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for $h(y,x)=\nu(y)\partial_y\delta(x-y)$ one has, upon partial integration: $$\int_{-\infty}^\infty u(y,t)h(y,x)\,dy=-\int_{-\infty}^\infty \delta(x-y)\partial_y[\nu(y)u(y,t)]\,dy=-\partial_x [\nu(x) …

No, entropy convexity and hyperbolicity are not equivalent conditions. A necessary and sufficient condition for the system of differential equations to possess a strictly convex entropy is that the sy …

Define the vector $X=(y,z)$ and matrices $\sigma_1={{0 1}\choose{1 0}}$, $\sigma_3={{1\; 0}\choose{0\, -1}}$, then the differential equations read $$X_t = -\sigma_3 X_x + \sigma_1 X.$$ No boundary con …

You could just try: $A\mapsto A+\nabla \lambda$ leaves the first two equations invariant if I transform $u\mapsto e^{i\lambda}u$ and leave $V$ the same, but then the third equation no longer holds, so …

The construction of adding $p$ as an additional element to the vector $\mathbf u$ only hides it in the vector $\mathbf v$, without actually eliminating it from the Euler equation. This can be easily d …

The terminology focusing versus defocusing comes from electromagnetic wave propagation in a material with a refractive index $n=\alpha|E|^2$ that depends on the energy density of the electric field (p …

If the time dependence of the potentials is periodic in time, then one enters the field of Floquet wave equations, which is a very active field of study with many real-world and even practical applica …

There is an extensive literature, this could be helpful entry point: Solving Navier-Stokes equations coupled with a heat transfer equation (2015) In this paper, the dynamics of an incompressible …

Gårding's differential operator (introduced in Extension of a Formula by Cayley to Symmetric Determinants) is discussed by Turnbull in Symmetric Determinants and the Cayley and Capelli Operator: The …