# Tagged Questions

Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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### geometric interpretation of the solution to PDE $u_x + yu_y =0$

The equation $$u_x + yu_y =0$$ has the general solution $$u(x,y)=f(ye^{-x})$$ The characteristic curves should look like I tried to plot the solution using google. Is this plot correct? ...
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### clarification to specific solution of the first order linear PDE problem $4u_{x}-3u_{y}=0, u(0,y)=y^3$

I am analyzing the following first order PDE problem and have difficulties with understanding the solution $$\left\{\begin{matrix} 4u_{x}-3u_{y}=0\\ u(0,y)=y^3 \end{matrix}\right.$$ I understand ...
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### Solutions to Laplace's equation between surfaces of a constant coordinate

In Physics (specifically electrostatics) it is often encountered that one must solve the Dirichlet problem for Laplace's equation $$\nabla^2 \varphi(x_1,x_2,x_3) = 0$$ ...
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### Math needed to study Navier-Stokes existence and smoothness problem [closed]

This is Navier-Stokes existence and smoothness problem. I think the main problem is that I am not familiar with the mathematics of the Navier-Stokes existence smoothness problem. What kind of math ...
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### comparison of two first order pde problems $u_x + cu_y =0$ vs. $u_x + cu_y =1$

I am analyzing two similar first order PDE problems. 1) $u_x + cu_y =0 \ \$ and $\ \ u(0,y)= sin(y)$ and 2) $u_t + cu_x =1 \ \$ and $\ \ u(0,y)= sin(y)$ As I understand 1) The solution ...
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### Dimension of boundary of a bounded domain; what to use for Sobolev inequalities

Let $\Omega$ be a bounded (at least) Lipschitz domain in $\mathbb{R}^{N}$. Its boundary $\partial\Omega$, if I'm right, is $(N-1)$-dimensional object embedded in $\mathbb{R}^N$. We can define Sobolev ...
what does the triangle sign used in PDE equations represent? example: $$u_{tt}- \Delta u =0$$ I took the below example from the Evan's book PDE p.4 (example 9). The equation is presented as the ...
I'm self-reading Evan's PDE book. It discusses elliptic and parabolic theories under the condition that the domain $U$ is open and bounded, and the functions vanishes on $\partial U.$ I'm wondering if ...