Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.

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19
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4answers
2k views

Classification of PDE

Recently I have been attending a course on PDE's. I was totally ignorant of the subject and wasn't that motivated to be honest. But I was intrigued and felt I had to take the course seriously both for ...
14
votes
11answers
4k views

Textbooks for PDE between Strauss and Folland

Walter A. Strauss's Partial Differential Equations: An Introduction is a classic PDE textbook for the undergraduate students. While Folland's Introduction to Partial Differential Equations, is a nice ...
10
votes
1answer
687 views

Regularity of the Maxwell equations

As is well-known, the Maxwell equations can be phrased vectorially as, \begin{align} \nabla \cdot \mathbf E &= \frac{\rho_f}{\varepsilon}, &\text{Gauss's law,}\\\ \nabla \cdot \mathbf ...
10
votes
3answers
4k views

Physical Interpretation of Robin Boundary Conditions

In a (bounded) domain $\Omega \subset \mathbb{R}^n$, if we're studying the Laplace equation or heat equation or such PDE's we can impose the Dirichlet $u|_{\partial\Omega} \equiv 0$, Neumann $D_{\nu} ...
9
votes
1answer
370 views

Special Second-Order PDE

Let $\Phi$ be a given smooth function on a neighborhood of zero in $\mathbb{R}^n$ with $$\Phi(0) = 0, ~~~~D \Phi(0) = 0, ~~~~ D^2\Phi(0) >0,$$ the latter meaning that the Hessian is positive ...
8
votes
1answer
198 views

Failure of Fredholm property of elliptic PDE systems

Roughly speaking, a PDE operator satisfies the Fredholm property if its principal symbol is elliptic and the information provided on the boundary satisfies the Shapiro-Lopatinskii condition. What can ...
7
votes
1answer
988 views

Short time existence on nonlinear parabolic PDE

I saw several papers that without proof accept the fact "Short time existence on nonlinear parabolic PDE" is there any affirmative proof of this fact? in which book we have this fact, the number of ...
7
votes
2answers
331 views

Fredholm alternative result for general elliptic system?

Now I have known that Fredholm alternative result is valid for the strong elliptic system. But I'm not sure that is it still valid for the general elliptic system, in which the second-order heading ...
7
votes
1answer
149 views

Mountain Pass theorem for minimization problems with constraints

Let $I[u]$ be a functional on a (possibly infinite dimensional) Hilbert space. Then, under some conditions, the Mountain Pass theorem guarantees the existence of a saddle point (see ...
7
votes
0answers
350 views

Regularity of solutions to a linear degenerate parabolic pde

I've encountered the following problem which is causing me some trouble : Let $(M^2,g)$ be a smooth compact Riemannian surface (with constant curvature for instance) and consider a smooth function ...
6
votes
3answers
835 views

Analytical solution to a Linear advection-reaction PDE

I am looking for an analytical solution for the linear PDE $(1)\qquad\qquad \qquad f_t+ A f_x + B f = 0, $ Where $A$ and $B$ are constant matrices and $f=f(x,t)$ is a vector. Clearly each one of ...
6
votes
2answers
292 views

Uniform bound on the eigenfunctions of the Laplacian

Is it possibly to have $L_\infty$ bounds on the eigenfunctions of the Laplacian operator on bounded regular domains with Dirichlet condition? I found several papers by Sogge but these are pretty ...
6
votes
1answer
209 views

Reference request: Optimal $L^p$-decay for nonhomogenous heat equation in $\mathbb R^d$

Let $u$ be a classical solution for the nonhomogeneous heat equation in $\mathbb R_+ \times\mathbb R^d$: $$ \begin{cases} \partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\ u(0,x)=u_0(x). \end{cases} $$ ...
6
votes
0answers
341 views

Lax Pairs for Linear PDEs

I'm trying to understand the discussion around equation (2.1) of the paper http://www.jstor.org/stable/53053. It says that the linear PDE $M(\partial_x,\partial_y)q=0$ with constant coefficients has ...
5
votes
2answers
172 views

Well-posedness of Fokker-Planck equation

Consider the following equation on $[0,T]\times\mathbb{R}^n$ \begin{eqnarray} &\partial_t\rho=\mathrm{div}(\rho\nabla V)+\Delta\rho\\ &\rho|_{t=0}=\rho^0, \end{eqnarray} where $V\in ...
5
votes
3answers
361 views

Structure of sign changes under the heat flow

Let $f$ be a smooth function on $R^2$, and define $N_f$ to be the set of points $p$ such that the nodal set of $f$ ($\{x\in R^2: f(x)=0\}$) divided every neighborhood of $p$ into four regions. Indeed, ...
5
votes
1answer
92 views

Evolution operator for a linear parabolic equation

Let $A(t)$ be a smooth family of positive definite operators on a Hilbert space $H$. Consider the operator $$D:= \frac{d}{dt}+A(t)$$ and let $U(t):H\to H$ be the evolution operator, i.e., $U(0)=I$ and ...
5
votes
1answer
146 views

Recognizing Schwartz regular distributions

Are there characterizations of Schwartz regular distributions other than being locally integrable (which does not lend itself to easy manipulations)? To be more detailed: if I want to show that some ...
5
votes
0answers
343 views

Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE

Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem $$ \partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x) $$ ...
4
votes
2answers
217 views

hodographic transformation

Let $\phi(x,t)$ be smooth function. Let $\zeta= x-t$ and $\eta= x+t$. $u=\phi_\zeta$, $v= \phi_\eta$. Let $u$, $v$ satisfies following equations: 1- $$u_\eta- v_\zeta= 0$$ ...
4
votes
2answers
381 views

Analytic solution of a system of linear, hyperbolic, first order, partial differential equations

In a try to solve a physical problem, I've faced a system of first-order partial differential equations of the form ...
4
votes
2answers
194 views

Solution to Schrödinger equation

I asked this question already on stackexchange, but I did not get any resonance at all, so maybe anybody here can give me a few hints about my problem. My goal is to solve this PDE for $f:[-1,1] ...
4
votes
2answers
484 views

Numerical solution to diffusion-like equation with negative diffusion coefficient region?

I am trying to numerically solve the initial value problem (see later discussion for ICs) $$ x \frac{\partial f}{\partial t} = \frac{\partial}{\partial x} (1-x^2) \frac{\partial f}{\partial x} - f$$ ...
4
votes
1answer
218 views

Please recommend some literature on the systematical theory of the elliptic systems!

Now I'm interested in the theory of elliptic systems, for example, both the linear and nonlinear case, the exsitence and regularity results, and is there a Fredholm alternative result for the linear ...
4
votes
1answer
109 views

Gradient estimate for elliptic equation

Given: 1)a bounded domain $\Omega$ in $\mathbb R^n$ of class $\mathcal{C}^{\infty}$ 2) the function $f\in L^{\infty}(\Omega)$ with $\int_{\Omega} f=0$ 3)$g=(g_i,\ldots,g_n)\in ...
4
votes
0answers
65 views

How do solutions of a PDE depend on parameters?

Let $\Omega\subset\mathbb R^n$ be a bounded smooth domain and $\sigma_1,\sigma_2:\Omega\to(c^{-1},c)$ measurable (for some constant $1<c<\infty$). Let $f\in ...
3
votes
2answers
225 views

Heat equation and evolution of number of critical points

Let $u_0$ be a smooth function on the unit sphere $S^1$ and assume that $u(t,x)$ is a smooth solution of the heat equation with initial data $u(0,x)=u_0(x)$. How one can apply the maximum principle to ...
3
votes
1answer
577 views

Stochastic Heat Equation

Given the heat equation: $$\partial_{t}{\varPhi(x,t)}=k^2\partial_{xx}{\varPhi(x,t)}$$ with the boundary conditions: $$\Phi(x,0)=\Phi_0$$ and a Neumann boundary condition of the kind: ...
3
votes
1answer
630 views

Long time behavior of the heat equation on R

Let $\mu\in\mathcal{S}'(R)$ be a Schwartz distribution. The solution of a heat equation with $\mu$ as the initial data is $$ u(t,x)= \int_R \frac{e^{-\frac{(x-y)^2}{2t}}}{\sqrt{2\pi t}} \mu(d y) $$ ...
3
votes
1answer
121 views

To give an estimate for the maximal function associated to the Schrödinger group by using a measurable selector function

I am consulting some papers (references below) about the Carleson's problem for the pointwise convergence of the Schrödinger group \begin{equation} S_t=e^{i t \Delta}. \end{equation} In this context ...
3
votes
1answer
203 views

Uniqueness of weak solutions of a heat equation

Let $M$ be a smooth compact closed manifold. Let $u \in H^1(0,T;H^{-1}(M)) \cap L^2(0,T;H^1(M))$ be a solution of $$u_t - \Delta u - u = 0$$ $$u(0)=u(T)$$ satisfying $\int_M u(t) = 0$ for all $t$. Is ...
3
votes
1answer
303 views

Upper bounds for the solution of an elliptic PDE depending on a parameter.

Suppose I have the following PDE on $[0,1]^n$ $$\mathcal{L}u = -\nabla \cdot \left(a(x, r)\nabla u\right) = f(x,r), \qquad x\in [0,1]^n,$$ with periodic boundary conditions and $\int f(x) dx =0$ . ...
3
votes
0answers
64 views

How can one do change of variables for solutions to a staochastic partial differential equation?

isHow can one do change of variables for solutions to a staochastic partial differential equation? For example, let us consider the following stochastic transport equation: $$ dy(t,x) + y_x(t,x) + ...
3
votes
0answers
267 views

well-posedness of the transport equation

I asked this question before on math exchange but did not have any luck with an answer. I would like to consider a simple example but get a thorough understanding of the theory behind it. I am ...
3
votes
0answers
125 views

Quadratic forms over symmetric $3\times3$ matrices

This question is motivated by a technique called compensated compactness (CC), which was elaborated by L. Tartar and F. Murat for the analysis of PDEs. The foreword of CC is a problem about quadratic ...
3
votes
0answers
380 views

Boundary regularity of the solution to the Beltrami equation

Hello, this question might sound a little vague, but I still dare to state , and I am basically requesting for some reference: Let us consider the orientation-preserving homeomorphic solutions $f: D ...
2
votes
1answer
221 views

Method of characteristics of a system of first order pdes

I asked the question on math.stackexchange.com, but didn't get any reply. So, I asked it again here. Any suggestion or hint is welcome, and thank you for your attention. Consider the system of first ...
2
votes
2answers
152 views

Some Questions from Reading on Wave Front Set from Hormander's Linear PDE Vol. 1

In Hormander's Linear PDE Vol. 1 (pg 252-253, before the definition of wave front set is introduced), Lemma $8.1.1$ says that if $\phi \in C_{0}^{\infty}$ and $v \in \mathcal{E}^{\prime}$, then ...
2
votes
2answers
126 views

Let $\mathrm{div}\,(A\,\mathrm{grad}\,u) + b u = f$. Is $(A\,\mathrm{grad}\,u)$ weakly differentiable?

Let us consider the basic linear elliptic PDE $$ \mathrm{div} (A\,\mathrm{grad}\,u) + bu = f, $$ with $f\in L^p,$ $A,b$ uniformly bounded. Do we have, for a weak solution $u\in W^{1,p}(\Omega')$, $$ ...
2
votes
1answer
180 views

Tempered distribution solution to a simple PDE

Let's consider the following PDE in $\mathbb R^d$ : $$\frac{\partial^d u}{\partial x_1...\partial x_d}=f$$ where $f$ is a tempered distribution with support in $\mathbb R^d_+$. There is a result by ...
2
votes
1answer
157 views

The centralizer of Lienard equation

Consider the lienard vector field $\cases{ x'=y -F(x) \\ y'=-x } $ in $\mathbb{R}^{2}$, where $F$ is a polynomial fuction with $F(0)=0$. Assume that $Y$ is a smooth vector field globally defined ...
2
votes
2answers
103 views

First order pde with characteristics [closed]

Consider a first order pde of the type $$u_y+b(x)u_x=0$$ and suppose that the coefficient $b$ is not necessairly continuous (for instance with a jump in some point). Is it still possible to apply in ...
2
votes
1answer
153 views

Heat transfer: boundary conditions with fluid velocity

The following equation is considered: $$ \frac{\partial u}{\partial t} - a\Delta u + \mathbf v \cdot \nabla u = f. $$ I have difficulties in formulating boundary conditions for this equation. If ...
2
votes
1answer
135 views

Extending a harmonic function in a ball to subharmonic in a larger ball

Consider the Laplace equation in a ball $B(r) \subset \mathbb{R}^n$ of radius $r$: $$ \begin{cases} -\Delta u &= 0, \quad \text {in} \quad B(r), \\ \ \ \ \ \ \, u&= g, \quad \text {in}\quad ...
2
votes
2answers
123 views

Is the left regularizer for elliptic BVP a left inverse for the principal part?

Take a differential operator with elliptic symbol, consider just the principal part of the operator. Can one invert this principal part with some parametrix type construction (at least construct a ...
2
votes
1answer
158 views

Differential finiteness of power series with hypergeometric coefficients

Let $f$ be a formal (or convergent near the origin, as you wish) power series with variables $x_1$, ... , $x_n$ and complex coefficients. $$ f = \sum_I f_I \mathbf x^I $$ This power series is said to ...
2
votes
1answer
117 views

elliptic boundary regularity, tangential regularity

A have a question related to the boundary regularity of a solution of a Poisson equation on a bounded domain. But to make the question easier to pose I will state it on $ R_+^2:=\{ x \in ...
2
votes
1answer
186 views

Pseudoinverse of Neumann-Laplacian

Suppose you have the following PDE: find $u \in H^1(\Omega)$ such that $$-\Delta u = f, \\ \frac{\partial u}{\partial n} = 0. $$ Further assume a solvability condition $$\int_\Omega f ...
2
votes
1answer
63 views

Solutions of a stochastic reduced wave equation

Given a spherical reference frame $\left(\rho,\phi,\theta\right)$, the reduced wave equation can be written as: $$\nabla^2U=k^2n^2U$$ in which: $U=U(\rho,\phi,\theta)$ The solutions of this equation ...
2
votes
1answer
168 views

Monotonicity preserving parabolic operators

Let $$ \mathcal{L}\equiv\sum_{i,j}^{n}a_{i,j}\frac{\partial^{2}}{\partial x_{i}\,\partial x_{j}}+\sum_{i}^{n}b_{i}\frac{\partial}{\partial x_{i}}+c $$ be uniformly elliptic on ...