Questions tagged [conservation-laws]

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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 ...
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1answer
118 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 ...
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1answer
35 views

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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135 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 ...
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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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3answers
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(...
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51 views

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 $...
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1answer
174 views

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 ...
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2answers
166 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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2answers
173 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 $f$ ...
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1answer
107 views

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 $r \in \mathbb{N}$. ...
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2answers
247 views

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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1answer
244 views

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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1answer
188 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} ...
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2answers
196 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) \...
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2answers
316 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 (...
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0answers
134 views

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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116 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 ...
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109 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}^{...
6
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1answer
233 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 ...
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5answers
2k 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?