, it is considered as an upper bound for the stratified-wavy flow pattern. Indeed, the data of stratified-wavy/annular transition follows the ZRC curve in the region of H < 0.5. 

Well-posedness boundary [llq Stratified - bubbly boundary tl ... 

In two-phase systems of > 1, surface tension contributes a dominant stabilizing term in the well-posedness criterion. Hence, the region of stable wavy stratified pattern extends and the transition to annular flow is delayed to higher gas rates, compared to those predicted by (infinite) long wave analysis (k —> 0). For instance, for air-water flow in a 1 inch pipe, < 0.01, while for D = 0.4cm, = 20. 

---