Robin Young's Research

Curriculum Vitae

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Current NSF award

Selected Recent Preprints

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Exact Solutions to Degenerate Conservation Laws

We consider large variation solutions to systems of conservation laws, for which the Glimm--Lax theory of decay breaks down. We identify and isolate geometric nonlinearities which are distinct from the usual genuine nonlinearity of each wave field, by describing some degenerate systems in which all nonlinearity is geometric, and is manifested in the coupling of the different wave families. We then construct exact explicit solutions to these equations, and examine properties of these solutions. We find a wide variety of phenomena, depending on the form of the nonlinearity. The most striking of these include strong nonlinear instability of solutions, and non-trivial time--periodic solutions. We also find solutions which grow or decay exponentially, and oscillating solutions which correspond to rotations by an irrational angle. These oscillating, periodic and exponential solutions can all appear in a single system with small initial data, demonstrating sensitive dependence on initial conditions.

Periodic Solutions to Conservation Laws

We consider periodic solutions to systems of \CL s, especially the full Euler equations of gas dynamics. We present evidence showing that solutions to the Euler equations are globally bounded. We also discuss the expected long term asymptotic behavior of solutions. The geometric nonlinearity, manifested in a nontrivial Lie algebra of vector fields, competes with genuine nonlinearity over large time scales, preventing decay to $N$-waves. Rather, solutions appear to decay to a stable continuous quasi-periodic asymptotic solution. Our approach is a detailed analysis of local wave interactions and the resonant accumulation of scattered waves. This point of view allows us to explain previously observed phenomena, including shockless solutions and strong instability of solutions. We describe a linearly degenerate model for the Euler equations, and use it to explain these nonlinear effects. All that remains for a rigorous proof of global bounds is the inclusion of genuine nonlinearity, which is itself stabilizing.

Sustained Solutions for Conservation Laws

We consider completely exceptional models for \x3 systems of \CL s having a given geometric interaction structure. We show that the model for the equations of gas dynamics has globally bounded solutions which do not decay, and conjecture that similar behavior occurs for the full Euler equations. Moreover, other \x3 systems with nontrivial Lie algebra have exponentially growing modes. Thus a \emph{necessary} condition for globally bounded solutions is that the geometric structure be the same as that of the gas dynamics equations, namely the system must be symmetric hyperbolic. We introduce a simple numerical scheme in which the approximations are exact weak solutions, so that there is no residual, and convergence follows easily. This allows us to explain most observed phenomena of periodic solutions.

Wave Interactions in Nonlinear Elastic Strings

We study a system modeling the dynamics of a nonlinear elastic string. This is a \x6 system of hyperbolic conservation laws, which is degenerate in that two wave families have multiplicity two. We construct the wave curves for this problem and solve the \RP. We then give a detailed analysis of elementary wave interactions, leading to a Glimm theorem, and describe features of the system when the total variation is large. There are complicated wave patterns, including infinitely many interactions in finite time, and both three- and four-resonances may be present.

Nonstrictly Hyperbolic Waves in Elasticity

We consider a system modeling the dynamics of a nonlinear elastic string. This \x6 system is nonstrictly hyperbolic, having two families with multiplicity two. Moreover, the distinct wavespeeds cross, giving a further degeneracy. It is essential to consider the multiplicity of eigenvalues when checking entropy conditions to ensure uniqueness. Because the nonlinearity appears through a single scalar function $T(u)$, the \RP\ can be analyzed in detail by a construction analogous to Oleinik's. We solve the \RP\ for this system with large data, and give a qualitative description of the interactions of nonlinear waves of arbitrary strength.

The p-system I: The Riemann problem

The $p$-system is the prototypical system of nonlinear hyperbolic conservation laws. It is the simplest nontrivial system, and appears as a subsystem of nearly all larger systems of physical importance. Thus a good understanding of the $p$-system is critical to understanding most interesting systems. The \RP\ is the building block of general solutions, and many features of solutions are apparent in the simplified context of \RS s. In this paper we solve the \RP\ for the $p$-system with a minimum of constitutive assumptions; in particular we do not assume convexity. We deliberately avoid the use of \RI s in the construction, in the expectation of extending our methods to larger systems.

The p-system II: The Vacuum

We consider the equations of isentropic gas dynamics in Lagrangian coordinates. We are interested in global interactions of large waves, and their relation to global solvability and well-posedness for large data. One of the main difficulties in this program is the possible occurrence of a vacuum, in which the specific volume is infinite. In this paper we show that the vacuum cannot be generated in finite time. More precisely, if the vacuum is present for some positive time, then it must be present in the initial data, in a precise sense which is given. We also discuss the annihilation of vacuums that are present in the initial data.

Interactions of Plane Waves in Nonlinear Elasticity with Wlodzimierz Domanski (DRAFT)

We describe plane waves and their interactions in nonlinear elasticity. The equations of nonlinear elasticity are known to be hyperbolic, so their solutions naturally exhibit wave-like behavior. We present a general method for understanding what types of nonlinear waves are present, and how they interact.

Last updated 8/27/01.