0
votes
0answers
6 views

Modified Symplectic Euler

Simple harmonic motion: $y'= -z $, $z'= f(y)$ and the modified Symplectic Euler equation are $$y'=-z+\frac {1}{2} hf(y)$$ $$y'=f(y)+\frac {1}{2} hf_y z$$ deduce that the coresponding approximate ...
0
votes
1answer
28 views

Implicit Euler Scheme and stability

Find the fixed points of the implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1}) \end{equation} when applied to the differential equation $y'=y(1-y)$ and investigate their ...
0
votes
0answers
20 views

A predictor-corrector method

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} ...
0
votes
1answer
19 views

$f, \hat{f} \in L^{1}(\mathbb R) \implies \widehat {\text{Re}(f)}, \widehat{\text{Im}(f)} \in L^{1}(\mathbb R)$?

Let $f:\mathbb R \to \mathbb C$ such that $f(x)= (f_{1}(x), f_{2}(x))$; where $f_{1}(x)=\text{Re}(f(x))=\text{the real part of} \ f $ and $f_{2}(x)=\text{Im}(f(x))= \text{ the imaginary part of} \ ...
0
votes
0answers
20 views

properties of frequency- decomostion operator $\square_{k}^{\sigma}=\sum_{|\ell|_{\infty}\leq 1}\square_{k}^{\sigma}\square_{k+\ell}^{\phi}$?

Let $\rho \in S(\mathbb R^{n})= \text{Schwartz space}, \ \rho:\mathbb R^{n}\to [0,1]$ be smooth radial function verifying $\rho(\xi)=1$ for $|\xi|_{\infty}\leq \frac{1}{2}$ and $\rho(\xi)=0$ for ...
0
votes
1answer
18 views

$\int_{|x|<t} |\mathcal{F}^{-1}f(x) |dx \leq C t^{\frac{n}{2}}\|f\|_{L^{2}}$?

Let $f\in L^{2}(\mathbb R^{n}).$ Fix $t>0,$ My Question:How to show, $\int_{|x|<t} |\mathcal{F}^{-1}f(x)| dx \leq C t^{\frac{n}{2}}\|f\|_{L^{2}}$ ? [We note $\mathcal{F}$ denotes the ...
1
vote
1answer
61 views

Imbedding theorem for Sobolev space $D^{k,p}$

I can't find any reference on the imbedding theorem for the Sobolev space $D^{k,p}$, which is defined by $$D^{k,p}(\Omega)=\{u\in L_{loc}^{1}(\Omega)|D^{k} u\in L^{p}(\Omega)\}.$$ Is it the same as ...
1
vote
0answers
21 views

Proving an elementary inequality of real vectors related to the p-Laplacian [duplicate]

How would you prove the following inequality? $$ \left|\left|a\right|^{p-2}a-\left|b\right|^{p-2}b\right|\leq C_p\left|a-b\right|^{p-1} $$ where $a,b\in\mathbb{R}^{n}$ and $C_p>0$ is some constant ...
0
votes
0answers
17 views

Thesis of Serge Resnick - Dynamical Problems in Non-Linear Advetive Partial Differential Equations

I'm seeking the thesis Dynamical Problems in Non-Linear Advetive Partial Differential Equations by Serge Resnick. In the virtual library of The University of Chicago ...
1
vote
0answers
26 views

Reference needed for the following sobolev inequalties

I'm reading a paper and the authors applied the following sobolev type estimates $$ ||(Dv)^{2}||_{H^{3k-2}(\Omega)}\leq C||v||_{H^{3k-1+\alpha}(\Omega)}^{2} $$ for $\alpha>\frac{1}{4}$, where $v$ ...
2
votes
1answer
69 views

Why is $-\Delta+1$ an isomorphism between Sobolev space $W^{2,p}$ and $L^p\,$?

Consider the linear elliptic operator $L: W^{2,p}(\mathbb{R}^N)\to L^p(\mathbb{R}^N)$ defined by $Lu=-\Delta u + u$. Can we prove that $L$ is an isomorphism for all $p\geq 1$? It can be proved that it ...
2
votes
0answers
24 views

Conservation of momentum for nonlinear Schrodinger equation

I am having trouble proving the following momentum conservation law. Given smooth compactly supported solution $u(x,t)\in \mathbb{C}$ where $(x,t)\in \mathbb{R}^n\times \mathbb{R}$ of $iu_t+\Delta u= ...
1
vote
1answer
28 views

Estimative with kernels of the Riesz transform

Let $x=(x_1,x_2)$ and $k(x)=\frac{x_1}{|x|^3}$, $|x|>0$. Condider $x,y\in\mathbb{R^2},\xi=|x-y|, \tilde{x}=\frac{x+y}{2}$ then $|k(x-t)-k(y-t)|\leqslant C\frac{|x-y|}{|\tilde{x}-t|^3}$ when ...
1
vote
0answers
16 views

Are pseudo(micro)-local operators pseudodifferential?

$\DeclareMathOperator{supp}{supp} \DeclareMathOperator{sing}{sing}$Let $\Omega$ be a domain with compact closure in $\mathbb R^n$. Consider a linear operator $A \colon X \to X$ satisfying one of the ...
2
votes
0answers
27 views

$(g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s}$ right or wrong?

I am proving something, and I may need the following inequality: $$ (g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s} $$ where $(g,f)_{L^2} $ is inner product and $f$ and $g$ have higher enough regularity ...
0
votes
0answers
54 views

What is changed in Navier-Stokes equations after Otelbaev's proof?

What is changed in Navier-Stokes equations by Otelbaev's Strong solution of Navier-Stokes equations? does it make any change to Navier-Stokes formulation?
1
vote
1answer
27 views

Approximation of continuous functions by a functions with vanishing second derivative

Denote by $C^n[-\infty,+\infty]$ the class of functions which: have finite limits at $\pm \infty$; and are differentiable $n$ times on the line, with all these derivatives bounded. Denote by $C^3_0$ ...
1
vote
1answer
31 views

Does $C([0,T];H^{s+1}) \cap C^1([0,T];H^{s})=C^1([0,T];H^{s+1})$?

I have asked this question with other problems, but about this part nobody answers. So I want to ask again, and put some details in it. My question is whether the following equality is correct. If it ...
2
votes
1answer
31 views

About the space $u \in C([0,T];H^{s+1}) \cap C^1([0,T];H^{s})$

I am reading Taylor's Partial differential equations III (nonlinear equations) (Section 1 of Chapter 16, Theorem 1.2), and Sogge's Lectures on Non-linear wave equatuions. I notice that in the energy ...
0
votes
1answer
26 views

To prove : $\lim_{t \rightarrow \infty} g_t \ast f = 0$.

Let $g_t(x) = e^{-x^2/2t^2}/t\sqrt{2\pi}$. To prove that for $f \in C_0(\mathbb{R})$, $$ \lim_{t \rightarrow \infty} g_t \ast f = 0.$$ Is the question valid ? Any hints on how to solve it ?
3
votes
1answer
59 views

Differences between $-\Delta: H_0^1(\Omega)\to H^{-1}(\Omega)$ and $-\Delta: H^2(\Omega)\cap H_0^1(\Omega)\to L^2(\Omega)$

I'll try to explain what I want to know: Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. When we look to $-\Delta: H^2(\Omega)\cap H_0^1(\Omega)\to L^2(\Omega)$, the meaning of $-\Delta$ is ...
2
votes
0answers
17 views

Estimation kernel

I just wonder if someone could just give me the proof for the following estimation: $$ \| \nabla (e^{t\Delta} f) \|^{2}_{L^{2}} \leq \|{f}\|_{L^{1}} \ \|e^{t\Delta} f\|_{L^{\infty}} $$ where ...
1
vote
1answer
47 views

Why the following is a seminorm rather than a norm

I really don't understand why the following is a seminorm rather than a norm? $$ p_k(u)=\sum_{|α|\le k}\sup_{x∈R^n}(1+|x|^2)^{k/2}|D^α u(x)|, $$ for all $u \in C^\infty$. I do understand if ...
0
votes
1answer
22 views

About an estimate of theorem 3 in Chapter 12 of Evans' book

This is the proof of theorem 3 in Chapter 12 of Evans' book as the following picture. I really don't understand why $|F(Du,u_t,u)|\le C(|Du|+|u_t|+|u|)$, because he didn't give us any restriction on ...
1
vote
1answer
24 views

On defining appropriate energy. Any principle?

I am reading Evans' book Partial differential equations. but I am really curious about how he define the appropriate energy? Is there any principle or rule to do this things? Because I notice that ...
1
vote
2answers
27 views

Showing $ u =0 $ for a particular laplace equation.

Given open $\Omega\subset \mathbb{R}^n $ with smooth enough boundary, if for $x_0 \in \Omega$ there exists $ u \in C^2(\Omega\setminus\{x_0\}) $( we may as well take $u$ as smooth as we want) that ...
0
votes
1answer
43 views

Equivalence between dual estimates

Let $H$ a Hilbert space and $X$ a measure space; let $U(t):H\to L^2(X)$ an operator defined for all real $t$. I'm considering the following estimates: $$\Vert U(t)f\Vert_{L_t^qL_x^r}\leq\Vert ...
0
votes
0answers
10 views

How to use $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in Banach sapace $C([0, T];M^{p,1})$?

(For details of this question you may see the paper,(proof of theorem 1, page.9), MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; Bull. Lond. Math. Soc. 41 ...
0
votes
1answer
34 views

when one can expect $\left \| |x|^{2}x- |y|^{2}y \right \| \leq \frac{1}{2} \left \| (x-y) \right \|$?

Suppose $(X, \left\|\cdot \right \|)$ is Banach algebra with the property that $\left\| (|x|^{2}x) \right\| \leq \left\|x\right \|^{3};$ for every $x\in X.$ Let $y_{0}\neq 0 \in X$ and fix it; and ...
1
vote
1answer
49 views

Uniqueness for second order PDE $-\Delta u = c(u)$

I'm trying to solve the following problem: Let $\Omega$ be an open set in $\mathbb R^n$ and consider the equation \begin{cases} -\Delta u = c(u) & \text{ on } \Omega \\ u = 0 & \text{ on } ...
2
votes
1answer
77 views

Why are so many inequalities and estimate in PDEs?

I want to study PDEs, but when I read some PDEs books, I notice that there are so many inequalities here and always do estimates. I really want to know why there are so many inequalities here and ...
2
votes
1answer
32 views

Harmonic map into sphere

Let $B$ be the unit ball and $S$ the unit sphere in $\mathbb{R}^3$. Consider the map $u: B\rightarrow S$ defined as: \begin{equation} u^j(x)=\frac{x_j}{|x|}\quad\forall \ j =1, 2, 3. \end{equation}I ...
1
vote
1answer
34 views

Solutions of a Poisson equation on an ellipsoid, with Neumann boundary condition.

I want to find the number of solutions of the following problem: Fix $n \in \mathbb{N}$, with $n \geq 2$. Define the domain $\Omega$ as the ellipsoid $$\Omega = \left\{x = (x_1,...,x_n) \in ...
0
votes
1answer
34 views

A Specific Example about Parabolic PDE

I am solving a PDE numerical problem. And I have already had a algorithm. However, it seems to be hard to find a specific example to test my solution. Could you please give me one? The equation ...
2
votes
0answers
26 views

The space of bounded mean oscillation $BMO(B_R)$, live in the Campanato space $\mathcal{L}^{1, n}(B_R)$

Let $B_R$ be an open bounded ball in $\mathbb{R}^n$. I am trying to show that if $u\in BMO(B_R)$ then $u\in \mathcal{L}^{1, n}(B_R)$ and that \begin{equation} \|u\|_{\mathcal{L}^{1, n}(B_R)}\leq ...
3
votes
0answers
47 views

$f, f|f| \in L^{1} (\mathbb R) \cap C_{0} (\mathbb R) \implies |f| \in H_{1} (\mathbb R)$?

Suppose $f \ \text{and} \ f|f|\in L^{1}(\mathbb R).$ Then, clearly, $|f|\in L^{2}(\mathbb R)$ and therefore by Plancheral theorem, we get, $\widehat{|f|} \in L^{2}(\mathbb R).$ Also, assume, $f, ...
0
votes
1answer
39 views

Domains of Lipschitz class are domains of type A.

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set. We say that $\Omega$ is of type $A$ if there exists a constant, $A$, such that \begin{equation} |\Omega\cap B_{\rho}(x_0)|\geq A\rho^n ...
3
votes
2answers
52 views

If a Laplacian eigenfunction is zero in an open set, is it identically zero?

This seems like it should be easy but I can't seem to figure it out. Suppose $f$ satisfies $\Delta f + \lambda f = 0$ in $\mathbb{R}^n$ (where $\lambda > 0$ is a constant), and in some open set ...
1
vote
1answer
32 views

Checking the solution of a first order pde

I need some help with this exercise. Given the following pde: $ \begin{cases} u_t + b(u)\cdot u_x=0\\[6pt] u(x, 0) = u_0(x) \end{cases} $ I have to check that its solution is $u(x, ...
2
votes
1answer
43 views

Weak Convergence Proof: Tried on My Own

This is related to a question asked here: What a proof of weak convergence is supposed to look like I asked what a proof of weak convergence was "supposed to look like". Specifically, I asked that if ...
0
votes
0answers
35 views

Convex integration techniques for Euler equation

This is more a question of curiosity to those working on the Euler equations with convex integration techniques, particularly in recent progress towards Onsager's conjecture, with the methods ...
3
votes
1answer
61 views

Applications of the theory of distributions outside of PDEs?

Are there any interesting, important or powerful mathematical applications to the Theory of Distributions besides those dealing with partial differential equations?
2
votes
1answer
19 views

Integral Inequality Question: Effect of Exponents

Is it true that, if $\alpha q < (1-\alpha)q$ then $\int|k(x,y)|^{\alpha q}dy \leq \int|k(x,y)|^{(1-\alpha)q}dy$, where $k(x,y)$ is a positive measurable function?
2
votes
0answers
47 views

Unexplained “It Suffices to Show”

For $u \in C^{\infty}(\Omega)$, for $\Omega$ convex, and $p$, $q$ such that $\frac{1}{p}-\frac{1}{q}<n$, then in order to show that $$||u-u_{\Omega}||_{L^{q}} \leq c_{n} \left[ ...
1
vote
1answer
53 views

Inhomogeneous diffusion equation and initial conditions inversion

While working on a physical diffusion process, I encountered the following Fokker-Planck equation $$ \frac{\partial F}{\partial t} = D (x) \frac{\partial^2 F}{\partial x^2} \tag1$$ where $D(x) > ...
1
vote
1answer
17 views

Help with inequality estimate, in $H^1$,

Given a bilinear form on $H^1 \times H^1$, where $H^1 = W^{1,2}$ \begin{align*} B[u,v] = \int_U \sum_{i,j}a^{i,j}(x)u_{x_i}v_{x_j} + \sum_ib^i(x)u_{x_i}v + c(x)uv \, \mathrm{d}x \end{align*} the book ...
3
votes
2answers
84 views

Laplacian identity.

Consider $f$ and $g$ smooth functions. How to prove the following identity: $$\Delta\left(\frac{f}{g}\right)=\frac{1}{g}\Delta f-\frac{2}{g}\nabla\left(\frac{f}{g}\right).\nabla g-\frac{f}{g^2}\Delta ...
5
votes
1answer
138 views

Why is it important to study the eigenvalues of the Laplacian?

Why is it important to study the eigenvalues of the Laplacian acting on regions in $\mathbb R^n$? What information does this give us? What problems does this information help us solve? In particular, ...
4
votes
3answers
106 views

Prove that there is a weak solution in $W^{1,2}_0$$(D)$ to following equation $\Delta u+\cos u=0$.

Let $D$ be the open bounded smooth subset in $\mathbb{R}^{n}$. Prove that there is a weak solution in $W^{1,2}_0$$(D)$ to following equation $$\Delta u+\cos u=0.$$ Help me some hints to start. ...
1
vote
1answer
57 views

Confused about class notes on gradient inequality and how to derive version for functions with compact support

I'm taking a grad analysis class, and I'm a little confused about how to show the basic gradient inequality for smooth functions with compact support. Here is my professor's statement of the basic ...