101,285 research outputs found
A wavefront tracking algorithm for N×N nongenuinely nonlinear conservation laws
No full textAbstractWe introduce a wavefront tracking algorithm for N×N hyperbolic systems of conservation lawsut+F(u)x=0, that admits characteristic fields that are neither genuinely nonlinear nor linearly degenerate in the sense of Lax. Instead we assume that, for any nongenuinely nonlinear ith characteristic family, the derivative of the ith eigenvalue λi(u) of DF(u) in the direction of the ith right eigenvector ri(u), vanishes on a single (N−1)-dimensional hypersurface in the u-space, transversal to the field ri(u). Systems that fulfill this type of assumptions are of particular interest in studying elastodynamic or rigid heat conductors at low temperature. The first proof of the existence of weak solutions for nongenuinely nonlinear systems was given by T. P. Liu (Mem. Amer. Math. Soc.30 (1981), no. 240), based on a Glimm scheme. Our construction here provides an alternative method for establishing the global existence of weak solutions for such systems. Moreover, relying on the stability analysis developed in Ancona and Marson, preprint S.I.S.S.A.-I.S.A.S. 27/99/11, 1999, and preprint, 2000, we show that these solutions are entropy admissible in the sense of Lax
On the attainable set for scalar nonlinear conservation laws with boundary control
No full textSummary:
The paper treats the initial boundary value problem for a scalar conservation law with strictly convex flux function. The boundary data is a Lebesgue-measurable and bounded function regarded as a control and constrained to remain in a prescribed set of admissible controls. A time being fixed, the authors characterize the set consisting of the corresponding entropy solutions at the time . Under natural assumptions on , it is proven that is a compact subset of . Such a compactness property provides the key information in order to establish the existence of solutions for a class of optimisation problems. Finally the results are applied by the authors to an optimisation problem concerning a model of traffic flow on a highway
Scalar non-linear conservation laws with integrable boundary data
No full textSummary: The paper deals with the initial-boundary value problem for a 1-D scalar conservation law in the case of initial and boundary data. The generalized solutions are constructed via semigroup theory, and a comparison principle as well as explicit (variational) representations for the solutions are obtained. Also the attainable set as is described for integrable boundary data considered as control. These results are applied to the concrete problem of optimal traffic fl
Well-posedness for general 2 x 2 conservation laws
No full textWe consider the Cauchy problem for a strictly hyperbolic
system of conservation laws in one space dimension
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which is neither linearly degenerate nor genuinely non-linear.
We make the following assumption on the characteristic fields.
If denotes the -th right eigenvector of
and the corresponding eigenvalue, then
the set is
a smooth curve
in the -plane that is transversal to the vector field
Systems of conservation laws that fulfill such assumptions arise in studying
elastodynamics
or rigid heat conductors at low temperature.
For such systems we prove the existence
of a closed domain \ \Cal D
\subset L^1, \ containing all functions with sufficiently
small total variation, and of a uniformly Lipschitz continuous
semigroup S:{\Cal D} \times [0,+\infty)\rightarrow \Cal D
with the following properties.
Each trajectory \ \ of is a weak
solution of (1). Viceversa, if a piecewise Lipschitz,
entropic solution of (1) exists for
then it coincides with the trajectory of ,
i.e.
This result yields the uniqueness and continuous dependence of
weak, entropy-admissible solutions of the Cauchy problem (1)
with small initial data, for systems satysfying the above assumption
Stability of front tracking solutions to the initial and boundary value problem for systems of conservation laws
No full textWe deal with the non characteristic initial and boundary value problem
for an strictly hyperbolic system of conservation laws in
one space dimension
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\partial_t u+ \partial_x F(u)=0,\qquad u(0,x) = \bar u (x)\,,\qquad
b\big( u(\psi(t),t) \big) = g(t)\,.\eqno (\ast)
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Here is a smooth vector field defined in an open, convex
neighborhood of the origin of , and are
functions with small total variation, is a non
characteristic Lipschitz boundary profile, and a
function. We prove that the front tracking solutions to ()
constructed by D. Amadori in \cite{Amadori} are stable for the
\elleuno topology. This implies the existence of a Standard Riemann
Semigroup and hence the well-posedness of ()
Sharp Convergence Rate of the Glimm Scheme for General Nonlinear Hyperbolic Systems
No full textConsider a general strictly hyperbolic,
quasilinear system, in one space dimension
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u_t+A(u) u_x=0,
\eqno (1)
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where , , is a smooth
matrix-valued map. Given an initial datum with small
total variation, let be the corresponding (unique)
vanishing viscosity solution of (1) obtained as limit of solutions to
the viscous parabolic approximation , as
. For every , we prove the a-priori bound
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\big\|u^\eps(T,\cdot)-u(T,\cdot)\big\|_{\elleuno}=o(1)\cdot\sqrt\eps\,|\log\eps|
\eqno (2)
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for an approximate solution u^\eps of (1) constructed by the Glimm
scheme, with mesh size \Delta x = \Delta t = \eps, and with a
suitable choice of the sampling sequence. This result provides for
general hyperbolic systems the same type of error estimates valid for
Glimm approximate solutions of hyperbolic systems of conservation laws
satisfying the classical Lax or Liu assumptions on the
eigenvalues and on the eigenvectors of the
Jacobian matrix .
The estimate (2) is obtained introducing a new wave interaction
functional with a cubic term that controls the nonlinear coupling of
waves of the same family and at the same time decreases at
interactions by a quantity that is of the same order of the product of
the wave strength times the change in the wave speeds. This is
precisely the type of errors arising in a wave tracing analysis of the
Glimm scheme, which is crucial to control in order to achieve an
accurate estimate of the convergence rate as~(2)
On the attainable set for scalar balance laws with distributed control
No full textThe paper deals with the set of attainable profiles of a solution u to a scalar balance law
in one space dimension with strictly convex flux function
∂tu + ∂xf(u) = z(t, x).
Here the function z is regarded as a bounded measurable control. We are
interested in studying the set of attainable profiles at a fixed time T > 0, both in case
z(t,·) is supported in the all real
line, and in case z(t,·) is supported in a compact
interval [a, b] independent on the time variable t
Well-posedness for General 2x2 Systems of Conservation Laws
No full textABSTRACT. We consider the Cauchy problem for a strictly hyperbolic 2x2 system of conservation laws in one space dimension
u_t+[F(u)]_x=0,
u(0,x)=u_0(x),
which is neither linearly degenerate nor genuinely non-linear.
We make the following assumption on the characteristic fields.
If r_i(u), i=1,2, denotes the i-th right eigenvector of DF(u) and lambda_i(u) the corresponding eigenvalue, then the set {u : D lambda_i . r_i (u) = 0} is a smooth curve in the u-plane that is transversal to the vector field r_i(u).
Systems of conservation laws that fulfill such assumptions arise in studying elastodynamics or rigid heat conductors at low temperature.
For such systems we prove the existence of a closed domain D
in L^1, containing all functions with sufficiently small total
variation, and of a uniformly Lipschitz continuous semigroup
S: Dx [0,+infty) --->D with the following properties.
Each trajectory t --> S_t (u_0) of S is a weak solution of (1). Viceversa, if a piecewise Lipschitz, entropic solution u= u(t,x)
of (1) exists for
Letter, [Author unclear] to Paulina T. Merritt
No full textHandwritten letter to Paulina Merritt from an unknown author, October 1, 1876.
A maximum principle for optimally controlled systems of conservation laws
No full textWe study a class of optimization problems
of Mayer form, for the strictly
hyperbolic nonlinear controlled system of conservation laws
,
where is the control variable. Introducing a family of
``generalized cotangent vectors", we derive necessary conditions for a solution
to be optimal, stated in the form of a Maximum
Principle
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