Boundary Control of PDEs

Edited by Joseph A. Ball (Blacksburg, VA, USA) Harry Dym (Rehovot, Israel) Marinus A. Kaashoek (Amsterdam, The Netherlands) Heinz Langer (Vienna, Austria) Christiane Tretter (Bern, Switzerland) Associate Editors: Vadim Adamyan, Wolfgang Arendt, Albrecht Bottcher, B. Malcolm Brown, Raul Curto, Fritz Gesztesy, Pavel Kurasov, Vern Paulsen, Mihai Putinar, Ilya M. Spitkovsky This series is devoted to the publication of current research in operator theory, with particular emphasis on applications to classical analysis and the theory of integral equations, as well as to numerical analysis, mathematical physics and mathematical methods in electrical engineering.

Operator Theory: Advances and Applications

I. Fundamental concepts and examples.- 1. Hyperbolicity, genuine nonlinearity, and entropies.- 2. Shock formation and weak solutions.- 3. Singular limits and the entropy inequality.- 4. Examples of diffusive-dispersive models.- 5. Kinetic relations and traveling waves.- 1. Scalar Conservation Laws.- II. The Riemann problem.- 1. Entropy conditions.- 2. Classical Riemann solver.- 3. Entropy dissipation function.- 4. Nonclassical Riemann solver for concave-convex flux.- 5. Nonclassical Riemann solver for convex-concave flux.- III. Diffusive-dispersive traveling waves.- 1. Diffusive traveling waves.- 2. Kinetic functions for the cubic flux.- 3. Kinetic functions for general flux.- 4. Traveling waves for a given speed.- 5. Traveling waves for a given diffusion-dispersion ratio.- IV. Existence theory for the Cauchy problem.- 1. Classical entropy solutions for convex flux.- 2. Classical entropy solutions for general flux.- 3. Nonclassical entropy solutions.- 4. Refined estimates.- V. Continuous dependence of solutions.- 1. A class of linear hyperbolic equations.- 2. L1 continuous dependence estimate.- 3. Sharp version of the continuous dependence estimate.- 4. Generalizations.- 2. Systems of Conservation Laws.- VI. The Riemann problem.- 1. Shock and rarefaction waves.- 2. Classical Riemann solver.- 3. Entropy dissipation and wave sets.- 4. Kinetic relation and nonclassical Riemann solver.- VII. Classical entropy solutions of the Cauchy problem.- 1. Glimm interaction estimates.- 2. Existence theory.- 3. Uniform estimates.- 4. Pointwise regularity properties.- VIII. Nonclassical entropy solutions of the Cauchy problem.- 1. A generalized total variation functional.- 2. A generalized weighted interaction potential.- 3. Existence theory.- 4. Pointwise regularity properties.- IX. Continuous dependence of solutions.- 1. A class of linear hyperbolic systems.- 2. L1 continuous dependence estimate.- 3. Sharp version of the continuous dependence estimate.- 4. Generalizations.- X. Uniqueness of entropy solutions.- 1. Admissible entropy solutions.- 2. Tangency property.- 3. Uniqueness theory.- 4. Applications.

Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves

The phenomena of concentration and cavitation and the formation of $\delta$-shocks and vacuum states in solutions to the Euler equations for isentropic fluids are identified and analyzed as the pressure vanishes. It is shown that, as the pressure vanishes, any two-shock Riemann solution to the Euler equations for isentropic fluids tends to a $\delta$-shock solution to the Euler equations for pressureless fluids, and the intermediate density between the two shocks tends to a weighted $\delta$-measure that forms the $\delta$-shock. By contrast, any two-rarefaction-wave Riemann solution of the Euler equations for isentropic fluids is shown to tend to a two-contact-discontinuity solution to the Euler equations for pressureless fluids, whose intermediate state between the two contact discontinuities is a vacuum state, even when the initial data stays away from the vacuum. Some numerical results exhibiting the formation process of $\delta$-shocks are also presented.

Formation of δ-shocks and vacuum states in the vanishing pressure limit of solutions to the euler equations for isentropic fluids

https://www.math.ntnu.no/conservation/2003/068.pdf

Dynamics of propagation and interaction of δ-shock waves in conservation law systems

Joseph, K.T., A Riemann problem whose viscosity solutions contain 8-measures, Asymptotic Analysis 7 (1993) 105-120. In this paper we study the Riemann problem for a 2 X 2 system of conservation laws, whose solution need not be real-valued functions, but may contain 8-measures concentrated along some curves. We consider a parabolic approximation of the Riemann problem and obtain an explicit formula. Then the solution of the Riemann problem is obtained as the limit of this approximate solution.

A Riemann problem whose viscosity solutions contain δ-measures

Explicit formula for the solution of convex conservation laws with boundary condition

. This paper is concerned with the initial‐boundary‐value problem for a nonlinear hyperbolic system of conservation laws. We study the boundary layers that may arise in approximations of entropy discontinuous solutions. We consider both the vanishing‐viscosity method and finite‐difference schemes (Lax‐Friedrichs‐type schemes and the Godunov scheme). We demonstrate that different regularization methods generate different boundary layers. Hence, the boundary condition can be formulated only if an approximation scheme is selected first. Assuming solely uniform 
$L^\infty$
 bounds on the approximate solutions and so dealing with 
$L^\infty$
 solutions, we derive several entropy inequalities satisfied by the boundary layer in each case under consideration. A Young measure is introduced to describe the boundary trace. When a uniform bound on the total variation is available, the boundary Young measure reduces to a Dirac mass. From the above analysis, we deduce several formulations for the boundary condition which apply whether the boundary is characteristic or not. Each formulation is based on a set of admissible boundary values, following the terminology of Dubois & LeFloch[15]. The local structure of these sets and the well‐posedness of the corresponding initial‐boundary‐value problem are investigated. The results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics.

Boundary Layers in Weak Solutions of Hyperbolic Conservation Laws

This paper is concerned with the boundary layers that arise in solutions of
a nonlinear hyperbolic system of conservation laws in presence of vanishing diffusion. We
consider self-similar solutions of the Riemann problem in a half-space, following a pioneering
idea by Dafermos for the standard Riemann problem. The system is strictly hyperbolic
but no assumption of genuine nonlinearity is made; moreover, the boundary is possibly
characteristic, that is, the wave speed do not have a specific sign near the (stationary)
boundary. 

First, we generalize a technique due to Tzavaras and show that the boundary Riemann
problem with diffusion admits a family of continuous solutions that remain uniformly
bounded in the total variation norm. Careful estimates are necessary to cope with waves
that collapse at the boundary and generate the boundary layer. 

Second, we prove the convergence of these continuous solutions toward weak solutions of
the Riemann problem when the diffusion parameter approaches zero. Following Dubois and
LeFloch, we formulate the boundary condition in a weak form, based on a set of admissible
boundary traces. Following Part I of this work, we identify and rigorously analyze the
boundary set associated with the zero-diffusion method. In particular, our analysis fully
justifies the use of the scaling $1/\varepsilon$ near the boundary (where 
$\varepsilon$ is the diffusion parameter),
even in the characteristic case as advocated in Part I by the authors.

Boundary layers in weak solutions of hyperbolic conservation laws II. self-similar vanishing diffusion limits

On etudie la limite pour e→0 des solutions du probleme aux valeurs limites et initiales: u t e+(ue 2 /2) x =(eu xx e)/2; ue(x, 0)=u 0 (x); ue(0, t)=u b (t), dans D={(x, t): x≥0, t≥0}. On suppose que u 0 (x)∈L 1 (0, ∞) et u b ∈C(0, ∞)

K. T. Joseph

Papers

A Riemann problem whose viscosity solutions contain δ-measures

Explicit formula for the solution of convex conservation laws with boundary condition

Boundary Layers in Weak Solutions of Hyperbolic Conservation Laws

Boundary layers in weak solutions of hyperbolic conservation laws II. self-similar vanishing diffusion limits

Burger's equation in the quarter plane, a formula for the weak limit