Well-Posedness Theory for a Nonconservative Burgers-Type System Arising in Dislocation Dynamics

In this work we study a system of nonconservative Burgers type in one space dimension, arising in modeling the dynamics of dislocation densities in crystals. Starting from physically relevant initial data that are of a special form, namely nondecreasing, periodic plus linear functions, we prove the global existence and uniqueness of a solution in $H^1_{loc}(\mathbb R\times[0,+\infty))$ that preserves the nature of the initial data. The approach is made by adding some viscosity to the system, obtaining energy estimates, and passing to the limit for vanishing viscosity. A comparison principle is shown for this system as well as an application in the case of the classical Burgers equation.

[1]  With Invariant Submanifolds,et al.  Systems of Conservation Laws , 2009 .

[2]  Gui-Qiang G. Chen,et al.  The Cauchy Problem for the Euler Equations for Compressible Fluids , 2002 .

[3]  Guy Barles,et al.  Nonlocal First-Order Hamilton–Jacobi Equations Modelling Dislocations Dynamics , 2006 .

[4]  Régis Monneau,et al.  Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution , 2006 .

[5]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[6]  P. Lax,et al.  Systems of conservation laws , 1960 .

[7]  J. Lions,et al.  Problèmes aux limites non homogènes et applications , 1968 .

[8]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[9]  L. Ambrosio.  Transport equation and Cauchy problem for BV vector fields , 2004 .

[10]  P. Floch.  Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form , 1988 .

[11]  J. Bogdanoff,et al.  On the Theory of Dislocations , 1950 .

[12]  István Groma,et al.  Investigation of dislocation pattern formation in a two-dimensional self-consistent field approximation , 1999 .

[13]  P. Lions,et al.  Ordinary differential equations, transport theory and Sobolev spaces , 1989 .

[14]  P. Cardaliaguet,et al.  Existence and uniqueness for dislocation dynamics with nonnegative velocity , 2005 .

[15]  Tai-Ping Liu,et al.  Existence theory for nonlinear hyperbolic systems in nonconservative form , 1993 .

[16]  A. El Hajj,et al.  A convergent scheme for a non-local coupled system modelling dislocations densities dynamics , 2007, Math. Comput..

[17]  Nicolas Forcadel,et al.  Dislocation dynamics with a mean curvature term: short time existence and uniqueness , 2008, Differential and Integral Equations.

[18]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .