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Publication# Ill-posedness of the quasilinear wave equation in the space $H^7/4 (ln H)^-Béta$ in dimension 2+1

Abstract

We are interested in the well posedness of quasilinear partial differential equations of order two. Motivated by the study of the Einstein equation in relativity theory, there are a number of works dedicated to the local well-posedness issue for the quasilinear wave equation. We will focus on local well-posedness for the wave equation ; more precisely we are looking at the smallest Sobolev index such that the local well-posedness holds true for initial data in this space. In 2005, D. Tataru and Hart. F. Smith provided the current best upper bound for the smallest index in low dimension. In 1998, Hans Lindblad constructed a counter example for s=3 in dimension three, thus revealing the sharpness of Tataru and Smith's criteria in this dimension. Here, our goal is to obtain sharp counterexamples to local well-posedness for quasilinear wave equations of geometric character. First, we check how the construction by Lindblad translates to dimension two. Next, we shall try to see if a similar breakdown result applies to the vanishing mean curvature problem in Minkowski space. Finally, as a more long term goal, we may try to find explicit singular solutions of this problem, starting with smooth data, by following the constructions of Krieger-Schlag-Tataru.

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Related concepts (17)

Related MOOCs (23)

Partial differential equation

In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similar to how x is thought of as an unknown number to be solved for in an algebraic equation like x2 − 3x + 2 = 0. However, it is usually impossible to write down explicit formulas for solutions of partial differential equations.

Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it - for example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on it - for example, both a latitude and longitude are required to locate a point on the surface of a sphere.

Wave equation

The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields - as they occur in classical physics - such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics. Single mechanical or electromagnetic waves propagating in a pre-defined direction can also be described with the first-order one-way wave equation, which is much easier to solve and also valid for inhomogeneous media.

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