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

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Separation of variables, when possible?

$$ \Delta \Psi(x, y, z) + V(x, y, z)\Psi(x, y, z) = E \Psi(x, y, z) $$ For which $V(x, y, z)$ can this partial differential equation (eigenproblem) be solved by separation of variables?
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2answers
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Separation of variables: when to have exponential solution and when sinusoidal?

In separation of variables, one can assume a solution of V(x,y) = X(x)Y(y) and after plugging this into Laplace's equation which is: ${{\partial^2 V} \over {\partial x^2}}$ + ${{\partial^2 V} \over ...
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1answer
43 views

Trace Theorem question

From PDE Evans, page 272. My question is towards the bootom of this post. THEOREM 1 (Trace Theorem). Assume $U$ is bounded and $\partial U$ is $C^1$. Then there exists a bounded linear operator ...
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0answers
21 views

Existence and Uniqueness of Solutions of PDEs

I have been looking into the Cauchy-Kovalevskaya Theorem where one can "establish the local existence of analytic solutions to a system of PDEs". I wanted to see an application (for example, see ...
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28 views

Basic exercise of real analysis and of p harmonic functions

I am studying the following definition of an article: Definition (blow up): Let $u$ a function (assuming real values) in the open ball $B(x_0,1)$. For $r>0$ define the function $u_r(x)$ in $B(0, ...
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1answer
42 views

Extension Theorem

From PDE Evans, 2nd edition, pages 268-270. My question is at the bottom of this post. THEOREM 1 (Extension Theorem). Assume $U$ is bounded and $\partial U$ is $C^1$. Select a bounded open set $V$ ...
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2answers
94 views

Partial Differential equations and applications- Reference request

I will be taking up a PDEs course next semester and would like to find some good references. The topics covered in the syllabus is given below. Partial differential equations: Conservation laws, ...
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1answer
23 views

Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$?

Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$, where $H^1(R;H^1(R^n))=\{f\colon R\to H^1(R^n)\colon \int \|f(t,\cdot)\|_{H^1}^2\,dt <\infty \quad \text{and} \int \|f'(t,\cdot)\|_{H^1}^2\,dt <\infty \}$ ...
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0answers
59 views

Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
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1answer
38 views

Third Order PDE written as a System of (Linear) First Order PDEs

I need to rewrite the PDE $$f_{y}+ff_{x}+f_{xxx}=0,$$ where $f=f(x,y)$ as a system of first order quasi-linear PDEs. I have no idea how to tackle this problem. Any form of help will be appreciated. ...
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1answer
16 views

About the weighted Sobolev norm

I'm wondering that the Sobolev norm with weight $$ \|f\|_{H^{s,k}(\mathbb R^n)} := \|(1+|x|^2)^{k/2} (\sqrt{1-\Delta})^s f \|_{L^2(\mathbb R^n)} $$ is equivalent to the norm $$ ...
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1answer
38 views

Solution to $u_t+\Delta^2u+\Delta u=0$

Suppose there exists a solution to *$u_t+\Delta^2u+\Delta u=0$ of the form $u(x,y,t)=c(t)e^{i\pi(x/4\pi+y/4\pi)}$. I need to find such a function $c(t)$. Plugging $u(x,y,t)$ into *, I got ...
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1answer
74 views

Can Dirichlet and Neumann eigenfunctions coincide for the Helmholtz equation?

We consider the null space (corresponding to the possible eigenvalue zero) of the linear indefinite elliptic PDE $\Delta u+k^2(x)u=0$ in $\Omega$ with $u=\partial_{\nu}u=0$ on the boundary. If the ...
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2answers
29 views

Finding polynomial satistying potential equation and boundary conditions

Can someone help me with this problem? I know that this polynomial is a solution of Poisson's equation.
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1answer
49 views

Can we conclude that $\Delta (\Phi\circ u)$ is a measure, given that $\Phi$ is a particular smooth function and $u$ is in some Sobolev space?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Assume that $u\in W_0^{1,2}(\Omega)$ is such that $\Delta u$ is a measure. Let $\Phi$ be a smooth function in $\mathbb{R}$, such that ...
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1answer
35 views

If $\lVert u(t) \rVert \leq \lVert u_0 \rVert$ for solution of PDE, is $\lVert u(t) \rVert \leq \lVert u(s) \rVert$ for all $t \leq s$?

Suppose I have some nonlinear PDE on some time interval $[0,T]$ $$u_t + A(u) = 0$$ $$u(0)=u_0$$ and I have managed to show existence and uniqueness of solution with $\lVert u(t) \rVert \leq \lVert u_0 ...
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0answers
19 views

Vanishes of second order

I encountered the following statement in an article: Let $f(X) = f(X_1, ..., X_N) = \sum_{|\alpha| \geq 2} {a_{\alpha}X^{\alpha}}$ be a convergent power series which vanishes of second order at ...
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131 views

Why is $a_{n}(x,y)=a_{n}(y)$?

This particular question is connected (with a slight variation in the definition of $g$) to an earlier question. The link is here. The specifics are: Given that $u(x,y)$ is the solution of a PDE ($x$ ...
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26 views

A problem of convergence in $L^2(0,T;L^6(\Omega))$

Why $u_m\to u\mbox{ in }L^2(0,T;L^2(\Omega))\Rightarrow u_m 1_{\{|u_m|\leq\lambda\}}\to u 1_{\{|u|\leq\lambda\}}\mbox{ in }L^2(0,T;L^6(\Omega))?$
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1answer
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Partial derivation of a population kinetic's equation

In reviewing my biophysics' course on population kinetics I am stuck in finding which equation was actually used to derive from. It uses an example to "explain" the analytical method, in order to ...
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62 views

heat kernel on n-sphere

I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way ...
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1answer
26 views

Well-Posedness PDE of the Form $\partial_t u = P(\partial_x) u$ for a Polynomial $P$

My question is to determine whether the PDE $\partial_t u = P(\partial_x) u$, with $2\pi$-periodic boundary conditions, for a polynomial $P$, is well-posed; this depends on the polynomial, and my ...
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1answer
31 views

Finite Difference Method Stability with diffusion equation

The diffusion equation is: $ \frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} \right) $ An explicit finite difference approach can be used to solve this, forward in ...
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0answers
29 views

Mollification of functions in $L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$

Let $u \in L^2(0,T;H^1(\Omega))\cap H^1(0,T;H^{-1}(\Omega))$ where $\Omega \subset \mathbb{R}^n$ is a bounded domain. I know that (eg. from Wloka or Hunter's PDE notes) that there is a mollification ...
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2answers
83 views

In three dimensions, the Laplacian of $1/r$ is $0$ outside the origin

Why does the following hold? $$\Delta_{3}\frac{1}{r}\Bigg\vert_{\mathbb{R}^3 \setminus \left\lbrace 0 \right\rbrace}=0$$
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1answer
32 views

Green's function ( Differential Equations)

I am pretty fine on solving the partial differential equations using the method of separation of variables, now I am trying to understand the concept of Green's function for solving the PDE.And, I am ...
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69 views

Implicit Function Theorem (Two Variables)

While reading through a paper I came a cross a result (due to the Implicit Function Theorem) that I cannot derive. Let $u(x,y)$ be the solution of a PDE ($x$ and $y$ are independent variables). We ...
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1answer
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Normal mode solutions

Why does the solution choose those highlighted in green. Don't you normally have to pick the more general $\displaystyle\eta=\hat\eta (e^{ik(x-ct)}+$complex conjugate$)$ and then consider $\hat\eta ...
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1answer
39 views

Linear Water Waves

When solving , $\displaystyle {\partial^2 \tilde\phi\over\partial x^2}+{\partial^2 \tilde\phi\over\partial y^2}=0$ Why is it that $-h<y<0$ as opposed to ...
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1answer
38 views

Surface tension

Consider deep water gravity waves but assume now that there are two fluids separated by the interface at $y=\eta(x,t)$ and suppose that the upper fluid extends to $y\to\infty$. Let the fluid in the ...
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1answer
63 views

Solve Burgers' Equation with side condition.

Solve Burgers' equation $$u_t + uu_x =0,$$ with $u=u(x,t)$ and the side condition $u(x,-1) = x^2$. Find the solution for $u=u(1,2)$ I can't figure out how to use the side condition in order to find ...
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1answer
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Weakly * continuous definition

What does it mean that $$t\to u(t,\cdot)$$ is waakly* continuous from $[0,T]$ to $L^\infty(\mathbb{R}^d)$? I guess that I have to see $L^\infty(\mathbb{R}^d)$ as the dual of $L^1(\mathbb{R}^d)$ and ...
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3answers
52 views

Solving $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0$ by changing variables

Transform the differential equation $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0 $ by introducing new variables $x = u+v$ and $y=u-v$. then solve it. I which I could show ...
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Hyperbolic Systems and Charactersitics

I am trying to follow the derivation of finding the characteristics of a hyperbolic system in the general case. I understand up till the equation, where I have highlighted the RHS in green. I ...
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1answer
75 views

Properties of weak derivatives in Sobolev spaces

In PDE Evans 2nd edition, pages 261-263, there is a theorem and its proof which concerns the four properties of weak derivatives. Unfortunately, I do not understand the fourth property, which I will ...
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Derivation of fundamental solution of heat equation by reduction to ODE - Question on integration factor

In the derivation of fundamental solution for heat equation ( as in PDE by L.Evans ), we come across the reduction to following ODE : $\alpha w + {1\over2}r w'+ w'' +{n-1\over{r}}w' = 0$ Set ...
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124 views

Eigenvalues problem for generalized Kuramoto-Sivashinsky equation

I been working on Kuramoto-Sivashinsky Equation. In the process of analysis, I need to solve the following eigenvalues problem $$ -u_{xxxx}-\lambda u_{xx}=\beta(\lambda)u $$ where $\lambda$ is a ...
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1answer
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How rigorous is multiplying both sides of an eqaution for the differential of a function?

I have to solve this equation: $$ -C_0 f + \frac{1}{2}f^2 +\frac{d^2 f}{d X^2}=A $$ where $C_0$ and $A$ are two real nonzero constant; $f:\mathcal{R}\to \mathcal{R}$ I have seen that the person who ...
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1answer
81 views

Rankine Hugoniot Jump Condition Derivation

I follow the majority of this derivation I just do not understand where the two terms highlighted in green come from.
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45 views

Mean value formula

Let $u(x)$ be an entire positive solution of the equation $$\Delta u - u = 0 ~~~on ~ R^{n} ~~n>1$$ (a) Can you find a mean value formula? (b) Let $\mu$ be a positive Radon Measure on the unit ...
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Partial Differential Equation with a flux term

I don't understand why $\phi=\frac{1}{2}u^2$ is the flux in this case? Recall how we derived all our equations: Take an interval $[a,b]$ and consider $$\dfrac{\mathrm d}{\mathrm ...
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1answer
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The range of the distributional laplacean, defined in $W_0^{1,1}(\Omega)$.

Let $\Omega\subset \mathbb{R}^N$ be a bounded, smooth domain. Assume that $u\in W_0^{1,1}(\Omega)$ and consider the distributional laplacean of $u$; $\Delta u$. My question is: when is $\Delta u\in ...
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1answer
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Change of variables from intinite to bounded support.

I may be missing something simple, but I am stuck. My question: I am solving a system of partial differential equations numerically, but one of the variables can take on any value, ie $x \in ...
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22 views

Maximum principle for linear elliptic operators of arbitrary order

What is known about maximum principles for strongly elliptic linear differential operators of even order (possibly higher than $2$)? By such an operator, I mean a linear differential operator with ...
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1answer
39 views

Weak solution of a parabolic equation when initial and boundary conditions are inconsistent

Consider the parabolic equation $$ u_t = u_{xx}, $$ $$ u(0,t)=u(1,t)=0, $$ $$ u(x,0) = 1. $$ Initial and boundary conditions are inconsistent but a weak solution exists. It belongs to ...
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1answer
46 views

When is separation of variables an acceptable assumption to solve a PDE?

We know that one of the classical methods for solving some PDEs is the method of separation of variables. It works for known types of PDEs and many examples of physical phenomena are successfully ...
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Non Linear general kinematic wave equation

I am rather confused by this section of my non-linear waves notes. In the parts I have underlined in green $c(u_0(\xi))$ is defined as a constant and then as a variable even though in both instances ...
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2answers
30 views

Fourier transform of a derivative

My PDE book gives the properties for the fourier transform of $u(x,t)$: $F(\frac{\partial^n}{\partial t^n}u(x,t))=\frac{\partial^n}{\partial t^n}\hat u(\xi,t)$, and $F(\frac{\partial^n}{\partial ...
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1answer
98 views

How do I apply this maximum principle?

I have the maximum principle: $$\text{If } \psi\geq 0 \text{ on }\Gamma. \text{Then }L\psi\geq0\text{ implies } \psi\geq 0 \text{ in } \bar{D},$$ where $D=(0,1)\times (0,T], \Gamma$ is parabolic ...
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1answer
25 views

Why are we discarding solutions to this heat equation?

Dirichlet problem on unit disc in polar: $u_{rr} + (1/r) + (1/r^2)u_{\theta\theta} = 0$ $u(1,\theta) = f$ Period in $\theta$ gives $u(r,\theta) = \sum R_n(r) e^{in\theta}$ Inserted into our PDE ...