TMA4305 Partial Differential Equations 2019

Consider the PDE with initial data: \[ u_t+u^2u_x=0,\quad u(0,x)=\frac1{1+x^2}. \] What is the largest \(T\) so that the problem has a classical solution for \(x\in\mathbb{R}\), \(t\in[0,T)\)?

Consider the PDE with initial data: \[ \begin{aligned} u_t+xuu_x&=0&&\text{for } x>0, t>0 u(0,x)&=g(x)&&\text{for } x>0 \end{aligned} \] Find the solution expressed by \(g\) (on implicit form). Why is there no need to specify a boundary value at \(x=0\)? Under what conditions does a classical solution exist for all \(t\in[0,T]\) and \(x\in[0,\infty)\)?

For week 37: An alternative approach to the one-dimensional wave equation: Fill in the details of the following outline.

Start with the equation \(u_{tt}-c^2u_{xx}=0\). Assuming that \(u\) is a solution, consider the two functions \(u_t \pm c u_x\). These satisfy (first order) transport equations, so each is a traveling wave with speed \(\pm c\). That is, there are functions \(w_\pm\) so that \[ \begin{aligned} u_t-cu_x &= -2c w_+'(x-ct)&&\text{(a right-traveling wave),}\\ u_t+cu_x &= 2c w_-'(x+ct)&&\text{(a left-traveling wave).} \end{aligned} \] The factors \(\pm 2c\) and the derivative on the right hand side are just for convenience in what follows. They are not essential. Add the two equations and integrate with respect to \(t\), then subtract them and integrate with respect to \(x\). Note that the first requires an integration “constant” \(C_1(x)\), whereas the second an integration “constant” \(C_2(t)\). Conclude that the two must be equal, hence an actual constant \(C\), and conclude that [\ u(t,x) = w_+(x-ct)+w_-(x+ct)+C. \] (But we might as well let one of the functions \(w_{\pm}\) absorb \(C\), so we can throw it away.)

Finally, substitute this into the initial data \[ u(0,x) = g(x), \quad u_t(0,x) = h(x) \] and derive the d'Alembert solution.

(a) Consider the cube \(Q=(0,\pi)\subset\mathbb{R}^n\) and the function \[w(t,\mathbf{x})=e^{-nt}\prod_{i=1}^{n} \sin x_i\qquad(t\ge0,\mathbf{x}\in\overline{Q}).\] Verify that \(w\) satisfies the standard heat equation.

(b) Let \(\Omega\) be a region with \(\overline\Omega\subset Q\), Assume that \(u \in C\bigl([0,\infty)\times\overline{\Omega}\bigr) \cap C^2\bigl((0,\infty)\times \Omega\bigr)\) satisfies \(u_t-\Delta u=0\) for \((t,\mathbf{x}) \in (0,\infty)\times \Omega\) and \(u(t,\mathbf{x})=0\) for \((t,\mathbf{x}) \in [0,\infty)\times\partial \Omega\). Show that \[\lim_{t\to\infty} u(t,\mathbf{x})=0\] uniformly in \(\mathbf{x}\).

Hint: Apply the maximum principle to \(mw \pm u\) where \(m\) is a suitably large constant.

(Corrected 2019-09-27) Most of the occurrences of \(\Omega\) in question (b) were unfortunately written \(Q\) instead, which unfortunately ruins the point.

If \(u\) is a harmonic function on the upper half space \(\mathbb{R}\times (0,\infty)\), and it is continuous on \(\mathbb{R}\times [0,\infty)\) with \(u(x,0)=0\) for all \(x\in\mathbb{R}\), extend \(u\) by setting \(u(x,-y)=-u(x,y)\) and show that the extension is harmonic on \(\mathbb{R}\).

Hint: The mean value property.

(2019-09-27) Note: More regularity is needed than I had realised at first. I'll get back to this.

Let \(\Omega\) be a bounded region with a piecewise \(C^1\) boundary, and assume that \(\Omega\) admits a Green's function.

1. For any \(\mathbf x\in\Omega\), show that \(G_{\mathbf x}(\mathbf y)\to\infty\) when \(\mathbf y \to \mathbf x\).

2. Show that \(G_{\mathbf x}(\mathbf y)>0\) for any \(\mathbf x,\mathbf y\in\Omega\) with \(\mathbf y\ne\mathbf x\). (Hint: Use the strong maximum principle on a suitable subregion of \(\Omega\).)

3. Conclude that \(-\partial_n G_{\mathbf x}\ge0\) at \(C^1\) boundary points, \(-\partial_n G_{\mathbf x}>0\) at \(C^2\) boundary points. At a \(C^2\) boundary point you can always place a ball inside \(\Omega\) whose boundary is tangent to \(\partial\Omega\) at this point.

4. Show that \[ -\int_{\partial\Omega} \partial_n G_{\mathbf x}\,dS = 1\].

5. (A challenge) Assuming that the Dirichlet problem is always solvable on \(\Omega\), show that \(-\partial_n G_{\mathbf x}\) behaves in a manner analogous to what we found for the Poisson kernel on the two-dimensional disk, in particular that it is an “approximate delta” as \(\mathbf x\) approaches the boundary of \(\Omega\). (Part of the challenge is to make this somewhat precise. Hint: Look at boundary values that are one at a given point on the boundary, then drop off rapidly to zero.)

Let \(\Omega\subseteq\mathbb{R}^n\) be a region, and \(\psi \in C_c(\Omega)\). Let \(\rho\) be a standard mollifier, and \(\rho_\delta(x)=\delta^{-n}\rho(x/\delta)\) as usual. Then \(\psi*\rho_\delta\to\psi\) uniformly. This follows from the uniform continuity of \(\psi\) and the usual properties of \(\rho\). You are not asked to prove this.

1. Assume instead that \(\psi\in C^k(\Omega)\). Conclude from the above that \(D^\alpha(\psi*\rho_\delta)\to D^\alpha\psi\) uniformly for every multi-index with \(\lvert\alpha\rvert\le k\).
Conclude further that \(C_c^\infty(\Omega)\) is dense in \(C_c^k(\Omega)\) in the following sense: For any \(\psi \in C_c^k(\Omega)\) there is a sequence of functions \(\psi_j \in C_c^\infty(\Omega)\) so that \(D^\alpha \psi_j \to D^\alpha\psi\) uniformly for every multi-index \(\lvert\alpha\rvert \le k\), and there exists a compact set \(K\subset\Omega\) with \(\operatorname{supp}(\psi_j)\subseteq K\) for all \(j\).
(This defines convergence in \(C_c^k(\Omega)\).)

2. Conclude from the above that the partial integration formula \(\int_\Omega (D^\alpha f) \psi \,d^nx = (-1)^{\lvert\alpha\rvert}\int_\Omega f D^\alpha \psi \,d^nx\) holds for any \(\psi\in C_c^k(\Omega)\), where \(f \in L^1_{\text{loc}}(\Omega)\) has a weak derivative \(D^\alpha f\).

3. Show the product rule \(\nabla(fg)=g\nabla f+f\nabla g\) where \(f \in W^{1,1}_{\text{loc}}(\Omega)\), \(g\in C^1(\Omega)\), and \(\nabla(fg)\) and \(\nabla f\) are weak gradients.

4. If \(f \in L^1_{\text{loc}}(\mathbb{R}^n)\) has a weak derivative \(D^\alpha f\), show that \(D^\alpha(f*\psi)=(D^\alpha f)*\psi\) for any \(\psi\in C_c^k(\mathbb{R}^n)\), provided \(\lvert\alpha\rvert\le k\).

If \(f \in C(\mathbb{T}^n)\) and \(u \in C^2(\mathbb{T^n})\) is a classical solution of \(-\Delta u=f\), show that \(\int_{\mathbb{T}^n}f\,d^x=0\).

Write \(H^m_{\perp}(\mathbb{T}^n)=\{f \in H^m(\mathbb{T}^n) | \int_{\mathbb{T}^n} f\,d^nx=0\}\). (Non-standard notation. I was tempted to write \(H^m_0\), but that would be too confusing. This space is the orthogonal complement of the constant functions.)

If \(f \in H^0_{\perp}(\mathbb{T}^n)\) write \(\mathcal{D}_f[u]=\int_{\mathbb{T}^n}(\tfrac12\lvert\nabla u\rvert^2+uf)\,d^nx\) for \(u \in H^1_{\perp}(\mathbb{T}^n)\). State and show an analogue of Dirichlet's principle for weak solutions of \(-\nabla u=f\) in \(H^1_{\perp}(\mathbb{T}^n)\), and show that this problem has a unique solution.

You “only” need to repeat the steps of the proof for the usual Dirichlet principles, adjusting to the new setting and checking that it all still works. There is only one major obstacle: There is no Poincaré inequality in \(H^1(\mathbb{T}^n)\). (The constant functions violate it.) However, there is such an inequality in \(H^1_{\perp}(\mathbb{T}^n)\), and that is all you need. You can prove it using Fourier analysis. It's easier than it looks at first – try it!