1,721,025 research outputs found
Surprising solutions to the isentropic Euler system of gas dynamics
Full text linkIn a recent paper, jointly with Elisabetta Chiodaroli and Ondřej Kreml we consider the Cauchy problem for the isentropic compressible Euler system in 2 space dimensions, with initial data which assume two different constant values and have a discontinuity across a line. If we consider selfsimilar solutions we then encounter a classical 1-dimensional Riemann problem for the corresponding hyperbolic system of conservation laws. We show that for some suitable choice of the pressure and of the initial data there exist infinitely many bounded admissible solutions which are not selfsimilar and indeed are genuinely 2-dimensional. We also show that some of these Riemann data are generated by a 1-dimensional compression wave. Our theorem leads therefore to Lipschitz initial data for which there are infinitely many global bounded admissible weak solutions. Each of these solutions coincide as long as the classical (Lipschitz) solution exists and they differentiate themselves immediately after the first blow-up time. Our approach is heavily influenced by a work of László Székelyhidi which provides a similar result in the case of the classical vortex-sheet problem for the incompressible Euler equations
Some results on the two-dimensional dissipative Euler equations
Full text linkWe make a review of some recent results concerning special solutions and behavior at infinity for 2D dissipative Euler equations. In particular, we give a simplified proof --in the space-periodic setting-- of the uniform space/time boundedness of the first derivatives of the velocity, under suitable assumptions on the external force and on the dissipation (damping) coefficient. This is used to sketch the proof of existence of almost-periodic solutions
Relative entropy methods for hyperbolic and diffusive limits
No full textWe review the relative entropy method in the context of hyperbolic and diffusive relaxation limits of
entropy solutions for various hyperbolic models. The main example consists of the convergence from
multidimensional compressible Euler equations with friction to the porous medium equation cite{LT12}.
With small modifications, the arguments used in that case can be adapted to the study of the
diffusive limit from the Euler-Poisson system with friction to the Keller-Segel system cite{LT13}.
In addition, the --system with friction and the system of viscoelasticity with memory are then reviewed,
again in the case of diffusive limits cite{LT12}.
Finally, the method of relative entropy is described for the multidimensional stress relaxation model converging to elastodynamics
cite[Section 3.2]{LT06}, one of the first examples of application of the method to hyperbolic relaxation limits
ON INVISCID LIMITS FOR THE NAVIER-STOKES EQUATIONS WITH SLIP BOUNDARY CONDITIONS INVOLVING THE VORTICITY
No full textIn this note we consider the inviscid limit for the Navier-Stokes equations under different slip boundary conditions of Navier's type and we show how this influences the convergence rate in the energy norm. The role of the initial data is also emphasized in connection with the vanishing viscosity limit
On quantitative compactness estimates for hyperbolic conservation laws
No full textWe are concerned with the compactness in L^1_loc of the semigroup (St)_{t>0} of entropy weak solutions generated by hyperbolic conservation laws in one space dimension. This note provides a survey of recent results establishing upper and lower estimates for the Kolmogorov "-entropy of the image through the mapping S_t of bounded sets in L^1 \cap L^\infty, both in the case of scalar and of systems of conservation laws. As suggested by Lax [16], these quantitative compactness estimates could provide a measure of the order of "resolution" of the numerical methods implemented for these equations
On inviscid limits for the navier-Stokes equations with slip boundary conditions involving the vorticity
No full textIn this note we consider the inviscid limit for the Navier-Stokes equations under different slip boundary conditions of Navier's type and we show how this influences the convergence rate in the energy norm. The role of the initial data is also emphasized in connection with the vanishing viscosity limit
On the asymptotic stabilization of the hyperelastic-rod wave equation
No full textWe discuss the problem of asymptotic stabilization of the hyper\-elastic-rod wave equation on the real line
\partial_t u-\partial_{txx}^3 u+3u \partial_x u=\gamma\left(2\partial_x u\, \partial_{xx}^2 u+u\, \partial_{xxx}^3 u\right),\quad t > 0,\>\>x\in \mathbb{R}.
We consider the equation with an additional force term of the form f:H^1(\mathbb{R})\to H^{-1}(\mathbb{R}),\, f[u]=-\lambda(u-\partial_{xx}^2 u),
for some \lambda>0. We resume the results of [F. Ancona and G. M. Coclite, 2015] on the existence of a semigroup of global weak dissipative solutions of the corresponding closed-loop system defined for every initial data u_0\in H^1(\mathbb{R}). Any such solution decays esponentially to 0 as t\to\infty
On the asymptotic stabilization of a generalized hyperelastic-rod wave equation
No full textWe discuss the problem of asymptotic stabilization of the hyper-elastic-rod wave equation on the real line
egin{equation*}
partial_t u-partial_{txx}^3 u+3u partial_x u=
gammaleft(2partial_x u, partial_{xx}^2 u+u, partial_{xxx}^3 u
ight),quad t > 0,>>xin mathbb{R}.
end{equation*}
We consider the equation with
an additional forcing term of the form
%
for some lambda>0. We resume the results of cite{AC} on the existence of a semigroup of global weak dissipative solutions of
the corresponding closed-loop system
defined for every initial data . Any such solution
decays exponentially to 0 as
Analysis of Oscillations and Defect measures in plasma physics
No full textWe perform a rigorous analysis of the quasineutral limit for a hy- drodynamical model of a viscous plasma represented by the Navier Stokes Poisson system in 3 − D in the general setting of ill prepared initial data. In general the limit velocity field cannot be expected to satisfy the incompress- ible Navier Stokes equation, indeed the presence of high frequency oscillations strongly affects the quadratic nonlinearities and we have to take care of self in- teracting wave packets. We provide a detailed mathematical description of the convergence process by using microlocal defect measures and by developing an explicit correctors analysis. Moreover we identify an explicit pseudo parabolic pde satisfied by the leading correctors terms
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