We prove the existence of a global weak solution to the Cauchy problem for a class of 2 × 2 equations which model one-dimensional multiphase flow, and which represent a natural generalization of the scalar Buckley-Leverett equation. Loss of strict hyperbolicity (coinciding wave speeds with a (~ I) normal form) occurs on a curve in state space, and waves in a neighborhood of this curve contribut...

We analyze the strength of Helly’s selection theorem (HST), which is the most important compactness theorem on the space of functions of bounded variation (BV ). For this we utilize a new representation of this space intermediate between L1 and the Sobolev space W , compatible with the—so called—weak∗ topology on BV . We obtain that HST is instance-wise equivalent to the Bolzano-Weierstraß prin...

Here, we rigorously prove some of the results obtained with asymp-totic methods for a modiied Burgers' equation, obtained by replacing the ux term in Burgers' equation considered in its hyperbolic form by the Hilbert transform. We show that, under the asumptions ju x j 1 C ?1=3 , jH(u x)j 1 C ?1=3 , and R 1 0 jju x jj 2 dt C, H(u) being the Hilbert transform of u and C a constant independent of...

In the paper we investigate local existence of a weak solution u to a degenerate parabolic boundary value problem. Degeneration occurs in the coeecient g(x; t; u) in front of the time derivative which is not assumed to be bounded below by some positive constant. Moreover, this coeecient depends on the solution u. We approximate the problem by semidiscretization in time (Rothe's method) and prov...

Graph cut algorithms [9], commonly used in computer vision, solve a first-order MRF over binary variables. The state of the art for this NP-hard problem is QPBO [1, 2], which finds the values for a subset of the variables in the global minimum. While QPBO is very effective overall there are still many difficult problems where it can only label a small subset of the variables. We propose a new a...

and Applied Analysis 3 Step 1. Let F {a} be singleton. Then, there are u ∈ U and v ∈ V such that ‖uv‖ < M, and ‖ u, v, a − a‖ < M 1 . 2.2 Letting w uv ◦ uv, then ‖ uv ◦ uv, a − a‖ ‖ u, v, u, v, a − a − u, v, a − a ‖ < . 2.3 Step 2. Let F {a1, a2}. There is a u1, v1 ∈ U × V such that ‖ u1, v1, a1 − a1‖ < / 1 M , and for u1, v1, a2 − a2 ∈ A there is a u2, v2 ∈ U × V such that ‖ u2, v2, u1, v1, a2...