All Questions
Tagged with conservation-laws hyperbolic-pde
9 questions
3
votes
1
answer
160
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Definitions of weak solutions for quasilinear wave equations
I am learning the shock problem for the balance system (perhaps not conserved, see, e.g., "Ingo Muller, Tommaso Ruggeri. Rational Extended Thermodynamics") and just have a question on the ...
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0
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0
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46
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Conservation law for generic linear hyperbolic PDEs?
Consider the wave equation:
$$
u_{tt} = \Delta u, \text{ on }U\times [0,T], u=0\text{ on }\partial U\times[0,T].
$$
To prove the only solution for the zero initial condition is zero, we only need to ...
- 1,755
3
votes
1
answer
404
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What does it mean by "converges boundedly"?
On page 92 of the book Hyperbolic Conservation Laws in Continuum Physics by C. M. Dafermos, there is a theorem 4.6.1 which says
Under some assumptions, suppose a sequence of solutions $U_{\mu_k}$ to ...
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1
vote
1
answer
79
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Assumptions on the flux of a conservation law required to obtain an entropy inequality
On page 87 of the book Hyperbolic Conservation Laws in Continuum Physics by C. M. Dafermos, there is a theorem which I summarise as follows
Theorem. (Theorem 4.5.2 in the book.) Let $U$ be a weak ...
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1
vote
1
answer
83
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Front tracking approximations and existence of solutions to conservation law PDEs
This question is about page 38-42 of these notes on censervation laws, more precisely PDEs of the form $u_t + [f(u)]_x =0.$ In this section of the note, the author provides a proof of the existence of ...
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1
vote
2
answers
376
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Banach space-valued test functions in the definition of a weak solution of a PDE problem
In the literature about PDEs it is easy to find books that talk about weak solutions of a partial differential equations. A short reminder of the usual definition is given bellow. More information ...
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7
votes
1
answer
895
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Comparing weak and strong solutions of a PDE problems
A few days ago I was reading the paper:
"Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system" - Feireisl, Jin, Novotny, 2012 [Arxiv].
...
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1
vote
2
answers
247
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Transformation from the PDE problem with a source to the PDE problem without it and viceversa
In the study of nonlinear conservation laws a lot of time I work on the two problems given bellow:
$$(1) \hspace{1cm} \begin{cases}
u_t+(f_{1}(u))_x=\lambda \cdot g(u) \\[2ex]
u(x,0)=h_{1}(x)
\...
- 657
1
vote
0
answers
162
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Reference request for a paper with Vanishing viscosity method and smooth approximation of initial data
I am trying to find the papers/books/notes that study problem (1),(3) given bellow using the vanishing viscosity method. I am especially interested in solutions in Sobolev spaces.
More detailed ...
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