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Questions tagged [conservation-laws]

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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 ...
lsb's user avatar
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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 ...
Ma Joad's user avatar
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1 answer
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Energy conservation for power nonlinear Schrödinger Equation

I'm trying to derive the energy conservation for the one dimensional Schrödinger Equation with power nonlinearity of p=5. The underlying Lagrange function is given by $$\cal{L}(u(t,x))=\cal{L}(t,x,u,u^...
find_me_in_a_Hilbertspace's user avatar
1 vote
0 answers
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How does a computer program recognize shocks given data of a solution to a conservation law?

Conservation laws are PDEs of the form $u_t +j_x=0.$ A discontinuous solution (for $u$ and $j$) to an equation like this can be easily found. Let's suppose that we are working with a piecewise ...
Ma Joad's user avatar
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3 votes
1 answer
356 views

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 ...
Ma Joad's user avatar
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1 vote
1 answer
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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 ...
Ma Joad's user avatar
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6 votes
2 answers
1k views

Definition of entropy and entropy flux of conservation laws: component-wise reasoning

Consider the conservation law $$\DeclareMathOperator{\dvg}{\operatorname{div}} \partial_t u(x,t) + \dvg G(u(x,t)) =0, \\ u \in U\subseteq \mathbb R^m, x\in X\subseteq \mathbb R^n, G \subseteq \mathbb ...
Ma Joad's user avatar
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1 vote
0 answers
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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 ...
Ma Joad's user avatar
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15 votes
3 answers
1k views

Where do some "energy identities" in PDE theory come from?

There are a lot of very complicated expression that helps us obtain useful estimates for PDEs. To just give one example, the following is one of the Virial identities: $$ \frac12\frac{d}{dt} \Im \left(...
Ma Joad's user avatar
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Parseval's equivalent of Norm that includes a Projection matrix

I need to optimize the norm, ${\bf x}^H {\bf P}_{\bf B} {\bf x} $, where, ${\bf P}_{\bf B} = {\bf B}^H({\bf B} {\bf B}^H)^{-1} {\bf B}$, ${\bf B}$ is a known $M \times N$ matrix, with $M < N$ and $...
ashaw2's user avatar
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Convolution, Fourier transforms, and area preservation [closed]

Consider the convolution of two functions, f * g. And let us assume, for practicality, some example case where an integral of f or g can be interpreted as the "area under the curve" (or the ...
david's user avatar
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5 votes
3 answers
609 views

What quantities are conserved under a general gradient-flow $\dot X(t) = -\nabla L(X(t))$?

Let $L:\mathbb R^N \to \mathbb R$ be a continuously differential function with gradient $x \mapsto \nabla L(x)$ and consider induced gradient-flow $$ \dot X(t) = -\nabla L(X(t)). $$ Question. Is ...
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2 answers
434 views

Examples of applications of hyperbolic conservation laws

I am giving a talk in front of my applied PDE research group on hyperbolic conservation laws, the most basic form of which is the PDE $$ u_t + f(u)_x = 0 $$ where $u$ is the conserved quantity and $f$ ...
groupoid's user avatar
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3 votes
1 answer
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How to find the conserved quantities of the Kirchhoff equation?

Consider the Kirchhoff equation, given by $$u_{tt}-\left(1+\int_{\mathbb{R}} u_x^2\;dx\right)u_{xx}+f(u)=0, (x,t) \in \mathbb{R}\times \mathbb{R}_+$$ where $f(u)=u-u^{2r+1}$, for $r \in \mathbb{N}$. ...
Guilherme's user avatar
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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 ...
Mark's user avatar
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7 votes
1 answer
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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]. ...
Mark's user avatar
  • 647
2 votes
1 answer
293 views

Comparing solutions of PDE problem with different initial conditions

My question(s) is about what happens with the solution of the problem if we change initial conditions. Let's say we have a PDE problem: $$ (1) \hspace{0.5cm} u_t+f(u)_x=0 $$ $$ (2) \hspace{0.5cm} ...
Mark's user avatar
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1 vote
2 answers
235 views

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) \...
Mark's user avatar
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5 votes
2 answers
478 views

How to use these higher symmetries and conservation laws?

For infinite dimensional integrable systems, there are usually infinite symmetries and conservation laws. For example, the KdV equation, the KP equation. However, unlike the classical symmetries (...
W. mu's user avatar
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1 vote
0 answers
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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 ...
Mark's user avatar
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0 votes
2 answers
171 views

On solutions of the continuity equation

Can all square integrable solutions $(\rho(t,x),j(t,x))$ of the homogeneous continuity equation $$\dot\rho(t,x)+\nabla \cdot j(t,x)=0$$ in 1+3 dimensions be approximated by solutions with compact ...
Arnold Neumaier's user avatar
4 votes
0 answers
122 views

Conservation laws for modified Degasperis-Procesi equation

It is known that the Korteweg-de Vries equation $$u_{t}+uu_{x}+u_{xxx} = 0,$$ with $u=u(x,t)$ smooth and with period equal to $L$, has important conservation laws, namely, $$E(u)=\frac{1}{2}\int_{0}^{...
Darós's user avatar
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6 votes
1 answer
276 views

Periodicity of KdV equation in relation to zero-curvature equation

In most of the resources that I have read, integrable systems described by a PDE posses a zero-curvature equation $$ \partial_t U - \partial_x V + [U,V] = 0 $$ which gives rise to the monodromy matrix ...
Hunter's user avatar
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19 votes
6 answers
3k views

reference for Noether's theorem

What is a good reference for a geometric version of Noether's theorem about Lagrangians, symmetries and conserved currents?
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